hcju/isabelleps huggingface dataset:
wave_lub: fixes Y :: "('a \<times> 'a \<Rightarrow> ennreal) set"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "\<And>f. f \<in> Y \<Longrightarrow> wave \<Gamma> f"   and "countable (support_flow (\<Squnion> Y))" shows "wave \<Gamma> (\<Squnion> Y)"
str_del_point_blocks_def: fixes p :: "'a" shows "fin.str_del_point_blocks p \<equiv> filter_mset ((\<notin>) p) (str_del_point_blocks p)"
update_by_fun_pair_eq_update_by_pair: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv_raw x xs" shows "update_by_fun_pair x k f xs = update_by_pair x (k, f (lookup_pair x xs k)) xs"
pos_emb_step_at_nested: fixes p :: "dir list"   and d1 :: "dir"   and q :: "dir list"   and d2 :: "dir"   and t :: "('c, 'd) tm" shows "position_of (emb_step_at (p @ [d1] @ q) d2 t) (p @ [d1]) = position_of t (p @ [d1])"
tabulation_axioms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (f \<star> g \<star> h) TfHgh.\<rho>\<sigma>.tab (tab\<^sub>0 (g \<star> h) \<star> TfHgh.p\<^sub>0) (tab\<^sub>1 f \<star> TfHgh.p\<^sub>1)"
items_le_idempotent: fixes k :: "nat"   and I :: "item set" shows "items_le k (items_le k I) = items_le k I"
simps_1: fixes f :: "'a" shows "Ide \<^bold>\<langle>f\<^bold>\<rangle> = C.ide f"
e_type_const_conv_vs: fixes ves :: "e list" assumes "const_list ves" shows "\<exists>vs. ves = $$* vs"
e_2: fixes i :: "nat" shows "partial_state.encode2 dims vars1 i = partial_state.encode1 dims vars2 i"
check_poly_strict_mono: fixes p :: "('c monom \<times> 'a) list"   and v :: "'c" assumes "check_poly_strict_mono power_mono p v"   and "check_poly_weak_mono_all p" shows "poly_strict_mono p v"
notnotx: shows "\<forall>w. P w (\<midarrow> (\<midarrow> x)) = (\<not> O w (\<midarrow> x))"
subdg_Obj_vsubset: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
hyps: shows "diagonal_mat (diag_block_mat (map (\<lambda>a. mat (Suc 0) (Suc 0) (\<lambda>_. a)) as))"
vertex_1_nat_def: shows "vertex_1 \<equiv> 1"
zig_zag: fixes r :: "'b tree"   and r1' :: "'b tree"   and r2' :: "'b tree"   and p :: "'b"   and s :: "'b tree"   and a :: "'b"   and u :: "'b tree"   and b :: "'b" assumes "size r = size r1' + size r2'"   and "real (T_part p r) + \<Phi> r1' + \<Phi> r2' - \<Phi> r \<le> 2 * \<phi> r + 1" shows "real (T_part p r + 1) + \<Phi> \<langle>s, a, r1'\<rangle> + \<Phi> \<langle>u, b, r2'\<rangle> - \<Phi> \<langle>s, a, \<langle>r, b, u\<rangle>\<rangle> \<le> 2 * \<phi> \<langle>s, a, \<langle>r, b, u\<rangle>\<rangle> + 1"
exec_dynCall_Normal_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>dynCall init p return c,Normal s\<rangle> \<Rightarrow> t" obtains "\<Gamma>\<turnstile> \<langle>call init (p s) return c,Normal s\<rangle> \<Rightarrow> t"
negligible_absI: fixes f :: "nat \<Rightarrow> real" assumes "Negligible.negligible f" shows "Negligible.negligible (\<lambda>x. \<bar>f x\<bar>)"
powers_84: fixes a :: "'a" shows "a\<^sup>2 = a * a"
sel_2: fixes x :: "complex" shows "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
must_syncE: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list"   and m :: "'heap" assumes "exec_mthr.must_sync P t x m" obtains taa :: "('addr, 'thread_id, 'heap) jvm_thread_action" and x' :: "'addr option \<times> 'addr frame list" and m' :: "'heap" and s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" where "exec_mthr.r_syntax P t x m taa x' m'" and "exec_mthr.actions_ok s t taa" and "m = shr s"
