hcju/isabelleps huggingface dataset:
cases: fixes x :: "('b \<Rightarrow> 'b list \<Rightarrow> bool) \<times> 'b list \<times> 'b list \<times> 'b list" obtains P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" and xa :: "'b" and xs :: "'b list" and y :: "'b" and ys :: "'b list" and z :: "'b" and zs :: "'b list" where "x = (Pa__, xa__ # xsa__, y__ # ysa__, z__ # zsa__)"   | P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" and xa :: "'b" and xs :: "'b list" and y :: "'b" and ys :: "'b list" where "x = (Pb__, xaa__ # xsb__, y # ys, [])"   | P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" and xa :: "'b" and xs :: "'b list" and z :: "'b" and zs :: "'b list" where "x = (P, xa # xs, [], z # zs)"   | uu_ :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" and uv_ :: "'b" and uw_ :: "'b list" where "x = (uu_, uv_ # uw_, [], [])"   | ux_ :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" and uy_ :: "'b" and uz_ :: "'b list" and va_ :: "'b list" where "x = (ux_, [], uy_ # uz_, va_)"   | vb_ :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" and vd_ :: "'b" and ve_ :: "'b list" where "x = (vb_, [], [], vd_ # ve_)"   | vf_ :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" where "x = (vf_, [], [], [])"
pp_def: shows "pp = endpoint_in_S (Rep_hyp2 a) (Rep_hyp2 b)"
invoke_pure: fixes ptr :: "(_) object_ptr"   and args :: "'o"   and h :: "(_) heap" shows "pure (invoke [] ptr args) h"
length_Suc_conv: fixes xs :: "'a list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
exhaust: fixes y :: "wall" obtains x1 :: "brick list" where "y = basic x1"   | x21 :: "brick list" and x22 :: "wall" where "y = x21 * x22"
un: shows "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m d"
\<Phi>_acts_elementwise: fixes a :: "'s" assumes "S.ide a" shows "S'.set (\<Phi> a) = \<Phi> ` S.set a"
supt_subset_WPO_S: assumes "\<And>f k. set (\<sigma> (f, k)) = {0..<k}" shows "{\<rhd>} \<subseteq> WPO_S"
h'_f: fixes x :: "'v option"   and n :: "nat" assumes "x \<in> A_n' n" shows "integral\<^sup>N (count_space (Y n)) (h' n x) = ennreal (f n x)"
simps_1: fixes h :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and xs :: "'d list" shows "interpret h TrueF xs = True"
vec_setE: fixes a :: "'c"   and v :: "'c vec" assumes "a \<in> set\<^sub>v v" obtains i :: "nat" where "v $ i = a" and "i < dim_vec v"
SetPC_Bind: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and p :: "'a \<Rightarrow> 'b \<Rightarrow> real" shows "SetPC a p = Bind p (\<lambda>p. SetPC a (\<lambda>_. p))"
simps_1: fixes ys :: "'a list list" shows "interact [] ys = concat ys"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "times_hom hom" shows "semigroup_mult_hom hom"
isOp_def: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" shows "isOp s X \<equiv> \<exists>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<and> s = stOf delta \<and> X = Op delta inp binp"
set_update_memI: fixes n :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "n < length xs" shows "x \<in> set (xs[n := x])"
takeWhile_not_last: fixes xs :: "'c list" assumes "distinct xs" shows "takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
simps: fixes x :: "'b"   and t :: "'b tree" shows "insert x t = (if t = \<langle>\<rangle> then \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> else case splay x t of \<langle>l, a, r\<rangle> \<Rightarrow> case cmp x a of LT \<Rightarrow> \<langle>l, x, \<langle>\<langle>\<rangle>, a, r\<rangle>\<rangle> | EQ \<Rightarrow> \<langle>l, a, r\<rangle> | GT \<Rightarrow> \<langle>\<langle>l, a, \<langle>\<rangle>\<rangle>, x, r\<rangle>)"
