hcju/isabelleps huggingface dataset:
geometric_bind_pmf_unfold: fixes p :: "real" assumes "p \<in> {0<..1}" shows "geometric_pmf p = bernoulli_pmf p \<bind> (\<lambda>b. if b then return_pmf 0 else map_pmf Suc (geometric_pmf p))"
Un_commute: fixes A :: "'l set"   and B :: "'l set" shows "A \<union> B = B \<union> A"
assms_3: shows "SAMS A' B' C' D' E' F'"
eheight_exp_reduce_bound: fixes n :: "nat" assumes "1 < n" shows "eheight_exp n \<le> 4 / real n * sum eheight_exp {..<n}"
inverse_pow_pow: fixes a :: "'a"   and r :: "nat" assumes "a \<in> carrier \<G>" shows "inv (a [^] r) = inv a [^] r"
set_cases: fixes e :: "'a"   and a :: "'a atom" assumes "e \<in> set_atom a" obtains z1 :: "nat" where "a = Leq z1__ e"   | z1 :: "nat" where "a = Geq z1 e"
f: fixes y :: "'b"   and y' :: "'b"   and x :: "'a" assumes "y < y'" shows "f y x < f y' x"
comp_cod_arr: fixes f :: "'b"   and b :: "'b" assumes "B.arr f"   and "B.cod f = b" shows "B b f = f"
pratt_triple_imp_prime: fixes p :: "nat"   and a :: "nat"   and x :: "nat" assumes "pratt_triple p a x"   and "1 < p"   and "x = p - 1"   and "[a ^ (p - 1) = 1] (mod p)" shows "prime p"
ghost_honorary_grey_empty_locs_def: shows "ghost_honorary_grey_empty_locs = - {mark_loop_mo_co_unlock, mark_loop_mo_co_won, mark_loop_mo_co_W}"
lim_F_at_top: shows "(F \<longlongrightarrow> m) at_top"
w_simps_6: 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 r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" assumes "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>" shows "partial_magma.cod V w = w"
fcomp_def: fixes f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" shows "fcomp f g = (\<lambda>x. g (f x))"
updVal_swapVal_eqBut: fixes val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and x :: "'var"   and sX :: "'b"   and xs :: "'varSort"   and y :: "'var" shows "eqBut (val (x := sX)_xs) (val (y := sX)_xs ^[y \<and> x]_xs) xs y"
insert_disjoint_1: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
translation_Compl: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and t :: "'c set" assumes "class.ab_group_add plus zero minus uminus" shows "plus a ` (- t) = - plus a ` t"
goodEnv_imp_qGoodEnv_pickE: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "goodEnv rho" shows "qGoodEnv (pickE rho)"
focus_uminus: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "focus X (- p) = - focus X p"
one_word_le: fixes a :: "1 word" shows "a \<le> 1"
power_one: fixes n :: "nat" shows "\<epsilon> \<^sup>@ n = \<epsilon>"
simps_3: fixes v :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" shows "assignment_rhs\<^sub>s\<^sub>t (send\<langle>v\<rangle>\<^sub>s\<^sub>t # S) = assignment_rhs\<^sub>s\<^sub>t S"
runit_coherence: fixes f :: "'a" assumes "C.ide f" shows "\<r>\<^sub>D[F f] = F \<r>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))"
exp_of_derived_is_subgroup: fixes H :: "'c set"   and n :: "nat" assumes "subgroup H H" shows "subgroup ((derived H ^^ n) H) H"
pos: fixes x :: "real" assumes "x \<in> S" shows "0 \<le> x"
prefix_length_def: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "prefix_length P xs = length (takeWhile P xs)"
sub_mod_div_eq_div: fixes n :: "nat"   and m :: "nat" shows "(n - n mod m) div m = n div m"
Mf: shows "f \<in> M \<rightarrow>\<^sub>M subprob_algebra M'"
starfun2_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "*f2* f \<equiv> \<lambda>x. Ifun (star_of f \<star> x)"