step: fixes C :: "'d \<Rightarrow> bool"   and r :: "'d"   and B :: "'d \<Rightarrow> 'd option"   and r' :: "'d"   and sr'' :: "'d option" assumes "C r"   and "B r = Some r'"   and "(Some r', sr'') \<in> option_while' C B" shows "(Some r, sr'') \<in> option_while' C B"
inj_int_encode: fixes A :: "int set" shows "inj_on int_encode A"
union: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z (M + N) = f (comm_monoid_mset.F f z M) (comm_monoid_mset.F f z N)"
mono_on_def: fixes f :: "'a \<Rightarrow> 'c"   and X :: "'a set" shows "order_class.mono_on f X = (\<forall>x\<in>X. \<forall>y\<in>X. x \<le> y \<longrightarrow> f x \<le> f y)"
diff_stabilizes: fixes x :: "'c"   and R :: "'a set"   and g1 :: "'a"   and g2 :: "'a" assumes "x \<in> E"   and "R \<in> rcosets stabilizer G \<phi> x"   and "g1 \<in> R"   and "g2 \<in> R" shows "g1 \<otimes> inv g2 \<in> stabilizer G \<phi> x"
assms_3: shows "\<I> \<turnstile>g gpv \<surd>"
inverse_eq2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b)" shows "((\<lambda>x. inverse (f x)) \<in> o[F](g)) = ((\<lambda>x. 1::'b) \<in> o[F](\<lambda>x. f x * g x))"
path2_split_ex': fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and x :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "x \<in> set ns" obtains ns\<^sub>1 :: "'node list" and ns\<^sub>2 :: "'node list" where "g \<turnstile> n-ns\<^sub>1\<rightarrow>x" and "g \<turnstile> x-ns\<^sub>2\<rightarrow>m" and "ns = butlast ns\<^sub>1 @ ns\<^sub>2"
prj_simps_11: shows "C.seq \<nu>\<pi>.prj\<^sub>1 \<p>\<^sub>0[\<mu>.leg0, \<nu>.leg1 \<cdot> \<nu>\<pi>.prj\<^sub>1]"
lotteries_on_def: fixes A :: "'a set" shows "lotteries_on A = {p. set_pmf p \<subseteq> A}"
simps_2: fixes y :: "'d" shows "flat_orA (allow (deny y)) = allow y"
NatMultiples_Factor: fixes k :: "nat"   and F :: "nat set" assumes "k \<in> F" shows "k \<in> NatMultiples F"
prob_space_axioms: fixes s :: "'s" shows "prob_space (T s)"
intro: fixes U\<^sub>V\<^sub>S :: "'d set"   and plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S :: "'d"   and minus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set" assumes "vector_space_ow U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale"   and "finite_dimensional_vector_space_ow_axioms U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis" shows "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S minus\<^sub>V\<^sub>S uminus\<^sub>V\<^sub>S scale basis"
assms_2: assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> A)" shows "P"
neg_one_even_power: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "even n" shows "power.power one times (uminus one) n = one"
order_pderiv2: fixes p :: "'a poly"   and a :: "'a"   and n :: "nat" assumes "pderiv p \<noteq> 0"   and "order a p \<noteq> 0" shows "(order a (pderiv p) = n) = (order a p = Suc n)"
drop_not_empty_conv: fixes n :: "nat"   and xs :: "'d list" shows "(xs \<up> n \<noteq> []) = (n < length xs)"
