hcju/isabelleps huggingface dataset:
step_abs: fixes s :: "'C"   and s' :: "'C" assumes "s \<in> wa_invar WAC"   and "s \<in> wa_cond WAC"   and "(s, s') \<in> wa_step WAC" shows "(\<alpha> s, \<alpha> s') \<in> wa_step WAA"
mat_eq_iff: fixes x :: "'a Matrix.mat"   and y :: "'a Matrix.mat" shows "(x = y) = (dim_row x = dim_row y \<and> dim_col x = dim_col y \<and> (\<forall>i j. i < dim_row y \<longrightarrow> j < dim_col y \<longrightarrow> x $$ (i, j) = y $$ (i, j)))"
max_factor_degree_aux: fixes ds :: "nat list"   and es :: "nat list" assumes "mset ds \<subseteq># mset es"   and "length ds + length ds \<le> length es"   and "sorted es" shows "sum_list ds \<le> sum_list (drop (length es div 2) es)"
sgn_sqrt: fixes x :: "real" shows "sgn (sqrt x) = sgn x"
span_li_not_depend_2: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and S :: "'d set"   and N :: "'d set" assumes "module R M"   and "S \<subseteq> N"   and "LinearCombinations.submodule R N M" shows "module.lin_dep R (M\<lparr>carrier := N\<rparr>) S = module.lin_dep R M S"
k_dfs_invariant: fixes a :: "'a" shows "KBP.k_invariant a (KBP.k_dfs a)"
arith_simps_63: fixes a :: "'a" shows "a * (0::'a) = (0::'a)"
dual_category_axioms: shows "dual_category (\<cdot>\<^sub>C)"
le_diff_iff': fixes a :: "nat"   and c :: "nat"   and b :: "nat" assumes "a \<le> c"   and "b \<le> c" shows "(c - a \<le> c - b) = (b \<le> a)"
timpl_closure_set_funs_term_subset: fixes FP :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" shows "\<Union> (funs_term ` timpl_closure_set FP TI) \<subseteq> \<Union> (funs_term ` FP) \<union> Abs ` snd ` TI"
mat_diag_mult_left: fixes A :: "'a mat"   and n :: "nat"   and nr :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "A \<in> carrier_mat n nr" shows "mat_diag n f * A = mat n nr (\<lambda>(i, j). f i * A $$ (i, j))"
Sup_lexord_def: fixes k :: "'c \<Rightarrow> 'd"   and c :: "'c set \<Rightarrow> 'c"   and s :: "'c set \<Rightarrow> 'c"   and A :: "'c set" shows "Sup_lexord k c s A = (let U = Sup (k ` A) in if \<exists>a\<in>A. k a = U then c {a \<in> A. k a = U} else s A)"
neg_equal_zero: fixes a :: "complex" shows "(- a = a) = (a = 0)"
simps_1: fixes s :: "'d stream" shows "sdrop 0 s = s"
steps_z_beta_alt: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set"   and l'' :: "'d"   and Z'' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta> \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l'', Z''\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l'', Z''\<rangle>"
product_functor_def: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B2 :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F1 :: "'a \<Rightarrow> 'd"   and F2 :: "'c \<Rightarrow> 'e" shows "product_functor A1 A2 B1 B2 F1 F2 \<equiv> ((category A1 \<and> category A2) \<and> category B1 \<and> category B2) \<and> (functor A1 B1 F1 \<and> functor A2 B2 F2) \<and> product_category A1 A2 \<and> product_category B1 B2"
distrib_1: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
existence_ivl_trans': fixes t :: "real"   and s :: "real"   and x0 :: "'a" assumes "t + s \<in> rev.existence_ivl0 x0"   and "t \<in> rev.existence_ivl0 x0" shows "s \<in> rev.existence_ivl0 (rev.flow0 x0 t)"
not_has_disc_normalize_match: fixes disc :: "'b \<Rightarrow> bool"   and neg :: "bool"   and m :: "'b match_expr"   and m' :: "'b match_expr" assumes "\<not> has_disc_negated disc neg m"   and "m' \<in> set (normalize_match m)" shows "\<not> has_disc_negated disc neg m'"