assms_1: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (s \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>)"
cSUP_subset_mono: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c multiset"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_above (g ` B)"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># g x" shows "Sup (f ` A) \<subseteq># Sup (g ` B)"
fraction_normal_of_Float: fixes x :: "Float.float" assumes "is_normal_Float TYPE(('a, 'b) IEEE.float) x" shows "fraction (normal_of_Float x) = nat \<bar>mantissa x\<bar> * 2 ^ (Suc LENGTH('b) - nat (bitlen \<bar>mantissa x\<bar>)) - 2 ^ LENGTH('b)"
heterogenous_cblinfun_id_def': shows "heterogenous_cblinfun_id = id_cblinfun"
prod_unfold_prod_mset: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "prod f A = \<Prod>\<^sub># (image_mset f (mset_set A))"
leg1_simps_4: shows "trg (tab\<^sub>1 r) = trg r"
restricted_agree_imp_agree: fixes s2 :: "('a, 'b) fmap"   and vs :: "'a set"   and s1 :: "('a, 'b) fmap" assumes "fmdom' s2 \<subseteq> vs"   and "agree (fmrestrict_set vs s1) s2" shows "agree s1 s2"
fis_singleton_alt: fixes f :: "'a fset" shows "fis_singleton f = (\<exists>e. f = {|e|})"
assms_1: shows "einterpret CX vs \<in> Csafe"
bij_signed_lconjaction: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "class.group_add minus plus zero uminus" shows "bij (signed_funaction (\<lambda>x y. minus (plus x y) x) s)"
np_lemma: fixes x :: "nat" shows "rec_exec rec_np [x] = Np x"
properfactor_mult_imp_properfactor: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "properfactor (Ring_Divisibility.mult_of R) b a" shows "properfactor R b a"
has_domain_iff_arr: fixes f :: "'a1 \<times> 'a2" shows "(A1xA2.domains f \<noteq> {}) = A1xA2.arr f"
assms_2: shows "f \<circ> pseudo_proj_True n = f"
Arg2pi_unique: fixes r :: "real"   and a :: "real"   and z :: "complex" assumes "complex_of_real r * exp (\<i> * complex_of_real a) = z"   and "0 < r"   and "0 \<le> a"   and "a < 2 * pi" shows "Arg2pi z = a"
chain_map_id_gen: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and f :: "'c \<Rightarrow> 'c" assumes "singular_chain p X c"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = x" shows "chain_map p f c = c"
pCons_1: shows "a \<noteq> 0 \<or> p \<noteq> 0"
dres_monad1: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'c dres" shows "dRETURN x \<bind> f = f x"
Sq: shows "set q \<subseteq> set A\<^sub>G \<times> set (q \<bullet> A\<^sub>G)"
cring0_open: fixes N :: "nat set set" shows "cring0.is_zariski_open N = (N = {})"
mconn_sym': fixes E :: "('b \<times> 'b) set" shows "sym (mconn E)"
find_ce_impl_refine: shows "find_ce_impl \<le> \<Down> Id gfind_ce"
init_InvInfoAtBetaUpright: fixes c :: "('c, 'd) configuration" assumes "init_config c" shows "InvInfoAtBetaUpright c"
f_2_lemma_3: fixes p :: "nat"   and q :: "nat"   and x :: "int"   and S :: "int set" assumes "QR p q"   and "QR.f_2 p q x \<in> S" shows "x \<in> QR.f_2 p q ` S"
strict_borderE: fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_border xs ys" obtains "border xs ys" and "length xs < length ys"
sum_over_product_left: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "CEM"   and "O y z" shows "P (x \<oplus> y \<otimes> z) ((x \<oplus> y) \<otimes> (x \<oplus> z))"