ideal: shows "connect_obsf (absorb \<A> (obsf_converter (w_adv_real |\<^sub>= w_usr))) (obsf_resource (sim |\<^sub>= 1\<^sub>C \<rhd> ideal_resource)) = connect_obsf \<A> (obsf_resource (w_adv_real \<odot> sim \<odot> w_adv_ideal_inv |\<^sub>= 1\<^sub>C \<rhd> w_adv_ideal |\<^sub>= w_usr \<rhd> ideal_resource))"
"1_1": shows "\<forall>y. (y, xs) \<in> measure length \<longrightarrow> y \<noteq> [] \<longrightarrow> factors_of_factor_tree (create_factor_tree_simple y) = mset y"
split3_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and p :: "'b \<times> 'c \<times> 'd" shows "split3 f p \<equiv> case p of (a, b, c) \<Rightarrow> f a b c"
ide_cod: fixes f :: "'c" assumes "Cop.arr f" shows "Cop.ide (Cop.cod f)"
emb_at_trans: fixes ps :: "(dir list \<times> dir) list"   and t :: "('c, 'd) tm"   and u :: "('c, 'd) tm"   and qs :: "(dir list \<times> dir) list"   and s :: "('c, 'd) tm" assumes "emb_at ps t = u"   and "emb_at qs u = s" shows "emb_at (qs @ ps) t = s"
simps_2: fixes a :: "'i"   and f :: "'i \<Rightarrow> 'g +\<^sub>\<bottom> 'h" shows "(try sum_bot.Left a catch f) = f a"
scaleR: fixes r :: "real"   and b :: "'d" shows "g (r *\<^sub>R b) = r *\<^sub>R g b"
summable_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "summable f" shows "summable (\<lambda>n. f (n + k))"
ASSERT_refine_right: fixes \<Phi> :: "bool"   and S :: "'b nres"   and R :: "('b \<times> 'c) set"   and S' :: "'c nres" assumes "\<Phi> \<Longrightarrow> S \<le> \<Down> R S'" shows "S \<le> \<Down> R (ASSERT \<Phi> \<bind> (\<lambda>_. S'))"
sub_block_repl: fixes j :: "'b"   and block :: "('c, 'b) fm list \<times> 'b"   and i :: "'b" assumes "j \<notin> block_nominals block" shows "i \<notin> block_nominals (sub_block (id(i := j, j := i)) block)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst: fixes s :: "'d strand_label \<times> ('e, 'f) stateful_strand_step"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (s \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p s \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>"
FinalAllow_approximating_in_doubt_allow: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and p :: "'d" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow" shows "(\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow"
cod_simps_1: shows "\<mu>\<nu>\<pi>.cod.apex = \<mu>\<nu>\<pi>.apex"
Union_eq: fixes A :: "'a set set" shows "\<Union> A = {x. \<exists>B\<in>A. x \<in> B}"
INT_subset_iff: fixes B :: "'a set"   and A :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "(B \<subseteq> \<Inter> (A ` I)) = (\<forall>i\<in>I. B \<subseteq> A i)"
insert_before_writes: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and child :: "(_) node_ptr"   and ref :: "(_) node_ptr option"   and h' :: "(_) heap" shows "writes (insert_before_locs ptr |h \<turnstile> get_parent child|\<^sub>r |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r |h \<turnstile> get_owner_document ptr|\<^sub>r) (insert_before ptr child ref) h h'"
domo_def: fixes S :: "'a Abs_State.st option" shows "domo S = (case S of None \<Rightarrow> {} | Some S' \<Rightarrow> set (Abs_State.dom S'))"
B: shows "x \<squnion> z = y \<squnion> z"
MkIde_Cod': fixes a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.ide a" shows "Cop_S.MkIde (Cop_S.Cod a) = a"
