hcju/isabelleps huggingface dataset:
lowEquiv_def: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option" shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2 \<equiv> \<forall>v\<in>dom \<Gamma>. \<Gamma> v = Some Low \<longrightarrow> s1 v = s2 v"
simps_2: fixes ts :: "'c \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and ta :: "('c, 'f, 'g) new_thread_action"   and tas :: "('c, 'f, 'g) new_thread_action list" shows "thread_oks ts (ta # tas) = (thread_ok ts ta \<and> thread_oks (redT_updT' ts ta) tas)"
rep_eq: fixes x :: "uint"   and xa :: "nat"   and xb :: "bool" shows "Rep_uint (set_bit_class.set_bit x xa xb) = set_bit_class.set_bit (Rep_uint x) xa xb"
l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas_axioms: shows "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas DocumentClass.type_wf"
2: shows "(\<Sum>x\<le>i. \<Sum>xa\<le>degree p. (if xa = x then 1::'a else (0::'a)) * coeff [:h (coeff p xa):] (i - x)) = h (coeff p i)"
intro: fixes T :: "real set"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and X :: "'d set" assumes "ll_on_open T f X" shows "ll_on_open_it T f X"
fmrel_on_fset_fmrel_restrict: fixes S :: "'d fset"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "('d, 'e) fmap"   and n :: "('d, 'f) fmap" shows "fmrel_on_fset S P m n = fmrel P (fmrestrict_fset S m) (fmrestrict_fset S n)"
no_sbintr_alt2: fixes n :: "nat" shows "signed_take_bit n = (\<lambda>w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n)"
Inv_preserves_Can_3: fixes t :: "'a local.term" assumes "Can t" shows "Cod (Inv t) = Dom t"
env_delete_noop: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd" assumes "x \<notin> edom m" shows "env_delete x m = m"
countable_iff_le_Aleph0: fixes A :: "V" shows "countable (elts A) = (vcard A \<le> \<aleph>0)"
dom_map_of_conv_image_fst: fixes xys :: "('a \<times> 'b) list" shows "dom (map_of xys) = fst ` set xys"
word_split_bl: fixes std :: "nat"   and c :: "'b word"   and b :: "'c word"   and a :: "'d word" assumes "std = size c - size b" shows "(a = of_bl (take std (to_bl c)) \<and> b = of_bl (drop std (to_bl c))) = (word_split c = (a, b))"
ptr_kinds_ptr_kinds_M: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap" shows "(ptr \<in> set |h \<turnstile> document_ptr_kinds_M|\<^sub>r) = (ptr |\<in>| document_ptr_kinds h)"
add_less_cancel_right: fixes a :: "'k multiset"   and c :: "'k multiset"   and b :: "'k multiset" shows "(a + c \<subset># b + c) = (a \<subset># b)"
join_left: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_upper_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> x \<squnion>\<^bsub>L\<^esub> y"
simps_2: fixes v :: "('d \<times> 'd) set"   and va :: "'c \<Rightarrow> 'd"   and vb :: "'d"   and vd :: "'c list"   and ve :: "unit" shows "ipurge_tr_rev_t_aux \<lparr>Pol = v, Map = va, Dom = vb, In = [], Out = vd, \<dots> = ve\<rparr> = \<lparr>Pol = v, Map = va, Dom = vb, In = [], Out = vd, \<dots> = ve\<rparr>"
prog_instrs_append_Write\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "prog_instrs (sb @ [Write\<^sub>s\<^sub>b volatile a sop v A L R W]) = prog_instrs sb"
assms_1: fixes n :: "nat" shows "norm (f n) \<le> g n"
le_option_refl: fixes x :: "'c option" shows "le_option x x"
gauge_integral_Fubini_curve_bounded_region_y: fixes f :: "'a \<times> 'b \<Rightarrow> 'c"   and s :: "('a \<times> 'b) set"   and g :: "'a \<times> 'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b"   and g1 :: "'b \<Rightarrow> 'a"   and g2 :: "'b \<Rightarrow> 'a" assumes "integrable lborel f"   and "\<And>x. (\<lambda>y. f (y, x)) integrable_on UNIV"   and "(\<lambda>x. integral UNIV (\<lambda>y. f (y, x))) \<in> borel_measurable lborel"   and "f = (\<lambda>x. if x \<in> s then g x else (0::'c))"   and "s = {(y, x). (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i) \<and> (\<forall>i\<in>Basis. g1 x \<bullet> i \<le> y \<bullet> i \<and> y \<bullet> i \<le> g2 x \<bullet> i)}" shows "integral s g = integral (cbox a b) (\<lambda>x. integral (cbox (g1 x) (g2 x)) (\<lambda>y. g (y, x)))"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "P \<langle>\<rangle>"   and "\<And>A a. P \<langle>A, a, \<langle>\<rangle>\<rangle>"   and "\<And>A a B b CD. (CD \<noteq> \<langle>\<rangle> \<Longrightarrow> P CD) \<Longrightarrow> P \<langle>A, a, \<langle>B, b, CD\<rangle>\<rangle>" shows "P a0"