f_accessible_mono: fixes Q :: "'a set"   and Q' :: "'a set"   and x :: "'a"   and \<delta> :: "('a, 'b) ta_rule set" assumes "Q \<subseteq> Q'"   and "x \<in> f_accessible \<delta> Q" shows "x \<in> f_accessible \<delta> Q'"
nonneg: fixes i :: "'i" assumes "i \<in> I" shows "0 \<le> \<mu> i"
assms_1: shows "Q = Set.filter (\<lambda>(A, uu_). \<not> Set.is_empty A) Q_pos"
tape_nat: fixes x :: "nat" shows "<[x]> = Oc \<up> Suc x"
rtranclp_trans: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'c"   and f2 :: "nat \<Rightarrow> pol \<Rightarrow> 'c"   and f3 :: "pol \<Rightarrow> nat \<Rightarrow> pol \<Rightarrow> 'c"   and pol :: "pol" shows "P (case pol of Pc x \<Rightarrow> f1 x | Pinj x xa \<Rightarrow> f2 x xa | PX x xa xb \<Rightarrow> f3 x xa xb) = (\<not> ((\<exists>x1. pol = Pc x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. pol = Pinj x21 x22 \<and> \<not> P (f2 x21 x22)) \<or> (\<exists>x31 x32 x33. pol = PX x31 x32 x33 \<and> \<not> P (f3 x31 x32 x33))))"
inj_on_subset: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'d set" assumes "inj_on f A"   and "B \<subseteq> A" shows "inj_on f B"
element_ptr_kinds_eq_h: shows "element_ptr_kinds h = element_ptr_kinds h2"
simps: fixes a1 :: "nat list"   and a2 :: "nat list" shows "(a1 \<lhd> a2) = (a1 = [] \<and> a2 = [] \<or> (\<exists>is ds i d. a1 = i # is \<and> a2 = d # ds \<and> is \<lhd> ds \<and> i < d))"
induct: fixes P :: "trilean \<Rightarrow> trilean \<Rightarrow> bool"   and a0 :: "trilean"   and a1 :: "trilean" assumes "\<And>uu_. P uu_ invalid"   and "P invalid true"   and "P invalid false"   and "P true true"   and "P true false"   and "P false false"   and "P false true" shows "P a0 a1"
sep_substate_disj_add: fixes x :: "'a"   and y :: "'a" assumes "x ## y" shows "x \<preceq> x + y"
diff_le_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (minus a b) c = less_eq a (plus c b)"
in_fst_imageE: fixes x :: "'g"   and S :: "('g \<times> 'h) set" assumes "x \<in> fst ` S" obtains y :: "'h" where "(x, y) \<in> S"
bl2nat_simps_2: fixes x :: "nat"   and n :: "nat" shows "bl2nat (Bk \<up> x) n = 0"
xlength: shows "\<forall>x. insertion (nth_default 0 (xs @ x # \<Gamma>)) (Var var) = x"
assms_5: shows "A\<^sub>P\<^sub>Q \<sharp>* P"
simps: fixes a1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and a2 :: "'d"   and a3 :: "'e \<Rightarrow> 'f"   and a4 :: "'d"   and a5 :: "'e \<Rightarrow> 'f" shows "a1 \<turnstile> \<langle>a2, a3\<rangle> \<rightarrow>* \<langle>a4, a5\<rangle> = ((\<exists>A l u. a1 = A \<and> a2 = l \<and> a3 = u \<and> a4 = l \<and> a5 = u) \<or> (\<exists>A l u l' u' l'' u''. a1 = A \<and> a2 = l \<and> a3 = u \<and> a4 = l'' \<and> a5 = u'' \<and> A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle> \<and> A \<turnstile> \<langle>l', u'\<rangle> \<rightarrow>* \<langle>l'', u''\<rangle>))"
map_of_map_apsnd: fixes t :: "'c \<Rightarrow> 'b"   and M :: "('a \<times> 'c) list" shows "map_of (map (apsnd t) M) = map_option t \<circ> map_of M"
finite_entropy_integrable: fixes S :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and Px :: "'b \<Rightarrow> real" assumes "finite_entropy S X Px" shows "integrable S (\<lambda>x. Px x * log b (Px x))"
vec_vec_Tensor_simpl2: fixes y :: "'a list"   and i :: "nat"   and z :: "'a"   and zs :: "'a list" shows "length y \<le> i \<longrightarrow> vec_vec_Tensor (z # zs) y ! i = vec_vec_Tensor zs y ! (i - length y)"
is_Less_iff: fixes f :: "atom" shows "is_Less f = (\<exists>r rs. f = Less r rs)"