ran_CompFun_is_not_SA: fixes Sas :: "('d, 'e, 'f) seqauto set"   and HA :: "('d, 'e, 'f) hierauto" assumes "\<not> Sas \<subseteq> SAs HA" shows "Sas \<notin> ran (CompFun HA)"
RDirProd_def: fixes R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" shows "RDirProd R S = monoid.extend (R \<times>\<times> S) \<lparr>zero = \<one>\<^bsub>add_monoid R \<times>\<times> add_monoid S\<^esub>, add = (\<otimes>\<^bsub>add_monoid R \<times>\<times> add_monoid S\<^esub>)\<rparr>"
subring_degree_one_associatedI: fixes K :: "'a list set"   and a :: "'a list"   and a' :: "'a list"   and b :: "'a list" assumes "subring K (poly_ring R)"   and "a \<in> K"   and "a' \<in> K"   and "b \<in> K"   and "a \<otimes>\<^bsub>poly_ring R\<^esub> a' = \<one>\<^bsub>poly_ring R\<^esub>" shows "[a, b] \<sim>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> [\<one>\<^bsub>poly_ring R\<^esub>, a' \<otimes>\<^bsub>poly_ring R\<^esub> b]"
right_diff_distrib: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ring plus zero minus uminus times" shows "times a (minus b c) = minus (times a b) (times a c)"
coeff_add_right_notin: fixes x :: "nat"   and p :: "'a mpoly" assumes "x \<notin> vars p" shows "MPoly_Type.coeff (Var x - p) (monomial (Suc 0) x) = (1::'a)"
w_reach_4: shows "w \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"
basic_pseudo_hoop_algebra_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.basic_pseudo_hoop_algebra times inf imp_l less_eq less one imp_r \<equiv> class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r \<and> class.basic_pseudo_hoop_algebra_axioms imp_l less_eq imp_r"
strip_while_Cons: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f"   and xs :: "'f list" assumes "\<not> P x" shows "strip_while P (x # xs) = x # strip_while P xs"
surjective_pairing: fixes t :: "'g \<times> 'h" shows "t = (fst t, snd t)"
rcs_Unit1: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "H \<bullet>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = H"
sp_spec_file_provider: shows "\<forall>p1 p2 f m1 m2. Policy.sp_spec_subj_obj p1 (FILEP f) m1 \<and> Policy.sp_spec_subj_obj p2 (FILEP f) m2 \<longrightarrow> Policy.sp_spec_subj_subj p1 p2"
joint_strategy_def: fixes \<sigma>\<^sub>e :: "'c list \<Rightarrow> 'b"   and \<sigma>\<^sub>o :: "'c list \<Rightarrow> 'b"   and p :: "'c list" shows "joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o p = (if even (length p) then \<sigma>\<^sub>e p else \<sigma>\<^sub>o p)"
absolutely_integrable_on_open_interval: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "(f absolutely_integrable_on box a b) = (f absolutely_integrable_on cbox a b)"
index_mat_multrow_mat_3: fixes n :: "nat"   and k :: "nat"   and a :: "'c" shows "dim_col (multrow_mat n k a) = n"
cSUP_le_iff: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset"   and u :: "'c multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_above (f ` A)" shows "(Sup (f ` A) \<subseteq># u) = (\<forall>x\<in>A. f x \<subseteq># u)"
normalize_positive_dst_ports_normalized_n_primitive: fixes m :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc_negated is_Dst_Ports False m" shows "\<forall>m'\<in>set (normalize_positive_dst_ports m). normalized_dst_ports m'"
assms_5: shows "prv (eqv \<chi>1 \<chi>2)"
ASSUMPTION_def: fixes A :: "bool" shows "ASSUMPTION A \<equiv> A"
setPair_noteq: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "{a, b} \<noteq> {c, d}" shows "\<not> (a = c \<and> b = d)"