gronwall_general: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and C :: "real"   and K :: "real"   and t :: "real" assumes "\<And>t. t \<in> {a..b} \<Longrightarrow> g t \<le> C + K * integral {a..t} g"   and "continuous_on {a..b} g"   and "\<And>t. t \<in> {a..b} \<Longrightarrow> 0 \<le> g t"   and "0 < C"   and "0 < K"   and "t \<in> {a..b}" shows "g t \<le> C * exp (K * (t - a))"
unitary11_gen_cis_blaschke_rev: fixes k' :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and \<phi> :: "real"   and a' :: "complex" assumes "k' \<noteq> 0"   and "M = k' *\<^sub>s\<^sub>m (cis \<phi>, 0, 0, 1) *\<^sub>m\<^sub>m (1, - a', - cnj a', 1)"   and "a' * cnj a' \<noteq> 1" shows "\<exists>k a b. k \<noteq> 0 \<and> mat_det (a, b, cnj b, cnj a) \<noteq> 0 \<and> M = k *\<^sub>s\<^sub>m (a, b, cnj b, cnj a)"
that: fixes \<pi> :: "Syntax.term list"   and eP' :: "Syntax.term"   and eV' :: "Syntax.term" assumes "\<lless>[], eP\<ggreater> P\<rightarrow>i \<lless>\<pi>, eP'\<ggreater>"   and "\<lless>\<pi>, eV\<ggreater> V\<rightarrow>i \<lless>[], eV'\<ggreater>"   and "{$$} \<turnstile> e', eP', eV' : \<tau>" shows "thesis"
induct: fixes P :: "('b, 'a) Term.term \<Rightarrow> ('a \<Rightarrow> ('b, 'c) Term.term) \<Rightarrow> bool"   and a0 :: "('b, 'a) Term.term"   and a1 :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "\<And>x \<sigma>. P (Var x) \<sigma>"   and "\<And>f ss \<sigma>. (\<And>x. x \<in> set ss \<Longrightarrow> P x \<sigma>) \<Longrightarrow> P (Fun f ss) \<sigma>" shows "P a0 a1"
Rep_Msg_inverse: fixes x :: "msg" shows "Abs_Msg (Rep_Msg x) = x"
Act: fixes P :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and P' :: "'state" assumes "P \<rightarrow>\<^sub>S \<langle>\<alpha>,P'\<rangle>" shows "S_transform.S_transition P \<langle>S_action.Act \<alpha>,P'\<rangle>"
\<epsilon>_uh_def: fixes seed_gen :: "'a spmf"   and hash :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "PRF_UHF.hash.\<epsilon>_uh seed_gen hash = (\<Squnion>w w'. spmf (PRF_UHF.hash.game_hash seed_gen hash w w') True)"
complete_if_size_min_height: fixes kdt :: "'a kdt" assumes "KD_Tree.size_kdt kdt = 2 ^ min_height kdt" shows "KD_Tree.complete kdt"
eq: shows "(\<Union>n. {a..S n}) = {a..<b}"
all_box: fixes F :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and sigma :: "behavior" shows "(sigma \<Turnstile> \<box>RAll F) = (\<forall>x. sigma \<Turnstile> \<box>F x)"
Gromov_product_at_diff3: fixes e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" shows "\<bar>Gromov_product_at e x y - Gromov_product_at e x z\<bar> \<le> dist y z"
CONST_INTF_def: fixes c :: "'a"   and I :: "interface" shows "c ::\<^sub>i I \<equiv> True"
simps_2: fixes n :: "nat"   and f :: "'g \<Rightarrow> 'g" shows "f ^^ Suc n = f \<circ> f ^^ n"
chop_singleton_right: shows "\<forall>r. \<exists>s. R_Chop(r,r,s)"
not_less_iff_gr_or_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "(\<not> less x y) = (less y x \<or> x = y)"
lfp_F_upperbound: shows "lfp (F ds) \<le> (fst XD_XH, dual (snd XD_XH))"
shuffle_to_top_Field: fixes Y :: "'x set"   and d :: "'d" assumes "inj_on Xd Y" shows "Field (shuffle_to_top Y d) = Field (Pd d) \<union> dX Y d"
ffun_upd_comm: fixes x :: "'a"   and y :: "'a"   and f :: "('a, 'b) ffun"   and u :: "'b"   and v :: "'b" assumes "x \<noteq> y" shows "f(y \<mapsto> u, x \<mapsto> v)\<^sub>f = f(x \<mapsto> v, y \<mapsto> u)\<^sub>f"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and f3 :: "'b list \<Rightarrow> 'b list \<Rightarrow> 'a"   and blue_witness :: "'b blue_witness" shows "P (case blue_witness of NO_CYC \<Rightarrow> f1 | REACH x xa \<Rightarrow> f2 x xa | CIRC x xa \<Rightarrow> f3 x xa) = ((blue_witness = NO_CYC \<longrightarrow> P f1) \<and> (\<forall>x21 x22. blue_witness = REACH x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32. blue_witness = CIRC x31 x32 \<longrightarrow> P (f3 x31 x32)))"