Ext_foo_member_in_Ext: shows "tprg \<turnstile> (Ext, mdecl Ext_foo) member_in Ext"
holf': shows "(\<lambda>u. f u - y) holomorphic_on ball z r"
inj_on_Un: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" shows "inj_on f (A \<union> B) = (inj_on f A \<and> inj_on f B \<and> f ` (A - B) \<inter> f ` (B - A) = {})"
llexord_LCons_LCons: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a llist"   and y :: "'a"   and ys :: "'a llist" shows "llexord r (LCons x xs) (LCons y ys) = (x = y \<and> llexord r xs ys \<or> r x y)"
ind_topology_def: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and S :: "'a set" shows "ind_topology X is_open S \<equiv> topological_space X is_open \<and> ind_topology_axioms X S"
MGF_complete: fixes G :: "prog"   and P :: "state assn"   and t :: "Term.term"   and Q :: "state assn" assumes "G,{}\<Turnstile>{P} t\<succ> {Q}"   and "G,{}\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}"   and "wf_prog G" shows "G,{}\<turnstile>{P} t\<succ> {Q}"
conjugate_one: shows "conjugate 1\<^sub>h = 1\<^sub>h"
semiring_1_cancel_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semiring_1_cancel_ow U plus minus zero one times \<equiv> (semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero) \<and> zero_neq_one_ow U one zero \<and> monoid_mult_ow U one times"
intro: fixes G :: "('a, 'b) pre_digraph" assumes "graph G"   and "non_empty_digraph G" shows "Designs_And_Graphs.non_empty_graph G"
Ri_subset_\<Gamma>: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and N :: "'b literal multiset set" assumes "redundancy_criterion \<Gamma> Rf Ri" shows "Ri N \<subseteq> \<Gamma>"
surj_swap_iff: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "surj (Fun.swap a b f) = surj f"
convex_affine_hull: fixes S :: "'b set" shows "convex (affine hull S)"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c" shows "(case Point x21 x22 of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
rhs_altdef: shows "rhs = 2 ^ T * (\<Prod>i = 1..n. 2 ^ i - 1)"
abstr_id3: fixes v :: "'c \<Rightarrow> nat"   and cc :: "('c, 'd) cconstraint"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and c1 :: "'c"   and c2 :: "'c" assumes "\<forall>c. 0 < v c" shows "abstr cc M v (v c1) (v c2) = M (v c1) (v c2)"
distinct_fst: fixes A :: "('b \<times> 'c) list" assumes "distinct (map fst A)" shows "distinct A"
less_minus_one_simps_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus one) zero"
capacity_const: shows "\<forall>e. (0::'capacity) \<le> augment f' e \<and> augment f' e \<le> c e"
pt_perm_fresh1: fixes a :: "'b"   and x :: "'a"   and b :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "\<not> a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x \<noteq> x"
of_nat_unat: fixes w :: "'l word" shows "semiring_1_class.of_nat (unat w) = unsigned w"
simps_2: fixes k :: "'b"   and k' :: "'b"   and v :: "'c"   and xs :: "('b \<times> 'c) list" shows "AList.delete_aux k ((k', v) # xs) = (if k = k' then xs else (k', v) # AList.delete_aux k xs)"
cod_simp_1: shows "partial_magma.cod (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) quiver.Zero = quiver.Zero"
order_iff: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (a = a \<squnion> b)"
inorder_delete2: fixes t :: "'b tree"   and x :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (delete2 x t) = del_list x (inorder t)"