in_arcs_del_arc_iff: fixes a :: "'b"   and u :: "'a" shows "in_arcs (del_arc a) u = (if head G a = u then in_arcs G u - {a} else in_arcs G u)"
that: fixes m :: "nat" assumes "\<sigma> [m..] \<approx> \<tau> [n..]"   and "\<forall>j<m. \<exists>k<n. \<sigma> [j..] \<approx> \<tau> [k..]" shows "thesis"
ResInput': fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x>x<y>.P \<equiv>\<^sub>e \<zero>"
bbind_strict1: shows "bbind\<cdot>\<bottom> = \<bottom>"
ring_simprules_6: shows "(1::'b) \<in> UNIV"
less_aexp_antisym: fixes x :: "'a aexp"   and y :: "'a aexp" shows "(x < y) = (\<not> y < x \<and> x \<noteq> y)"
group_race_def: fixes T :: "nat set"   and \<gamma> :: "nat \<Rightarrow> thread_state option" shows "group_race T \<gamma> \<equiv> \<exists>j\<in>T. \<exists>k\<in>T. j \<noteq> k \<and> W (the (\<gamma> j)) \<inter> (R (the (\<gamma> k)) \<union> W (the (\<gamma> k))) \<noteq> {}"
image_map_indets_Polys: fixes f :: "'d \<Rightarrow> 'b"   and X :: "'d set" shows "map_indets f ` P[X] = P[f ` X]"
Entry_label_incr: fixes n :: "label"   and i :: "nat" assumes "Entry = n \<oplus> i" shows "n = Entry"
list_of_set_def: fixes s :: "'a set" shows "list_of_set s = set_choose (list_of_set_set s)"
Suc_i_mod_s_0_1: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "i mod s = s - Suc 0" shows "Suc i mod s = 0"
as0_12: shows "AAs0 \<cdot>\<cdot>aml \<eta>s0 = AAs"
inverse_finite_3_def: shows "inverse = (\<lambda>x. x)"
intro: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "Dmonoid prod one"   and "Dgrp_axioms prod one" shows "Dgrp prod one"
fst_of_sp_is_vert: fixes g :: "('a, 'b) rgraph_scheme"   and v1 :: "'a"   and es :: "'a edge list"   and v2 :: "'a" assumes "subpath g v1 es v2" shows "v1 \<in> vertices g"
composeES_yields_ES: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec" assumes "ES_valid ES1"   and "ES_valid ES2" shows "ES_valid ES1 \<parallel> ES2"
obj_at_def: fixes P :: "('e, 'f, 'g) object \<Rightarrow> bool"   and r :: "'g" shows "obj_at P r \<equiv> \<lambda>s. case sys_heap s r of None \<Rightarrow> False | Some obj \<Rightarrow> P obj"
in_minus_Un_1: fixes i :: "'a"   and I :: "'a set" assumes "i \<in> I" shows "I - {i} \<union> {i} = I"
not: fixes bs :: "real list"   and p :: "fm" shows "Ifm bs (not p) = Ifm bs (NOT p)"
inf_nil: fixes x :: "'a" shows "x \<sqinter> \<bottom> = \<bottom>"
weakenpreR: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} c { Q}"   and "\<And>s. P' s \<Longrightarrow> P s" shows "\<turnstile>\<^sub>3\<^sub>a {P'} c { Q}"
s_UnauthP_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Unauth e\<ggreater> P\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Unauth e'a" and "\<lless>\<pi>, e\<ggreater> P\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"   | h :: "hash" where "e = Hashed h e'" and "\<pi>' = \<pi> @ [\<lparr>e'\<rparr>]" and "value e'"
assms_3: shows "- L = L' \<cdot>l \<sigma>"
hom_inv: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd"   and x :: "'b" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h (inv\<^bsub>G\<^esub> x) = inv\<^bsub>H\<^esub> h x"
PLUS_inject: fixes xs :: "'a rexp list"   and ys :: "'a rexp list" assumes "\<forall>x\<in>set xs \<union> set ys. \<not> is_Zero x \<and> \<not> is_Plus x"   and "sorted xs"   and "sorted ys" shows "(PLUS xs = PLUS ys) = (xs = ys)"
Reception: fixes evsr :: "event list"   and A :: "agent"   and B :: "agent"   and X :: "msg" assumes "evsr \<in> set_pur"   and "Says A B X \<in> set evsr" shows "Gets B X # evsr \<in> set_pur"