indRelR_preserves_pred_iff_indRelL_reflects_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "rel_preserves_pred indRelR Pred = rel_reflects_pred indRelL Pred"
fps_dvd1_trivial_unit_factor: fixes f :: "'a fps" assumes "f dvd 1" shows "unit_factor f = f"
bind_def: fixes M :: "'d nres"   and f :: "'d \<Rightarrow> 'c nres" shows "Refine_Basic.bind M f \<equiv> case M of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> Sup (f ` X)"
divides_ff_mult_cancel_right: fixes x :: "'b fract"   and z :: "'b fract"   and y :: "'b fract" shows "divides_ff (x * z) (y * z) = (z = 0 \<or> divides_ff x y)"
partial_deliverTE: fixes s :: "'c net_state"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state"   and S :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and T :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set" assumes "(s, i:deliver(d), s') \<in> pnet_sos S T" obtains s :: "'c net_state" and s' :: "'c net_state" and t :: "'c net_state" where "sb__ = SubnetS sc__ t__" and "s'b__ = SubnetS s' t__" and "(sc__, i:deliver(d), s') \<in> S"   | t :: "'c net_state" and t' :: "'c net_state" and s :: "'c net_state" where "sb__ = SubnetS s t" and "s'b__ = SubnetS s t'" and "(t, i:deliver(d), t') \<in> T"
otoy_trans: fixes i :: "nat" shows "automaton.trans (optoy i) = oseqp_sos \<Gamma>\<^sub>T\<^sub>O\<^sub>Y i"
assms_2: fixes n :: "'a"   and m :: "'a" assumes "n \<in> S"   and "m \<in> S"   and "n \<noteq> m"   and "f n \<inter> f m = {}" shows "g n \<inter> g m = {}"
group_append: fixes xs :: "'a list"   and ys :: "'a list" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys\<rbrakk> \<Longrightarrow> f x \<noteq> f y" shows "local.group (xs @ ys) = local.group xs @ local.group ys"
chain_boundary_of: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" shows "chain_boundary p (frag_of f) = (if p = 0 then 0 else \<Sum>k\<le>p. frag_cmul ((- 1) ^ k) (frag_of (singular_face p k f)))"
path0: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c" assumes "IFC prob" shows "IFC_def.path prob \<sigma> 0 = IFC_def.entry prob"
strict_trans: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "strict_border b a"   and "strict_border c b" shows "strict_border c a"
elims_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'b \<Rightarrow> 'd list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'd"   and xa :: "('a, 'b) atm" assumes "M.Struct wtFsym wtPsym arOf resOf intT intF intP"   and "Struct.satA intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and T1 :: "'a trm" and T2 :: "'a trm" where "x = \<xi>__" and "xa = Eq T1 T2" and "Struct.int intF \<xi>__ T1 = Struct.int intF \<xi>__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and p :: "'b" and Tl :: "'a trm list" where "x = \<xi>" and "xa = Pr p Tl" and "intP p (map (Struct.int intF \<xi>) Tl)"
bz: shows "(p, q, u, v, d) = bezout (A $ i $ k) (A $ j $ k)"
Sup_no_Red_in_Liminf: shows "Sup_llist Ns - Red_F (Sup_llist Ns) \<subseteq> Liminf_llist Ns"
simple_fw_no_interfaces: fixes rs :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme"   and y :: "char list"   and x :: "char list" assumes "simple_firewall_without_interfaces rs" shows "simple_fw rs p = simple_fw rs (p\<lparr>p_iiface := x, p_oiface := y\<rparr>)"
lift_trimmed_neg: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "trimmed_neg C" shows "trimmed_neg (MS (MSLCons (C, e) xs) f)"
card_rcosets_equal: fixes R :: "'a set"   and H :: "'a set" assumes "R \<in> rcosets\<^bsub>add_monoid G\<^esub> H"   and "H \<subseteq> carrier G" shows "card H = card R"
