hcju/isabelleps huggingface dataset:
ins3_def: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree" shows "ins3 n d x t = Time_Monad.val (ins3_tm n d x t)"
is_some_is_not_None: fixes x :: "'b option" shows "is_some x = (x \<noteq> None)"
Bloch: fixes f :: "complex \<Rightarrow> complex"   and a :: "complex"   and r :: "real"   and r' :: "real" assumes "f holomorphic_on ball a r"   and "0 < r"   and "r' \<le> r * cmod (deriv f a) / 12" obtains b :: "complex" where "ball b r' \<subseteq> f ` ball a r"
elims_2: fixes x :: "atom"   and xa :: "'b list" assumes "I\<^sub>d\<^sub>l\<^sub>o x xa" obtains i :: "nat" and j :: "nat" and xs :: "'b list" where "x = Eq i__ j__" and "xa = xsa__" and "xsa__ ! i__ = xsa__ ! j__"   | i :: "nat" and j :: "nat" and xs :: "'b list" where "x = Less i j" and "xa = xs" and "xs ! i < xs ! j"
compat2: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and default :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "order_pair gt default"   and "gt x y"   and "z \<le> y" shows "gt x z"
that: fixes U :: "'a set set"   and s' :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and t' :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "is_zariski_open U"   and "\<pp> \<in> U"   and "s' \<in> \<O> U"   and "s = st.class_of U s'"   and "t' \<in> \<O> U"   and "t = st.class_of U t'" shows "thesis"
FrP: shows "extractFrame P = \<langle>A\<^sub>P', \<Psi>\<^sub>P\<rangle>"
ffmember_filter: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool"   and A :: "'b fset" shows "(x |\<in>| ffilter P A) = (x |\<in>| A \<and> P x)"
size_gen_o_map: fixes f :: "'h \<Rightarrow> nat"   and g :: "'g \<Rightarrow> 'h" shows "size_list f \<circ> map g = size_list (f \<circ> g)"
sscan_scons_snth: fixes a :: "'b"   and f :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'d stream"   and i :: "nat" shows "(a ## sscan f xs a) !! i = fold f (stake i xs) a"
\<I>1: shows "\<I>1 \<le> map_\<I> f1 g1 \<I>1'"
Nmlize_Hcomp: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Arr t"   and "Arr u" shows "\<^bold>\<lfloor>t \<^bold>\<star> u\<^bold>\<rfloor> = \<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<star> \<^bold>\<lfloor>u\<^bold>\<rfloor>\<^bold>\<rfloor>"
m_assoc: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b"   and z :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)"   and "y \<in> carrier (DirProds G I)"   and "z \<in> carrier (DirProds G I)" shows "x \<otimes>\<^bsub>DirProds G I\<^esub> y \<otimes>\<^bsub>DirProds G I\<^esub> z = x \<otimes>\<^bsub>DirProds G I\<^esub> (y \<otimes>\<^bsub>DirProds G I\<^esub> z)"
a_get_ancestors_di_locs_def: shows "get_ancestors_di_locs = Shadow_DOM.get_parent_locs \<union> get_host_locs \<union> get_disconnected_document_locs"
assms_4: shows "B\<^sub>2 \<notin> P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2"
conversep_conversep: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<inverse>\<inverse> = r"
rel_funI: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "(A ===> B) f g"
point_in_vector_or_complement_iff: fixes p :: "'a"   and v :: "'a" assumes "point p"   and "is_vector v" shows "(p \<le> v) = (\<not> p \<le> - v)"
left_lang_def: fixes M :: "'b dfa"   and q :: "hf" assumes "dfa M" shows "dfa.left_lang M q \<equiv> {u. dfa.nextl M (dfa.init M) u = q}"
bounded_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "order.greater_eq (inf b c) a = (order.greater_eq b a \<and> order.greater_eq c a)"
assms: fixes k :: "'a"   and v :: "'b" assumes "(k, v) \<in> set xs" shows "P k v"