nonzero_coeffsD: fixes a :: "('a \<times> 'b) multiset"   and x :: "'a"   and n :: "'b" assumes "nonzero_coeffs a"   and "(x, n) \<in># a" shows "n \<noteq> (0::'b)"
bad: fixes s' :: "'d"   and x :: "'a" assumes "\<not> bad s'"   and "enat (count s') < n + enat (count s)"   and "consider x"   and "x \<in> outs_\<I> \<I>" shows "spmf (map_spmf (bad \<circ> snd) (callee s' x)) True \<le> k"
moebius_similarity_plus: fixes a :: "complex"   and c :: "complex"   and b :: "complex"   and d :: "complex" assumes "a \<noteq> 0"   and "c \<noteq> 0" shows "moebius_similarity a b + moebius_similarity c d = moebius_similarity (a * c) (a * d + b)"
add_left_cancel: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cancel_semigroup_add_ow U plus"   and "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "(plus a b = plus a c) = (b = c)"
lub_spmf_upper: fixes p :: "'a spmf" assumes "p \<in> Y" shows "ord_spmf (=) p local.lub_spmf"
r_find_k_1: fixes b :: "nat" shows "eval r_find_k [b] \<down>"
assms_5: fixes x :: "real" assumes "a \<le> ereal x"   and "ereal x \<le> b" shows "0 \<le> g' x"
parallel_oracle_def: fixes left :: "'a \<Rightarrow> 'c \<Rightarrow> ('e \<times> 'a) spmf"   and right :: "'b \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'b) spmf" shows "left \<ddagger>\<^sub>O right = (\<lambda>(s1, s2). case_sum (\<lambda>a. map_spmf (map_prod Inl (\<lambda>s1'. (s1', s2))) (left s1 a)) (\<lambda>b. map_spmf (map_prod Inr (Pair s1)) (right s2 b)))"
sup_derivable_fp: fixes p :: "'a" assumes "antisymmetric A (\<sqsubseteq>)"   and "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"   and "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x"   and "extreme_bound A (\<sqsubseteq>) (Collect derivable) p" shows "f p = p"
sin_npi2: fixes n :: "nat" shows "sin (pi * real n) = 0"
AE_I: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool"   and N :: "'c set" assumes "{x \<in> space M. \<not> P x} \<subseteq> N"   and "emeasure M N = 0"   and "N \<in> sets M" shows "almost_everywhere M P"
vcpower_two_ftimesI: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> A" shows "x \<in>\<^sub>\<circ> A ^\<^sub>\<times> 2\<^sub>\<nat>"
\<Z>_srng_\<Z>_sgrp_vmult: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "\<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ>"
\<bb>: shows "ideal \<bb> R (+) (\<cdot>) \<zero> \<one>"
K_def: shows "K \<equiv> {uu_. \<exists>x X. uu_ = x \<and> X \<in> XS \<and> \<not> qAFresh xs x X}"
count_rem_components_def: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> 'e) list" shows "count_rem_components bs = length (remdups (map component_of_term (Keys_to_list (map fst bs)))) - count_const_lt_components (filter (\<lambda>b. fst b \<noteq> 0) bs)"
lm073: fixes x :: "'a set"   and X :: "'a set set" assumes "x \<in> X" shows "finestpart x \<subseteq> finestpart (\<Union> X)"
minus_def: fixes x :: "'a"   and y :: "'a" shows "comm.additive.minus x y \<equiv> x - y"
read_def: fixes c :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and P :: "'c \<Rightarrow> 'b process" shows "read c A P \<equiv> Mprefix (c ` A) (P \<circ> inv c)"
nhds_set_def: fixes X :: "'b set" shows "nhds_set X = Inf (principal ` {S. open S \<and> X \<subseteq> S})"
decode_Var_def: fixes x :: "hf" shows "decode_Var x \<equiv> name_of_nat (nat_of_ord (pred x))"
ext: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