inj_comm_semiring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_comm_semiring_hom hom \<equiv> comm_semiring_hom hom \<and> inj_semiring_hom hom"
funionI2: fixes c :: "'a"   and B :: "'a fset"   and A :: "'a fset" assumes "c |\<in>| B" shows "c |\<in>| A |\<union>| B"
vector_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_vector x" shows "(x \<cdot> y) ; z = x \<cdot> y ; z"
seq_cons: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and t :: "state"   and rs :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, t\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r # rs, Undecided\<rangle> \<Rightarrow> t'"
assms_2: shows "H \<equiv> Hermite_reduce_above M (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res"
intro: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e" assumes "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "continuous_map S is_open S' is_open' f" shows "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f"
bisim_val_id: fixes u :: "'c \<Rightarrow> real"   and l :: "'s"   and l' :: "'s"   and u' :: "'c \<Rightarrow> real" assumes "u \<in> V"   and "(l, u) ~ (l', u')" shows "[u]\<^sub>\<R> = [u']\<^sub>\<R>"
ce_star_Cons: fixes \<sigma>1 :: "'c \<Rightarrow> 'd"   and l :: "('c, 'd) label"   and ls :: "('c, 'd) label list"   and \<sigma>2 :: "'c \<Rightarrow> 'd" shows "ce_star \<sigma>1 (l # ls) \<sigma>2 = (\<exists>\<sigma>. ce \<sigma>1 l \<sigma> \<and> ce_star \<sigma> ls \<sigma>2)"
effect_tapI: fixes h' :: "heap"   and h :: "heap"   and r :: "'a"   and f :: "heap \<Rightarrow> 'a" assumes "h' = h"   and "r = f h" shows "effect (tap f) h h' r"
collapse_2: fixes status :: "status" assumes "is_success status" shows "status = SUCCESS"
append_eq_append_conv: fixes xs :: "'c list"   and ys :: "'c list"   and us :: "'c list"   and vs :: "'c list" assumes "length xs = length ys \<or> length us = length vs" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
P_mod_minus: fixes I :: "'a set"   and p :: "'a" assumes "ideal S I"   and "p \<in> carrier R"   and "P_mod R S X I p" shows "P_mod R S X I (-\<^sub>a p)"
comp_inf_vector: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> (z \<sqinter> top)) = x \<sqinter> (top \<sqinter> id z) \<sqinter> y"
iso_assoc: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "C.iso \<a>\<^sub>C[f, g, h]"
in_listsD: fixes xs :: "'c list"   and A :: "'c set" assumes "xs \<in> lists A" shows "\<forall>x\<in>set xs. x \<in> A"
Re_stereographic: fixes z :: "complex" shows "Re (2 * z / (1 + z * cnj z)) = 2 * Re z / (1 + (cmod z)\<^sup>2)"
dvd_add_left_iff: 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 a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.comm_semiring_1_cancel plus minus zero times one"   and "dvd.dvd times a c" shows "dvd.dvd times a (plus b c) = dvd.dvd times a b"
absorb2: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" shows "a \<sqinter> b = b"
State_Idle_step: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d" assumes "State_Idle localState output_fun trans_fun (localState c)" shows "State_Idle localState output_fun trans_fun (localState (trans_fun NoMsg c))"
0: shows "i < length (x # xs) \<longrightarrow> i < length xs \<or> i = length xs"
A_def: shows "A = dghm_up I \<AA> \<CC> \<phi>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>"