f_range_onD: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set"   and B :: "'d set" assumes "f_range_on A f"   and "B \<subseteq> A" shows "f B \<subseteq> B"
sum_assn_ctxt: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and x :: "'a + 'c"   and y :: "'b + 'd"   and z :: "assn" assumes "(A +\<^sub>a B) x y = z" shows "hn_ctxt (A +\<^sub>a B) x y = z"
infinite_imp_bij_betw2: fixes A :: "'a set"   and a :: "'a" assumes "infinite A" shows "\<exists>h. bij_betw h A (A \<union> {a})"
s_r: shows "s r = Some (\<sigma>, \<tau>, \<E> [Read (VE (Loc l))])"
degree_1_ipoly: fixes p :: "int poly"   and x :: "real" assumes "degree p = Suc 0" shows "(poly (real_of_int_poly p) x = 0) = (x = real_of_rat (Rat.Fract (- coeff p 0) (coeff p 1)))"
irreflp_on_converse: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "irreflp_on P A" shows "irreflp_on P\<inverse>\<inverse> A"
changes_le_der_def: fixes b :: "real"   and p :: "real poly" shows "changes_le_der b p = int (degree p) - changes_poly_at (pders p) b"
le_iff_inf: fixes x :: "'c multiset"   and y :: "'c multiset" shows "(x \<subseteq># y) = (x \<inter># y = x)"
coerce_prod: shows "COERCE('f \<times> 'g, 'h \<times> 'i) = prod_map\<cdot>COERCE('f, 'h)\<cdot>COERCE('g, 'i)"
wary_\<zeta>ss: shows "wary (apps (Hd \<zeta>) ss)"
counit_simps_1: shows "arr \<epsilon>\<^sub>1"
maxsimp_preimage: fixes C :: "'a set" assumes "C \<in> X"   and "SimplicialComplex.maxsimp (f \<turnstile> X) (f ` C)" shows "domain.chamber C"
ts_inv_ok_conv_dom_eq: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'd option" shows "ts_inv_ok ts I = (dom ts = dom I)"
assms: shows "\<forall>i. (S i, S (Suc i)) \<in> r"
subst_sends_fv_to_img: fixes t :: "('d, 'c) Term.term"   and s :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv (t \<cdot> s) \<subseteq> fv t \<union> range_vars s"
upreservesq: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and m :: "'c" assumes "\<forall>j. U (\<sigma> j) (\<sigma>' j)"   and "R \<sigma> m" shows "R \<sigma>' m"
selectlenhelp3: fixes l :: "bit list" shows "\<forall>j. select l 0 j = selecthelp l j"
intro: fixes x :: "'d"   and X :: "'d set" assumes "x \<in> X" shows "ind_mor_btw_stalks_axioms X x"
simIChainFresh: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "eqvt Rel"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* Q"   and "\<And>\<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* subject \<alpha>; distinct (bn \<alpha>); bn \<alpha> \<sharp>* C; xvec \<sharp>* \<alpha>; xvec \<sharp>* Q'\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
edge_preserving: fixes R :: "('d \<times> 'e) set"   and E :: "('f \<times> 'd \<times> 'd) set"   and G :: "('f \<times> 'e \<times> 'e) set" shows "edge_preserving R E G = (on_triple R `` E \<subseteq> G)"
fst_apfst: fixes f :: "'f \<Rightarrow> 'd"   and x :: "'f \<times> 'e" shows "fst (apfst f x) = f (fst x)"
a'_def: shows "a' = real (max (nat \<lceil>a\<rceil>) 0) + 1"
preR: shows "pre_adjust \<langle>l, (a, lv), delete x r\<rangle>"
UU: shows "homotopic_with (\<lambda>x. True) U U (f2 \<circ> f1 \<circ> (g1 \<circ> g2)) id"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and f5 :: "char list \<Rightarrow> 'b"   and f6 :: "'b"   and f7 :: "char list \<Rightarrow> 'b"   and f8 :: "'b"   and f9 :: "'b"   and action :: "action" shows "P (case action of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2 | Log \<Rightarrow> f3 | Reject \<Rightarrow> f4 | Call x \<Rightarrow> f5 x | Return \<Rightarrow> f6 | Goto x \<Rightarrow> f7 x | Empty \<Rightarrow> f8 | Unknown \<Rightarrow> f9) = ((action = Accept \<longrightarrow> P f1) \<and> (action = Drop \<longrightarrow> P f2) \<and> (action = Log \<longrightarrow> P f3) \<and> (action = Reject \<longrightarrow> P f4) \<and> (\<forall>x5. action = Call x5 \<longrightarrow> P (f5 x5)) \<and> (action = Return \<longrightarrow> P f6) \<and> (\<forall>x7. action = Goto x7 \<longrightarrow> P (f7 x7)) \<and> (action = Empty \<longrightarrow> P f8) \<and> (action = Unknown \<longrightarrow> P f9))"