strategyproof_an_sds_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "strategyproof_an_sds agents alts sds \<equiv> strategyproof_sds agents alts sds \<and> an_sds agents alts sds"
jj: shows "\<forall>x\<in>rvars (\<T> (l ! j)). \<langle>\<V> (l ! j)\<rangle> x = \<langle>\<V> (l ! (j - 1))\<rangle> x \<or> \<langle>\<V> (l ! j)\<rangle> x = the (\<B>\<^sub>l (l ! (j - 1)) x) \<or> \<langle>\<V> (l ! j)\<rangle> x = the (\<B>\<^sub>u (l ! (j - 1)) x)"
sep_conj_left_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_conj plus sep_disj Q R) = sep_algebra.sep_conj plus sep_disj Q (sep_algebra.sep_conj plus sep_disj P R)"
wf_\<Gamma>_append: fixes rs1 :: "'a rule list"   and rs2 :: "'a rule list" shows "wf_\<Gamma> (rs1 @ rs2) = (wf_\<Gamma> rs1 \<and> wf_\<Gamma> rs2)"
leg_simps_3: shows "C.arr assoc_fgh.dom.leg1"
exhaust: fixes y :: "\<upsilon>" obtains x1 :: "\<omega>" where "y = \<omega>\<upsilon> x1"   | x2 :: "\<sigma>" where "y = \<sigma>\<upsilon> x2"
exec_gpv_lift_resumption: fixes r :: "('c, 'call, 'ret) Resumption.resumption"   and s :: "'s" shows "exec_gpv callee (lift_resumption r) s = local.execp_resumption r s"
ys'_def: shows "ys' \<equiv> map (\<lambda>(x, n). x ^ Suc n) ys"
maximalI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>c. 0 \<le> c \<Longrightarrow> t (\<lambda>_. c) = (\<lambda>_. c)" shows "maximal t"
simps_1: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "mult.Tensor f [] xs = []"
has_field_der_id: fixes x :: "'a" shows "((\<lambda>x. x\<^sup>2 / (2::'a)) has_field_derivative x) (at x)"
Prefix0_def: fixes c :: "'c"   and P :: "('c, 'd) action" shows "Prefix0 c P \<equiv> action_of (R (true \<turnstile> do (\<lambda>_. c) \<and> (\<lambda>(A, A'). alpha_rp.more A' = alpha_rp.more A))) `;` P"
arctan_inverse: fixes x :: "real" assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
nsum_split: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>Suc (n + m). f j \<in> carrier A" shows "\<Sigma>\<^sub>e A f Suc (n + m) = \<Sigma>\<^sub>e A f n \<plusminus> \<Sigma>\<^sub>e A cmp f (slide (Suc n)) m"
translation_good: fixes F :: "((Class \<times> Method) \<times> Var list \<times> Expr) list"   and \<Sigma> :: "((Class \<times> Method) \<times> nat) list" assumes "compileProg F"   and "TP_MST \<Sigma>"   and "TP \<Sigma> F" shows "Sig_good \<Sigma>"
fix_def: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set" shows "fix f A = {x \<in> A. f x = x}"
pareto_loser_in_alts: fixes x :: "'b" assumes "x \<in> pareto_losers R" shows "x \<in> alts"
has_chain_integral_chain_integral3: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" assumes "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i"
assms_2: shows "k \<le> 2 ^ (2 * n + 2)"
break_def: shows "break \<equiv> \<lambda>\<sigma>. Some ((), \<sigma>\<lparr>break_status := True\<rparr>)"
rev_shiftl: fixes w :: "'b word"   and n :: "nat" shows "word_reverse w << n = word_reverse (w >> n)"
unfold: shows "subst_term (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (subst_term (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) t) = fold (\<lambda>single. subst_term [single]) (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (fold (\<lambda>single. subst_term [single]) (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) t)"
less_induct: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "class.wellorder less_eq less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