sel_1: fixes x21 :: "'c"   and x22 :: "'c list" shows "list.hd (x21 # x22) = x21"
l4: shows "z \<notin> path_image (linepath (Complex (Re lb) (Im ub)) lb)"
disjoint_iff: fixes A :: "'x set"   and B :: "'x set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
Un_assoc: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
distinct1_rotate: fixes xs :: "'b list" shows "cycle (rotate1 xs) = cycle xs"
arc_fst_in_verts: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b" assumes "pair_wf_digraph G"   and "e \<in> parcs G" shows "fst e \<in> pverts G"
nsubst_dterm': fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "'a \<Rightarrow> ('sf, 'sz) trm"   and \<theta> :: "('sf + 'a, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "TadmitFO \<sigma> \<theta>"   and "dfree \<theta>"   and "\<And>i. dsafe (\<sigma> i)" shows "dterm_sem I (TsubstFO \<theta> \<sigma>) \<nu> = dterm_sem (adjointFO I \<sigma> \<nu>) \<theta> \<nu>"
nat_div_distrib: fixes x :: "int"   and y :: "int" assumes "0 \<le> x" shows "nat (x div y) = nat x div nat y"
nsumMulEleL: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and n :: "nat" assumes "\<forall>i. f i \<in> carrier R"   and "x \<in> carrier R" shows "x \<cdot>\<^sub>r \<Sigma>\<^sub>e R f n = \<Sigma>\<^sub>e R (\<lambda>i. x \<cdot>\<^sub>r f i) n"
DBnd: shows "DBnd = fact n * max (1::'a) Bnd ^ n"
act_emp: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c" assumes "unital_quantale_module act" shows "act x (\<Squnion>{}) = \<bottom>"
sub_1: shows ".\<not> \<psi> \<notin> (Cn A \<div> .\<not> \<phi>) \<oplus> \<phi>"
FixVars_axioms: shows "FixVars TYPE('var) TYPE('varSort)"
size_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "euclidean_size zero = 0"
path_component_eq_connected_component: fixes S :: "'b set"   and x :: "'b" assumes "locally path_connected S" shows "path_component S x = connected_component S x"
subst_dnf'_def: fixes \<A> :: "'b ltln list set"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "subst_dnf' \<A> m = (\<Union>\<Phi>\<in>\<A>. subst_clause' \<Phi> m)"
tcoalg: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2" shows "tcoalg s1 s2"
max: fixes S :: "'b set" assumes "independent S"   and "S \<subseteq> V"   and "B \<subseteq> S" shows "S = B"
proj2_Col_coeff_unique: fixes a :: "proj2"   and r :: "proj2"   and i :: "real"   and j :: "real" assumes "a \<noteq> r"   and "proj2_abs (i *\<^sub>R proj2_rep a + proj2_rep r) = proj2_abs (j *\<^sub>R proj2_rep a + proj2_rep r)" shows "i = j"
simps_1: fixes x :: "vrs"   and y :: "ty" shows "tyvrs_of (VarB x y) = {}"
dAB: shows "\<forall>i<n. \<forall>j<n. i \<noteq> j \<longrightarrow> (A * B) $$ (i, j) = 0"
leadsTo_subset: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto> B" shows "A \<subseteq> wlt F B"
P0: shows "Suc (nat (ord_Zp x)) \<le> n"
hadd_def: fixes x :: "hf" shows "(@+) x \<equiv> hmemrec (\<lambda>f z. x \<squnion> RepFun z f)"
vinsert_vempty: fixes a :: "V" shows "vinsert a []\<^sub>\<circ> = ZFC_in_HOL.set {a}"
clmNextRes: fixes ts :: "traffic"   and c :: "cars" shows "clm ts c \<noteq> bot \<longrightarrow> (\<exists>n. Rep_nat_int (res ts c) \<union> Rep_nat_int (clm ts c) = {n, n + 1})"
supportAB: fixes A :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and B :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (\<lambda>l s. A l s \<and> B s) \<subseteq> support A"
f_set: fixes S :: "'a set"   and n :: "nat" assumes "S \<subseteq> T"   and "n = card S" shows "set (f S n) = S"