one_not_zero: shows "principal_dchar n 1 \<noteq> 0"
bv_prepend_append2: fixes x :: "nat"   and y :: "nat"   and b :: "'a"   and xs :: "'a list" assumes "x < y" shows "bv_prepend y b xs = bv_prepend x b [] @ bv_prepend (y - x) b [] @ xs"
EncB_partsE: fixes SK :: "nat"   and K :: "nat"   and EK :: "nat"   and M :: "msg"   and b :: "msg"   and H :: "msg set" assumes "EncB SK K EK M b \<in> parts H" obtains "Crypt K (sign SK \<lbrace>M, Hash b\<rbrace>) \<in> parts H" and "Crypt EK (Key K) \<in> parts H" and "b \<in> parts H"
trans_MapOps_update_def: shows "trans_MapOps_update \<equiv> \<lambda>k v m. trie_odlist_update_with (fst k) m Mapping.empty (Mapping.update (snd k) v)"
zero_row: shows "is_zero_row_upt_k (last_nonzero_row + (1::'rows)) k A"
simps: fixes xs :: "nat list" shows "strt xs = (let ys = map Suc xs in strt' ys (length ys))"
injective_map_axioms: shows "injective_map \<eta> M M'"
mult_mono_nonpos_nonpos: fixes c :: "real"   and a :: "real"   and d :: "real"   and b :: "real" assumes "c \<le> a"   and "a \<le> 0"   and "d \<le> b"   and "d \<le> 0" shows "a * b \<le> c * d"
lt_lower_eq_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "(lt (lower p v) = lt p) = (lt p = min_term \<or> lt p \<prec>\<^sub>t v)"
cos_series_stream_def: shows "cos_series_stream = sin_series_stream_aux False 1 0"
fix_least: fixes F :: "'a \<rightarrow> 'a"   and x :: "'a" assumes "F\<cdot>x = x" shows "fix\<cdot>F \<sqsubseteq> x"
map_F_id0: shows "map_F id id id id id id id id id = id"
Aexp_subst: fixes e :: "exp"   and y :: "Vars.var"   and x :: "Vars.var"   and a :: "Arity" shows "\<A>\<^bsub>a\<^esub> e[y::=x] \<sqsubseteq> env_delete y (\<A>\<^bsub>a\<^esub> e) \<squnion> esing x\<cdot>(up\<cdot>0)"
hom_deg_setD_1: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and z :: "nat"   and A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "p \<in> hom_deg_set z A" shows "homogeneous p"
powr_int: fixes x :: "real"   and i :: "int" assumes "0 < x" shows "x powr real_of_int i = (if 0 \<le> i then x ^ nat i else 1 / x ^ nat (- i))"
assms_2: shows "wrap_Id f n\<^sub>1 = wrap_Id f n\<^sub>2"
special_nsg_e: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "\<natural>\<^bsub>G\<^esub>H \<triangleright> {\<one>\<^bsub>G\<^esub>}"
pwd_spy_mult_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_mult n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_mult_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
fresh_swap_exchange2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and x1 :: "'var"   and x2 :: "'var"   and var :: "'varSort \<Rightarrow> 'var set"   and xs :: "'varSort" assumes "good X"   and "{x1, x2} \<subseteq> var xs" shows "fresh xs x2 (X #[x2 \<and> x1]_xs) = fresh xs x1 X"
rep_eq: fixes x :: "('c, 'd) mapping_rbt" shows "entries x = RBT_Impl.entries (impl_of x)"
simps: fixes n :: "nat"   and P :: "pol" shows "Commutative_Ring.pow n P = (if n = 0 then Pc 1 else if even n then Commutative_Ring.pow (n div 2) (sqr P) else P \<langle>*\<rangle> Commutative_Ring.pow (n div 2) (sqr P))"