sum_Sigma_product: fixes S :: "'a set"   and T :: "'a \<Rightarrow> 'b set"   and x :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite S"   and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)" shows "(\<Sum>i\<in>S. sum (x i) (T i)) = (\<Sum>(i, j)\<in>Sigma S T. x i j)"
runit'_simps_1: fixes f :: "'a" assumes "S.ide f" shows "S.arr (S.runit' f)"
matrix_inv_unique: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and B :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" assumes "A ** B = mat (1::'d)"   and "B ** A = mat (1::'d)" shows "matrix_inv A = B"
content_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "content (p * q) = content p * content q"
1: shows "(\<Union>f\<in>injections X Y. {f +< {(x, y)} |y. y \<in> Y - Range f}) = \<Union> {{f +< {(x, y)} |y. y \<in> Y - Range f} |f. f \<in> injections X Y \<and> x \<notin> Domain f}"
in_set_in_alist: fixes t :: "('c, 'd) bintree"   and xs :: "('c, 'd) bintree option list" assumes "Some t \<in> set xs" shows "(val t, priority t) \<in> set (elements xs)"
ex_assn_def: fixes P :: "'b \<Rightarrow> assn" shows "\<exists>\<^sub>Ax. P x \<equiv> Abs_assn (\<lambda>h. \<exists>x. h \<Turnstile> P x)"
at_prm_fresh: fixes c :: "'d"   and pi :: "('d \<times> 'd) list" assumes "at TYPE('d)"   and "c \<sharp> pi" shows "pi \<bullet> c = c"
B_fin: shows "finite {k. k < length xs \<and> k < j \<and> xs ! k = i}"
prog_add_clock: fixes env :: "v sem_env"   and s :: "'b state"   and prog :: "top0 list"   and s' :: "'b state"   and res :: "(v sem_env, v) result" assumes "evaluate_prog False env s prog (s', res)" shows "\<exists>c. evaluate_prog True env (update_clock (\<lambda>_. c) s) prog (update_clock (\<lambda>_. 0) s', res)"
simps_2: fixes s :: "'d net_state"   and t :: "'d net_state" shows "netmap (SubnetS s t) = netmap s ++ netmap t"
Val_abs_axioms: shows "Val_abs \<gamma>_const Const plus_const"
less_le: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.order less_eq less" shows "less x y = (less_eq x y \<and> x \<noteq> y)"
inject_1: fixes x11 :: "agent"   and x12 :: "agent"   and x13 :: "msg"   and y11 :: "agent"   and y12 :: "agent"   and y13 :: "msg" shows "(Says x11 x12 x13 = Says y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
tfr\<^sub>s\<^sub>e\<^sub>t_if_comp_tfr\<^sub>s\<^sub>e\<^sub>t: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "comp_tfr\<^sub>s\<^sub>e\<^sub>t arity Ana \<Gamma> M" shows "tfr\<^sub>s\<^sub>e\<^sub>t (set M)"
word_sle_eq: fixes a :: "'a word"   and b :: "'a word" shows "(a \<le>s b) = (sint a \<le> sint b)"
prems_1: shows "pre_strong_term_class.wellformed (name $$ ts)"
fwi_canonical_extend: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and I :: "nat set"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "linordered_ab_monoid_add.canonical_subs plus less_eq n I m"   and "I \<subseteq> {0..n}"   and "less_eq zero (m k k)"   and "k \<le> n" shows "linordered_ab_monoid_add.canonical_subs plus less_eq n (I \<union> {k}) (linordered_ab_monoid_add.fwi plus less_eq m n k n n)"
inv_stereographic_circline_stereographic_circline: shows "inv_stereographic_circline \<circ> stereographic_circline = id"
Units_inv_inv: fixes x :: "int" assumes "x \<in> Units R" shows "inv (inv x) = x"
coeff_in_ring: fixes a :: "'e \<Rightarrow> 'a"   and A :: "'e set"   and x :: "'e" assumes "a \<in> A \<rightarrow> carrier K"   and "x \<in> A" shows "a x \<in> carrier K"
is_digit_char_simps_1: shows "is_digit_char CHR ''0''"