triangle_right: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "V (H g \<epsilon>) (V (\<a> g f g) (H \<eta> g)) = V (category.inv V (bicategory.runit V H \<a> \<i> src g)) (bicategory.lunit V H \<a> \<i> src trg g)"
\<tau>inv_flip: shows "\<tau>inv trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1"
simps_5: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and X :: "'b ltln set" shows "(\<psi>\<^sub>1 R\<^sub>n \<psi>\<^sub>2)[X]\<^sub>\<nu> = \<psi>\<^sub>1[X]\<^sub>\<nu> R\<^sub>n \<psi>\<^sub>2[X]\<^sub>\<nu>"
mat_col_first_def: fixes A :: "'b mat"   and n :: "nat" shows "mat_col_first A n \<equiv> (mat_row_first A\<^sup>T n)\<^sup>T"
subseteq_zmset_def: fixes A :: "'c zmultiset"   and B :: "'c zmultiset" shows "(A \<subseteq>#\<^sub>z B) = (\<forall>a. zcount A a \<le> zcount B a)"
neg: fixes s1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "Modules.module_hom s1 s2 f" shows "f (- x) = - f x"
Kripke_def: fixes S :: "'a set"   and S0 :: "'a set"   and R :: "('a \<times> 'a) set"   and L :: "'a \<Rightarrow> 'b set option" shows "Kripke S S0 R L = (S0 \<subseteq> S \<and> R \<subseteq> S \<times> S \<and> Domain R = S \<and> dom L = S)"
singleton_lst_proj_4: fixes l'' :: "'e"   and a :: "'d" shows "unlabel [(l'', a)] = [a]"
int_of_nat_def: shows "int_of_nat \<equiv> (id ---> abs_int) int_of_nat_raw"
minus_sub_one_diff_one: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and m :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "minus (uminus (neg_numeral.sub minus plus one m num.One)) one = uminus (numeral.numeral one plus m)"
abs_eq: fixes x :: "('e, 'f, 'g) Tree" shows "Not\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tNot x)"
fv\<^sub>s\<^sub>n\<^sub>d_def: fixes x :: "('d, 'c) strand_step" shows "fv\<^sub>s\<^sub>n\<^sub>d x \<equiv> case x of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> fv x | _ \<Rightarrow> {}"
comp_inj_on: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'d \<Rightarrow> 'e" assumes "inj_on f A"   and "inj_on g (f ` A)" shows "inj_on (g \<circ> f) A"
assms_5: shows "ord_term_lin.is_le_rel sing_reg"
Ord_mem_iff_lt: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" shows "(k \<in> elts l) = (k < l)"
HSem_def': fixes ESem :: "'f \<Rightarrow> ('d \<Rightarrow> 'e) \<rightarrow> 'e"   and \<Gamma> :: "('d \<times> 'f) list"   and \<rho> :: "'d \<Rightarrow> 'e" shows "has_ESem.HSem ESem \<Gamma>\<cdot>\<rho> = (\<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> evalHeap \<Gamma> (\<lambda>e. ESem e\<cdot>x))"
\<phi>_cancel_right: fixes \<gamma> :: "V" assumes "\<gamma> \<in> elts (tp D)" shows "ordermap D VWF (\<phi> \<gamma>) = \<gamma>"
cbv_star_type_preservation: fixes t :: "lam"   and t' :: "lam"   and \<Gamma> :: "(name \<times> ty) list"   and T :: "ty" assumes " t \<longrightarrow>cbv* t'"   and "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile> t' : T"
join_self: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.cpo below" shows "cpo.join below x x = x"
s1: shows "(\<Sum>a\<in>F. 0) = 0"
principal_Ax: fixes r :: "(C, BD) sequent list \<times> (C, BD) sequent"   and A :: "(C, BD) form"   and R :: "((C, BD) sequent list \<times> (C, BD) sequent) set" assumes "r \<in> Ax"   and "rightPrincipal r (\<box> A) R" shows "( \<Empt> \<Rightarrow>* \<LM> A  \<RM>) \<in> set (fst r)"
write_reg_privilege: fixes s' :: "'b sparc_state"   and w1 :: "word32"   and w2 :: "'b word"   and w3 :: "word5"   and s :: "'b sparc_state" assumes "s' = snd (fst (write_reg w1 w2 w3 s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
leg_in_hom_1: shows "\<guillemotleft>T.leg0 : T.apex \<rightarrow> T.src\<guillemotright>"
n: shows "(\<Sum>i\<le>n. p i * i) = n"
prems_1: shows "di + Suc len \<le> length lsrc"
nat_less_le: fixes m :: "nat"   and n :: "nat" shows "order.greater n m = (order.greater_eq n m \<and> m \<noteq> n)"
assms_2: shows "w\<^sup>\<star> * e\<^sup>T = e\<^sup>T"