select_convs_3: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'a" shows "monoid.more \<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr> = more"
FirstFreshVariable_def: fixes istate :: "'a istate" shows "FirstFreshVariable istate = (case istate of IState x1 x2 x3 x4 x5 \<Rightarrow> x1)"
power_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times one n = one"
simps_5: fixes b :: "bexp"   and c :: "com"   and X :: "char list set" shows "bury (WHILE b DO c) X = WHILE b DO bury c (L (WHILE b DO c) X)"
Deriv_Lang: fixes c :: "'b"   and R :: "'b rexp set" shows "Deriv c (Lang R) = Lang (Pderiv c R)"
real_root_one: fixes n :: "nat" assumes "0 < n" shows "root n 1 = 1"
unit_simps_1: fixes a :: "'a" assumes "C.obj a" shows "D.arr (F.unit a)"
simps: fixes sub :: "(String.literal \<times> String.literal) set"   and tcs :: "'a \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" shows "wf_osig (sub, tcs) = (wf_subclass sub \<and> wf_tcsigs sub tcs)"
non_empty: fixes m :: "int" assumes "poly_mod_2 m" shows "{0..<m} \<noteq> {}"
h9: shows "[(ainv * b') ^ (4 * k + 3 - 1) = 1] (mod 4 * k + 3)"
replace_comp_cls_val_process: fixes Mkt :: "('c, 'd) discrete_market"   and x :: "'d"   and pf2 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "\<forall>n w. prices Mkt x n w = cls_val_process Mkt pf2 n w"   and "portfolio pf1"   and "portfolio pf2" shows "\<forall>n w. cls_val_process Mkt (qty_replace_comp pf1 x pf2) n w = cls_val_process Mkt pf1 n w"
assms: fixes x :: "'a" assumes "x \<in> space M \<inter> A" shows "f x \<le> g x"
subfield_m_inv_1: fixes K :: "'a set"   and k :: "'a" assumes "subfield K R"   and "k \<in> K - {\<zero>}" shows "inv k \<in> K - {\<zero>}"
shiftl_eq_push_bit: 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 push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.shiftl push_bit a n = push_bit n a"
expand: fixes prod :: "'b \<times> 'c"   and prod' :: "'b \<times> 'c" assumes "hamlet prod = hamlet prod' \<and> time prod = time prod'" shows "prod = prod'"
cases: fixes x :: "nat" obtains "x = 0"   | i :: "nat" where "x = Suc i"
ND_complete: fixes F :: "'b formula" assumes "\<Turnstile> F" shows "{} \<turnstile> F"
br'_algo_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "br'_algo \<delta> \<equiv> \<lparr>wa_cond = br'_cond, wa_step = br'_step \<delta>, wa_initial = br'_initial \<delta>, wa_invar = br'_invar \<delta>\<rparr>"
simple_blocks_eq_sup: shows "mset_set design_support = arcs_blocks"
hyperset_defs_1: fixes A :: "'f set option"   and B :: "'f set option" shows "A \<squnion> B \<equiv> case A of None \<Rightarrow> None | \<lfloor>A\<rfloor> \<Rightarrow> case B of None \<Rightarrow> None | \<lfloor>B\<rfloor> \<Rightarrow> \<lfloor>A \<union> B\<rfloor>"
13: shows "(w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) * selected_edge h j g\<^sup>T = bot"
outside_same_component: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "connected_component (- S) x y"   and "x \<in> outside S" shows "y \<in> outside S"
alluopairs_bex: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<forall>y\<in>set xs. P x y = P y x" shows "(\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y) = (\<exists>(x, y)\<in>set (alluopairs xs). P x y)"
B'_def: shows "B' = (\<lambda>x. B x \<inter> set_pmf (p x))"
eNorm_zero: fixes N :: "'b quasinorm" shows "eNorm N (0::'b) = 0"
upd_inj: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat"   and j :: "nat" assumes "kuhn_simplex p n base upd s"   and "i < n"   and "j < n" shows "(upd i = upd j) = (i = j)"
gcd_int_nonpos_iff: fixes x :: "int"   and y :: "int" shows "(gcd x y \<le> 0) = (x = 0 \<and> y = 0)"