AssertAllState''Induct: fixes t :: "(nat \<times> linear_poly) list"   and P :: "('i \<times> 'a atom) set \<Rightarrow> ('i, 'a) state \<Rightarrow> bool"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t"   and "P {} (init t)"   and "\<And>as bs t. \<lbrakk>as \<subseteq> bs; P as t\<rbrakk> \<Longrightarrow> P bs t"   and "\<And>s a ats. \<lbrakk>\<not> \<U> s; \<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s; \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s; \<triangle> (\<T> s); \<nabla> s; \<diamond> s; P (set ats) s; index_valid (set ats) s\<rbrakk> \<Longrightarrow> P (insert a (set ats)) (ass_bnd a s)" shows "P (set ats) (assert_bound_loop ats (init t))"
neg_test_bit: fixes x :: "'b word"   and n :: "nat" shows "NOT x !! n = (\<not> x !! n \<and> n < LENGTH('b))"
foreach_mono: fixes f\<^sub>1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and f\<^sub>2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'c set"   and a\<^sub>0 :: "'d"   and a\<^sub>1 :: "'d" assumes "\<And>x a\<^sub>0 a\<^sub>1. f\<^sub>1 x a\<^sub>0 a\<^sub>1 \<Longrightarrow> f\<^sub>2 x a\<^sub>0 a\<^sub>1"   and "foreach f\<^sub>1 S a\<^sub>0 a\<^sub>1" shows "foreach f\<^sub>2 S a\<^sub>0 a\<^sub>1"
simps_1: shows "rel_of Leq_Rel = (\<le>)"
Mp_Mp: fixes f :: "int poly" shows "r.eq_m (r.Mp f) f"
enumdom_nonempty_domain_1: fixes s :: "nat \<Rightarrow> nat option"   and i :: "nat"   and x\<^sub>0 :: "nat"   and y :: "nat" assumes "total_cons s"   and "\<phi> i x\<^sub>0 \<down>" shows "eval r_enumdom [i, y] \<down>"
iso_assoc': fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.iso \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h]"
Quotient_freeword: shows "Quotient (eq_onp proper_signed_list) Abs_freeword freeword cr_freeword"
choose_row_sum: fixes n :: "nat" shows "sum ((choose) n) {..n} = 2 ^ n"
Lcm_subset: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "Lcm A dvd Lcm B"
divisionTr0: fixes p :: "'a"   and c :: "'a"   and n :: "nat" assumes "Idomain S"   and "p \<in> carrier R"   and "c \<in> carrier S"   and "c \<noteq> \<zero>\<^bsub>S\<^esub>" shows "lcf R S X (c \<cdot>\<^sub>r X^\<^bsup>R n\<^esup> \<cdot>\<^sub>r p) = c \<cdot>\<^sub>r\<^bsub>S\<^esub> lcf R S X p"
that: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> B"   and "ys = [a, b]" shows "thesis"
t0Hyp: shows "t\<^sub>0 \<in> U (X t\<^sub>0)"
sigma_finite_measure: fixes M :: "'b measure" assumes "finite_measure M" shows "sigma_finite_measure M"
path_image_part_circlepath_subset: fixes s :: "real"   and t :: "real"   and r :: "real"   and z :: "complex" assumes "s \<le> t"   and "0 \<le> r" shows "path_image (part_circlepath z r s t) \<subseteq> sphere z r"
dvd_triv_right: fixes a :: "'a"   and b :: "'a" shows "a dvd b * a"
sem_derived_4: fixes \<phi> :: "('U, 'C, 'Out) Noninterference.aprop dfmla"   and \<psi> :: "('U, 'C, 'Out) Noninterference.aprop dfmla"   and env :: "pvar \<Rightarrow> nat" shows "sem (Eq \<phi> \<psi>) env = eq (sem \<phi> env) (sem \<psi> env)"
rep_eq: fixes x :: "'d"   and xa :: "('b, 'c, 'd) formula" shows "Rep_formula (Act x xa) = Act\<^sub>\<alpha> x (Rep_formula xa)"
iso_iff: shows "((\<lambda>x. finprod G x I) \<in> Group.iso (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) (G\<lparr>carrier := J\<rparr>)) = (carrier (G\<lparr>carrier := J\<rparr>) \<subseteq> (\<lambda>x. finprod G x I) ` carrier (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) \<and> (\<forall>x\<in>carrier (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I). finprod G x I = \<one>\<^bsub>G\<lparr>carrier := J\<rparr>\<^esub> \<longrightarrow> x = \<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub>))"