iffs: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a"   and getF' :: "Literal list list"   and getM' :: "(Literal \<times> bool) list"   and getConflictFlag' :: "bool"   and getC' :: "Literal list"   and more' :: "'a" shows "(\<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = \<lparr>getF = getF', getM = getM', getConflictFlag = getConflictFlag', getC = getC', \<dots> = more'\<rparr>) = (getF = getF' \<and> getM = getM' \<and> getConflictFlag = getConflictFlag' \<and> getC = getC' \<and> more = more')"
fps_shift_times_fps_X': fixes f :: "'a fps" shows "fps_shift 1 (f * fps_X) = f"
path0: fixes E :: "('b \<times> 'b) set"   and u :: "'b" shows "path E u [] u"
vsubset_VPowD: fixes a :: "V"   and b :: "V" assumes "VPow a \<subseteq>\<^sub>\<circ> VPow b" shows "a \<subseteq>\<^sub>\<circ> b"
nub_body''_def: shows "nub_body'' \<equiv> \<Lambda> f r. case nextR\<cdot>r of Nothing \<Rightarrow> lnil | Just\<cdot>(x, xs) \<Rightarrow> x :@ f\<cdot>(c2a\<cdot>(lfilter\<cdot>(neg oo Abs_cfun ((=\<^sub>B) x))\<cdot>(a2c\<cdot>xs)))"
cobounded1: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "a \<preceq> linordered_monoid.max a b"
iMODb_imirror_ident: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "imirror [ r, mod m, c ] = [ r, mod m, c ]"
follow_mrexps: fixes r :: "(bool \<times> 'b) rexp"   and s :: "'b rexp"   and b :: "bool" assumes "r \<in> mrexps s" shows "follow b r \<in> mrexps s"
COMB_step_def: fixes s :: "nat list \<times> (bool list \<times> nat list + nat list)"   and q :: "nat" shows "COMB_step s q = (case snd s of Inl b \<Rightarrow> map_pmf (\<lambda>((a, b), c). ((a, b), Inl c)) (BIT_step (fst s, b) q) | Inr b \<Rightarrow> map_pmf (\<lambda>((a, b), c). ((a, b), Inr c)) (return_pmf (TS_step_d (fst s, b) q)))"
monoidI: fixes G :: "('e, 'f) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x" shows "Group.monoid G"
open_core_eq: fixes T :: "'b set set"   and a :: "'b set" assumes "a open\<^bsub>T\<^esub>" shows "core\<^bsub>T\<^esub> a = a"
bits_div_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide zero a = zero"
ab: shows "A = a\<^sup>2 + N * b\<^sup>2"
atMost_def: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and u :: "'e" shows "ord.atMost less_eq u \<equiv> {x. less_eq x u}"
np_specD3: fixes np :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and a :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and b :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "np_spec np"   and "a \<in> set hs"   and "b \<in> set hs"   and "a \<noteq> b" shows "(a, b) \<in>\<^sub>p snd ` set (np gs bs hs data)"
index_zero_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat" assumes "i < nr"   and "j < nc" shows "0\<^sub>m nr nc $$ (i, j) = (0::'a)"
absorb1: fixes b :: "'a multiset"   and a :: "'a multiset" assumes "b \<subseteq># a" shows "a \<union># b = a"
defs_uses_disjoint': fixes n :: "'node"   and g :: "'g"   and v :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> defs g n"   and "v \<in> uses g n" shows "False"
dir1: shows "\<forall>s\<^sub>1 s\<^sub>2. (P s\<^sub>1||Q s\<^sub>2) \<sqsubseteq> S (s\<^sub>1, s\<^sub>2)"
ennreal_mult_divide_eq: fixes b :: "ennreal"   and a :: "ennreal" assumes "b \<noteq> 0"   and "b \<noteq> top" shows "a * b / b = a"
cond_spmf_fst_def: fixes p :: "('b \<times> 'a) spmf"   and a :: "'b" shows "cond_spmf_fst p a = map_spmf snd (cond_spmf p ({a} \<times> UNIV))"
Implementation_def: shows "Implementation = ((\<forall>p. Init \<not> Calling memCh p \<and> \<box>[ENext p]_(e p)) \<and> MClkISpec memCh crCh cst \<and> RPCISpec crCh rmCh rst \<and> IRSpec rmCh mm ires)"
le_f: fixes n :: "nat" shows "emeasure M (E n) \<le> f n"
lift_generator_function_def: fixes f :: "'c \<Rightarrow> 'd" shows "lift_generator_function f = map (map_prod id f)"
AbstrLevels_S13_A92: fixes i :: "AbstrLevelsID" assumes "sS13 \<in> AbstrLevel i" shows "sA92 \<notin> AbstrLevel i"