rsapss_verify_def: fixes m :: "bit list"   and s :: "bit list"   and d :: "nat"   and n :: "nat" shows "rsapss_verify m s d n = (if length s \<noteq> length (nat_to_bv n) then False else let em = nat_to_bv_length (rsa_crypt (bv_to_nat s) d n) (roundup (length (nat_to_bv n) - 1) 8 * 8) in emsapss_decode m em (length (nat_to_bv n) - 1))"
inner_node_def: fixes prog :: "cmd"   and n :: "w_node" shows "While_CFGExit.inner_node prog n \<equiv> While_CFG.valid_node prog n \<and> n \<noteq> (_Entry_) \<and> n \<noteq> (_Exit_)"
card_cartesian_product: fixes A :: "'c set"   and B :: "'d set" shows "card (A \<times> B) = card A * card B"
assms: fixes \<pi> :: "perm"   and x :: "'a"   and y :: "'b" assumes "r (\<pi> \<bullet> x) (\<pi> \<bullet> y)" shows "r x y"
cod_comp: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.cod C (C g f) = partial_magma.cod C g"
rotate1_hd_tl: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "rotate1 xs = tl xs @ [hd xs]"
in_language_\<dd>s: fixes w :: "'b list"   and L :: "'b language"   and v :: "'b list" shows "in_language (\<dd>s w L) v = in_language L (w @ v)"
assms_2: shows "\<forall>AA\<in>AAA. finite AA"
R_def: fixes x :: "('a \<times> 'a) set"   and p :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "(x \<subseteq> rel_R p q) = (rel_aka.dom_op p \<subseteq> rel_aka.wp x q)"
freshAbs_vsubstAbs_ident: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and xs :: "'varSort"   and x :: "'var"   and x1 :: "'var" assumes "goodAbs A"   and "freshAbs xs x A" shows "A $[x1 // x]_xs = A"
Units_l_inv_ex: fixes x :: "'a" assumes "x \<in> Units G" shows "\<exists>y\<in>carrier G. y \<otimes> x = \<one>"
app_unapp: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and u :: "'c"   and u\<^sub>1 :: "'c"   and u\<^sub>2 :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "unapp u = Some (u\<^sub>1, u\<^sub>2)" shows "u = app u\<^sub>1 u\<^sub>2"
reverse_iterateoi_rule: fixes m :: "'s" assumes "invar m" shows "set_iterator_rev_linord (reverse_iterateoi m) (\<alpha> m)"
ereal_open_affinity_pos: fixes S :: "ereal set"   and m :: "ereal"   and t :: "ereal" assumes "open S"   and "m \<noteq> \<infinity>"   and "0 < m"   and "\<bar>t\<bar> \<noteq> \<infinity>" shows "open ((\<lambda>x. m * x + t) ` S)"
elims: fixes x :: "'a \<Rightarrow> 'a"   and xa :: "'a list"   and y :: "'a list" assumes "map_hd x xa = y" obtains f :: "'a \<Rightarrow> 'a" where "x = f__" and "xa = []" and "y = []"   | f :: "'a \<Rightarrow> 'a" and a :: "'a" and S :: "'a list" where "x = f" and "xa = a # S" and "y = f a # S"
try_swap_eq: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and p :: "'b"   and c\<^sub>1 :: "'a"   and c\<^sub>2 :: "'a"   and w :: "'b \<Rightarrow> 'a"   and x :: "'b"   and w' :: "'b \<Rightarrow> 'a" assumes "distinct [blue, brown, green, red]"   and "finite UNIV" shows "(Blue_Eyes.try_swap blue p c\<^sub>1 c\<^sub>2 w x = Blue_Eyes.try_swap blue p c\<^sub>1 c\<^sub>2 w' x) = (w x = w' x)"
f1: shows "\<not> (fst instr = sreg_type RDPSR \<or> fst instr = sreg_type RDWIM \<or> fst instr = sreg_type RDTBR \<or> fst instr = sreg_type RDASR \<and> privileged_ASR (get_operand_w5 (snd instr ! 0)))"
abs_summable_on_empty: fixes f :: "'d \<Rightarrow> 'e" shows "f abs_summable_on {}"
summable_norm_exp: fixes x :: "'a" shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R fact n))"
null_eqI: fixes n :: "'c" assumes "\<And>f. C n f = n \<and> C f n = n" shows "n = C.null"
list_to_poly_Nil: fixes cs :: "'d list" shows "list_to_poly [] cs = 0"