chain_def: fixes w :: "nat \<Rightarrow> 'a list" shows "chain w \<equiv> mono w \<and> (\<forall>k. \<exists>l. k < length (w l))"
elim_singleton: fixes x :: "'b"   and s :: "'b"   and x' :: "'c"   and t :: "'c" assumes "x \<in> {s} \<and> x' \<in> {t}" obtains "x = s \<and> x' = t"
zero_above: shows "\<forall>j s. 1 \<le> j \<and> j < ncols A - mod_type_class.to_nat s \<longrightarrow> U $ s $ (s + mod_type_class.from_nat j) = (0::'a)"
nb\<^sub>2: fixes v :: "'a" assumes "v \<in> set (\<Pi>\<^sub>\<V>)" shows "(\<Phi>\<^sub>S\<inverse> \<Pi> \<A> t) v = Some (\<A> (State t (index (\<Pi>\<^sub>\<V>) v)))"
cases: fixes a1 :: "trm"   and a2 :: "stack" assumes "a1 \<rhd> a2" obtains t :: "trm" and k :: "stack" where "a1 = t \<star> k" and "a2 = k"
is_jump_def: fixes instr :: "('k, 'l, 'm, 'n, 'o, 'p, 'q, 'r) instr" shows "is_jump instr = (\<exists>x12. instr = ICJump x12)"
antipar: shows "S.antipar (\<Phi>.map yx) (\<Psi>o yx)"
abs_eq: fixes xa :: "perm"   and x :: "('b, 'c, 'd) Tree" shows "xa \<bullet> abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> (xa \<bullet> x)"
comp_of_ords_def: fixes le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "comp_of_ords le lt x y = (if lt x y then Lt else if le x y then Eq else Gt)"
prems_2: shows "set (m # ms) \<subseteq> M"
closed_comp: fixes f :: "int \<Rightarrow> int"   and g :: "int \<Rightarrow> int" assumes "f \<in> CofGroups.Ex1 \<and> g \<in> CofGroups.Ex1" shows "f \<circ> g \<in> CofGroups.Ex1"
tab_simps_2: shows "src \<rho> = src r\<^sub>0"
lifted_less_eq_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" shows "ord.lifted_less_eq less_eq f g \<equiv> \<forall>x. less_eq (f x) (g x)"
productive_imp_INTERP: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "production N C \<noteq> {}" shows "INTERP N \<Turnstile> C"
ProcNoRec1: fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'f, 'h) com option"   and \<Theta> :: "('g set \<times> 'f \<times> 'g set \<times> 'g set) set"   and F :: "'h set"   and P :: "'e \<Rightarrow> 'g set"   and p :: "'f"   and Q :: "'e \<Rightarrow> 'g set"   and A :: "'e \<Rightarrow> 'g set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P Z) the (\<Gamma> p) (Q Z),(A Z)"   and "p \<in> dom \<Gamma>" shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P Z) Call p (Q Z),(A Z)"
s_AuthV: fixes v :: "Syntax.term"   and \<pi> :: "Syntax.term list" assumes "closed v"   and "value v" shows "\<lless>\<pi>, Auth v\<ggreater> V\<rightarrow> \<lless>\<pi>, Hash (hash v)\<ggreater>"
zero_dagger: shows "(0::'c)\<^sup>\<star> = (1::'c)"
orecvmsg_simps_5: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and m :: "'e" shows "orecvmsg P \<sigma> (send m) = True"
rel_flip: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c llist"   and b :: "'d llist" shows "lrel R\<inverse>\<inverse> a b = lrel R b a"
power_eq_1_iff: fixes w :: "'c"   and n :: "nat" assumes "w ^ n = (1::'c)" shows "norm w = 1 \<or> n = 0"
of_rat_ceiling: fixes r :: "rat" shows "\<lceil>of_rat r\<rceil> = \<lceil>r\<rceil>"
comm_inv_char: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> x = y"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>x y. (y \<noteq> 0 \<Longrightarrow> P y (x mod y)) \<Longrightarrow> P x y" shows "P a0 a1"
unary_wt_sym_0_imp_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: fixes f :: "'s" assumes "arity_sym f = 1"   and "wt_sym f = 0" shows "\<delta>\<^sub>h = \<epsilon>\<^sub>h"
norm_less_p1: fixes x :: "'a" shows "norm x < norm (of_real (norm x) + (1::'a))"
num_double: fixes n :: "num" shows "num.Bit0 num.One * n = num.Bit0 n"