Spec: fixes P :: "'d set"   and r :: "('d \<times> 'd) set"   and Q :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and A :: "'d set" assumes "P \<subseteq> {s. (\<forall>t. (s, t) \<in> r \<longrightarrow> t \<in> Q) \<and> (\<exists>t. (s, t) \<in> r)}" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Spec r Q,A"
ms_3: shows "set ms \<subseteq> set ns\<^sub>1 \<union> set (tl ns')"
distinct_29: fixes "term" :: "'d"   and input :: "('d, 'e, 'f) input"   and psi1' :: "('d, 'e, 'f) psi"   and psi2' :: "('d, 'e, 'f) psi" shows "term\<lparr>input \<noteq> psi1' \<parallel> psi2'"
ordertype_WW_seg: fixes n :: "nat" shows "ordertype (WW_seg n) (lenlex less_than) = \<omega> \<up> ord_of_nat n"
onet_nhop_quality_increases: fixes p :: "net_tree" shows "oclosed (opnet (\<lambda>i. opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg) p) \<Turnstile> (\<lambda>_ _ _. True, other quality_increases (net_tree_ips p) \<rightarrow>) global (\<lambda>\<sigma>. \<forall>i\<in>net_tree_ips p. \<forall>dip. let nhip = the (nhop (rt (\<sigma> i)) dip) in dip \<in> vD (rt (\<sigma> i)) \<inter> vD (rt (\<sigma> nhip)) \<and> nhip \<noteq> dip \<longrightarrow> rt (\<sigma> i) \<sqsubset>\<^bsub>dip\<^esub> rt (\<sigma> nhip))"
naturality: fixes f :: "'a" assumes "arr f" shows "\<ll>.map (cod f) \<cdot> L f = local.map f \<cdot> \<ll>.map (local.dom f)"
iteratesp_def: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.ccpo Sup less_eq less" shows "ccpo.iteratesp Sup less_eq \<equiv> \<lambda>f. lfp (\<lambda>p x. (\<exists>xa. x = f xa \<and> p xa) \<or> (\<exists>M. x = Sup M \<and> Complete_Partial_Order.chain less_eq M \<and> (\<forall>x\<in>M. p x)))"
eucl_rel_poly_by0: fixes x :: "'b poly" shows "eucl_rel_poly x 0 (0, x)"
w2: shows "h k \<cdot>\<^sub>v gs ! k \<in> carrier_vec n"
ordermap_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "ordermap A r \<equiv> wfrec r (\<lambda>f x. set (f ` {y \<in> A. (y, x) \<in> r}))"
length_replicate: fixes n :: "nat"   and x :: "'b" shows "List.length (List.replicate n x) = n"
cSUP_le_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and u :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (f ` A)" shows "less_eq (Sup (f ` A)) u = (\<forall>x\<in>A. less_eq (f x) u)"
The_opt_def: fixes p :: "'b \<Rightarrow> bool" shows "The_opt p = (if \<exists>!x. p x then Some (The p) else None)"
2: shows "fmlookup (f ++ g) v = fmlookup f v"
word_add_less_mono1: fixes i :: "'b word"   and j :: "'b word"   and k :: "'b word" assumes "i < j"   and "unat j + unat k < 2 ^ LENGTH('b)" shows "i + k < j + k"
g_bigo: shows "g \<in> O(\<lambda>x. real x powr p')"
Gen_union: fixes A :: "(symbol \<times> character list) list set"   and B :: "(symbol \<times> character list) list set" shows "Gen (A \<union> B) = Gen A \<union> Gen B"
offs_next_zero: fixes i :: "nat"   and ns :: "nat list"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and ub :: "nat" assumes "i < length ns"   and "offs_num (length ns) xs index key mi ma i = 0"   and "offs_set_prev ns xs index key mi ma i = {}" shows "offs_next ns ub xs index key mi ma 0 = offs_next ns ub xs index key mi ma i"
order_iff_strict: fixes b :: "'a Word.word"   and a :: "'a Word.word" shows "(b \<le>s a) = (b <s a \<or> a = b)"
stutstep_def: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "stutstep s n \<equiv> s n = s (Suc n)"
ras_rel_def_internal: fixes R :: "('c \<times> 'd) set" shows "ras_rel R \<equiv> br ras_raw_\<alpha> ras_raw_invar O \<langle>R\<rangle>list_rel"
space_closed: fixes i :: "nat" shows "sets.pullback_algebra M (\<lambda>f. f i) (space P) \<subseteq> Pow (space P)"
is_strongly_dom_component_safe_def: shows "Shadow_DOM_DOM_Components.is_strongly_dom_component_safe \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe Shadow_DOM.get_root_node Shadow_DOM.to_tree_order"