map_add_le_mapE: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option"   and h :: "'c \<Rightarrow> 'd option" assumes "f ++ g \<subseteq>\<^sub>m h" shows "g \<subseteq>\<^sub>m h"
vec_eq_iff: fixes x :: "('d, 'e) Finite_Cartesian_Product.vec"   and y :: "('d, 'e) Finite_Cartesian_Product.vec" shows "(x = y) = (\<forall>i. x $h i = y $h i)"
class_add_sees_method: fixes P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "jvm_method"   and D :: "char list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> C\<^sub>0 sees M\<^sub>0, b :  Ts\<rightarrow>T = m in D"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> C\<^sub>0 sees M\<^sub>0, b :  Ts\<rightarrow>T = m in D"
coeff_add: fixes p :: "'a mpoly"   and q :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "coeff (p + q) m = coeff p m + coeff q m"
finprod_Un_disjoint: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> int" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier R"   and "g \<in> B \<rightarrow> carrier R" shows "finsum R g (A \<union> B) = finsum R g A \<oplus> finsum R g B"
s_init_foreach_def: fixes U0 :: "'e set" shows "s_init_foreach U0 \<equiv> FOREACH U0 (\<lambda>x (U0, ws). RETURN (insert x U0, ws(x \<mapsto> []))) ({}, Map.empty) \<bind> (\<lambda>(U0, ws). RETURN (None, U0, ws))"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "Ferrack.num" shows "nozerocoeff (CN n c t) = (c \<noteq> 0 \<and> nozerocoeff t)"
recognizeOtherPPA: shows "\<forall>c d. \<lfloor>PPA (Agent d)\<rfloor>\<^sub>d \<longrightarrow> \<lfloor>PPA (Agent d)\<rfloor>\<^sub>c"
simps_1: fixes xs :: "('b, 'c) Combinators list" shows "noDenyAll1 (DenyAll # xs) = noDenyAll xs"
Nonce_supply1: fixes evs :: "event list" shows "\<exists>N. Nonce N \<notin> used evs"
irfb_inv2: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "(fb\<^sub>\<R> \<circ> fb\<^sup>-\<^sub>\<R>) \<phi> = \<phi>"
GB_alt_3_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m" shows "is_Groebner_basis G = (\<forall>f\<in>pmdl G. f \<noteq> 0 \<longrightarrow> (\<exists>g\<in>G. g \<noteq> 0 \<and> lt g adds\<^sub>t lt f))"
chine_hcomp_props_1: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "\<guillemotleft>chine_hcomp \<nu> \<mu> : Leg0 (Dom \<nu>) \<down>\<down> Leg1 (Dom \<mu>) \<rightarrow>\<^sub>C Leg0 (Cod \<nu>) \<down>\<down> Leg1 (Cod \<mu>)\<guillemotright>"
respection_of_barbs_and_closures_3: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_respects_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_respects_binary_pred (Rel\<^sup>+) (\<lambda>a. hasBarb a CWB)"
divide_fps_const': fixes f :: "'b fps"   and c :: "'b" shows "f / fps_const c = f * fps_const (inverse c)"
derangements_inv: fixes d :: "nat \<Rightarrow> nat"   and S :: "nat set" assumes "d \<in> derangements S" shows "inv d \<in> derangements S"
E: fixes f :: "'c" assumes "\<guillemotleft>f : a' \<rightarrow> a\<guillemotright>" shows "Cones.FUN f (\<iota> \<chi>) = \<Phi>.FUN a' (\<phi> (a', a) f)"
63: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (x \<sqinter> - y \<squnion> x \<sqinter> - - y) = x"
pr_simps_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "local.dom (pr0 a b) = HF'.prod a b"
PSimQ: shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
simps_5: fixes Inv :: "'b"   and b :: "bexp"   and c :: "'b acom"   and P :: "'b"   and Inv' :: "'b"   and b' :: "bexp"   and c' :: "'b acom"   and P' :: "'b" shows "({Inv} WHILE b DO c {P} \<le> {Inv'} WHILE b' DO c' {P'}) = (b = b' \<and> c \<le> c' \<and> Inv \<le> Inv' \<and> P \<le> P')"