cSum1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and C :: "'a"   and Q :: "pi" assumes "P \<longmapsto> a<x> \<prec> P'"   and "\<And>C. F C P a x P'" shows "F C (P \<oplus> Q) a x P'"
rsk_free_asset_axioms: shows "rsk_free_asset Mkt r risk_free_asset"
neighbourhood_base_of: fixes P :: "'a set \<Rightarrow> bool"   and X :: "'a topology" shows "neighbourhood_base_of P X = (\<forall>W x. openin X W \<and> x \<in> W \<longrightarrow> (\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W))"
padic_is_comm_monoid: fixes p :: "int" assumes "prime p" shows "Group.comm_monoid (padic_int p)"
top_rule_def: fixes t :: "'d" shows "top_rule t = (LG {} {0, 1}, LG {(t, 0, 1)} {0, 1})"
defAss'E: fixes g :: "'g"   and m :: "'node"   and v :: "'var"   and ns :: "'node list" assumes "defAss' g m v"   and "g \<turnstile> Entry g-ns\<rightarrow>m" obtains n :: "'node" where "n \<in> set ns" and "v \<in> defs g n"
simps_4: fixes r :: "'b rexp"   and s :: "'b rexp" shows "rexp_empty (Plus r s) = (rexp_empty r \<and> rexp_empty s)"
pointermap_insert_def: fixes a :: "'c"   and m :: "('c, 'd) pointermap_scheme" shows "pointermap_insert a m \<equiv> \<lparr>entries = entries m @ [a], getentry = getentry m(a \<mapsto> length (entries m))\<rparr>"
param_nfoldli: fixes Ra :: "('c \<times> 'e) set"   and Rb :: "('d \<times> 'f) set" shows "(nfoldli, nfoldli) \<in> \<langle>Ra\<rangle>list_rel \<rightarrow> (Rb \<rightarrow> bool_rel) \<rightarrow> (Ra \<rightarrow> Rb \<rightarrow> \<langle>Rb\<rangle>nres_rel) \<rightarrow> Rb \<rightarrow> \<langle>Rb\<rangle>nres_rel"
simps_6: fixes f :: "'f"   and b :: "'d set"   and c :: "('d, 'e, 'g) com" shows "mark_guards f (While b c) = While b (mark_guards f c)"
eq_occurs_in_cl_def: fixes t :: "'a trm"   and s :: "'a trm"   and C :: "'a literal set"   and \<sigma> :: "('a \<times> 'a trm) list" shows "eq_occurs_in_cl t s C \<sigma> = (\<exists>L t' s'. L \<in> C \<and> orient_lit_inst L t' s' pos \<sigma> \<and> t = t' \<lhd> \<sigma> \<and> s = s' \<lhd> \<sigma>)"
poincare_on_line_def: fixes p :: "complex_homo"   and a :: "complex_homo"   and b :: "complex_homo" shows "poincare_on_line p a b = poincare_collinear {p, a, b}"
a: shows "obj a \<and> Src t = \<^bold>\<langle>a\<^bold>\<rangle>\<^sub>0"
PrimRec1'p_PrimRec1'_eq: shows "PrimRec1'p = (\<lambda>x. x \<in> PrimRec1')"
assms: shows "range_vars \<delta> \<inter> bvars\<^sub>s\<^sub>s\<^sub>t S = {}"
mult_imp_div_pos_le: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c"   and z :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less_eq x (times z y)" shows "less_eq (divide x y) z"
rquot_invariant_def: fixes M :: "nat bdd list \<times> bool list" shows "rquot_invariant M = (\<lambda>l. length l = length (fst M))"
nth_shiftr: fixes w :: "'b word"   and m :: "nat"   and n :: "nat" shows "(w >> m) !! n = w !! (n + m)"
local_const: fixes es :: "e list"   and n :: "nat"   and i :: "nat"   and vs :: "v list" assumes "const_list es"   and "length es = n" shows "\<lparr>[Local n i vs es]\<rparr> \<leadsto> \<lparr>es\<rparr>"
that: fixes g :: "'a set \<Rightarrow> 'b" assumes "g \<in> hom (G Mod N) H"   and "\<And>x. x \<in> carrier G \<Longrightarrow> g (N #>\<^bsub>G\<^esub> x) = h x" shows "thesis"
SNF_2x2_works: shows "\<forall>A\<in>carrier_mat 2 2. is_SNF A (Smith_2x2_JNF A)"
elims_2: fixes x :: "'a Line"   and xa :: "'a Line" assumes "x \<parallel>  xa" obtains lineA :: "'a Line" and lineB :: "'a Line" where "x = lineA" and "xa = lineB" and "direction lineA = 0 \<or> direction lineB = 0 \<or> (\<exists>k. k \<noteq> (0::'a) \<and> direction lineB = k ** direction lineA)"
irreducible_linear_poly: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)"   and "coprime a b" shows "irreducible [:a, b:]"
greaterThanLessThan_def: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a"   and u :: "'a" shows "ord.greaterThanLessThan less l u \<equiv> ord.greaterThan less l \<inter> ord.lessThan less u"
lookup_append_none: fixes env :: "('f, 'g, 'h) env"   and xs :: "'h list"   and ys :: "'h list" assumes "lookup env xs = None" shows "lookup env (xs @ ys) = None"
zth: shows "\<forall>f\<in>{f. (\<forall>i\<in>UNIV. f i \<in> UNIV) \<and> (\<forall>i. i \<notin> UNIV \<longrightarrow> f i = i)} - {p. p permutes UNIV}. det (\<chi>i. A $ i $ f i *s B $ f i) = (0::'a)"
bound: shows "\<forall>\<^sub>F i in at_top. AE x in M. norm (s i x) \<le> w x"
assms: shows "\<pi> = \<Phi>\<inverse> \<Pi> \<nu> t"
vars_of_type_eq_type_max_eq: fixes t :: "'BExp set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<forall>x\<in>vars_of_type t. mem\<^sub>1 x = mem\<^sub>2 x" shows "type_max t mem\<^sub>1 = type_max t mem\<^sub>2"
periodic_set_multiple: fixes k :: "int"   and S :: "real set"   and \<delta> :: "real" assumes "k \<noteq> 0" shows "periodic_set S \<delta> = periodic_set S (real_of_int k * \<delta>)"
FOREACHi_def: fixes \<Phi> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'a set" shows "FOREACH\<^bsup>\<Phi>\<^esup> S \<equiv> FOREACH\<^sub>C\<^bsup>\<Phi>\<^esup> S (\<lambda>_. True)"
sum_image: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite X" shows "comm_monoid_add_class.sum f X = \<Sum> (f ` X)"
finite_EqOnGL: fixes p :: "pvar"   and p' :: "pvar" shows "finite {EqOnUOut p p' u ou |u ou. (u, ou) \<in> GL \<times> UNIV}"
simps: fixes b :: "bool"   and r :: "(bool \<times> 'b) rexp" shows "hom_ab (b, r) = (follow b r, final_a (b, r))"
intro: fixes S :: "('a, 'b) eq_object_scheme" assumes "\<And>x. x \<in> carrier S \<Longrightarrow> x .=\<^bsub>S\<^esub> x"   and "\<And>x y. \<lbrakk>x .=\<^bsub>S\<^esub> y; x \<in> carrier S; y \<in> carrier S\<rbrakk> \<Longrightarrow> y .=\<^bsub>S\<^esub> x"   and "\<And>x y z. \<lbrakk>x .=\<^bsub>S\<^esub> y; y .=\<^bsub>S\<^esub> z; x \<in> carrier S; y \<in> carrier S; z \<in> carrier S\<rbrakk> \<Longrightarrow> x .=\<^bsub>S\<^esub> z" shows "equivalence S"
length_map: fixes f :: "'g \<Rightarrow> 'f"   and xs :: "'g list" shows "length (map f xs) = length xs"
absorb_iff2: fixes b :: "'e"   and a :: "'e" shows "(b \<le> a) = (inf a b = b)"
nstut_unch: fixes v :: "'a \<Rightarrow> 'b" shows "NSTUTINV Unchanged v"
sort_poly_spec_id: fixes p :: "(char list list \<times> int) list"   and p' :: "(char list multiset \<times> int) multiset" assumes "(p, p') \<in> unsorted_poly_rel" shows "sort_poly_spec p \<le> \<Down> (sorted_repeat_poly_list_rel (rel2p (Id \<union> term_order_rel))) (RETURN p')"
find_first_le: fixes a :: "'b"   and xs :: "'b list" assumes "a \<in> set xs" shows "find_first a xs < length xs"
fbspec: fixes A :: "'d fset"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "fBall A P"   and "x |\<in>| A" shows "P x"
insert_lemma: fixes x :: "nat"   and u :: "nat" assumes "x \<notin> nat_to_set u" shows "set_to_nat (nat_to_set u \<union> {x}) = u + 2 ^ x"
rep_eq: fixes x :: "moebius_mat" shows "unit_disc_fix_mmat x = unit_disc_fix_cmat (Rep_moebius_mat x)"
Tn_nn_integral_preserving: fixes f :: "'a \<Rightarrow> ennreal"   and n :: "nat" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f ((T ^^ n) x) \<partial>M = integral\<^sup>N M f"