ifp_def: fixes a :: "'a"   and b :: "'a" shows "ifp a b \<equiv> sp_spec_subj_subj a b \<or> sp_spec_subj_subj b a \<or> (\<exists>c y. sp_spec_subj_subj a c \<and> sp_spec_subj_obj c (PAGE y) WRITE \<and> sp_spec_subj_obj b (PAGE y) READ) \<or> a = b"
lt_minus_lessE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p \<prec>\<^sub>t lt (p - q)" shows "lt p \<prec>\<^sub>t lt q"
steps: shows "\<Gamma>\<turnstile> (r, s) \<rightarrow>\<^sup>* (r', s')"
cInput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and Tvec :: "'a list"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "set xvec \<subseteq> supp N"   and "xvec \<sharp>* Tvec"   and "length xvec = length Tvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K" shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<longmapsto> K\<lparr>N[xvec::=Tvec]\<rparr> \<prec> P[xvec::=Tvec]"
strand_sem_subst_1: fixes \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and M :: "('fun, 'var) Term.term set"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"   and "\<lbrakk>M; S\<rbrakk>\<^sub>c (\<delta> \<circ>\<^sub>s \<theta>)" shows "\<lbrakk>M; S \<cdot>\<^sub>s\<^sub>t \<delta>\<rbrakk>\<^sub>c \<theta>"
unit_factor_dvd: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero" shows "dvd.dvd times (unit_factor a) b"
Bii: fixes i :: "nat" assumes "i < n" shows "B $$ (i, i) \<in> \<real>"
smult_distrib_right: fixes r :: "'r"   and s :: "'r"   and m :: "'m" assumes "r \<in> R"   and "s \<in> R"   and "m \<in> M" shows "(r + s) \<cdot> m = r \<cdot> m + s \<cdot> m"
ordinal_number_ord: fixes X :: "'c Order set"   and Y :: "'c Order set" assumes "X \<in> carrier ODnods"   and "Y \<in> carrier ODnods" shows "X \<sqsubset> Y \<or> X = Y \<or> Y \<sqsubset> X"
singleton_conv2: fixes a :: "'e" shows "{x. a = x} = {a}"
axioms_1: fixes \<Omega> :: "'b set"   and M :: "'b set set" assumes "ring_of_sets \<Omega> M" shows "semiring_of_sets \<Omega> M"
set_quicksort: fixes xs :: "'c list" shows "set (quicksort xs) = set xs"
tuple_in_hom: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and h :: "('a, 'a set) Maps.arr"   and k :: "('a, 'a set) Maps.arr" assumes "Maps.commutative_square f g h k" shows "Maps.in_hom (Maps.tuple h f g k) (Maps.dom h) (Maps.pbdom f g)"
rcsTr1: fixes H :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "x \<in> H \<bullet> a"   and "H \<bullet> a = H \<bullet> b" shows "x \<in> H \<bullet> b"
lead_coeff_pCons_2: fixes p :: "'b poly"   and a :: "'b" assumes "p = 0" shows "lead_coeff (pCons a p) = a"
BExprSound: fixes e :: "BExpr"   and tp :: "TP" assumes "(e, tp) \<in> VS_Bexpr" shows "SemBExpr e tp"
foldr_map: fixes g :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and f :: "'i \<Rightarrow> 'h"   and xs :: "'i list"   and a :: "'g" shows "foldr g (map f xs) a = foldr (g \<circ> f) xs a"
monad_reader_writerT': assumes "monad_reader Monad_Overloading.return Monad_Overloading.bind ask" shows "monad_reader Monad_Overloading.return Monad_Overloading.bind ask"
join_above2: fixes x :: "'a"   and y :: "'a" assumes "compatible x y" shows "y \<sqsubseteq> x \<squnion> y"
prems_4: shows "((\<mu> R\<^sub>r \<eta> \<in> old (fst x) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst x) \<union> new (fst x) \<or> \<eta> \<in> old (fst x) \<union> new (fst x) \<and> \<mu> R\<^sub>r \<eta> \<in> next (fst x)) \<and> (\<forall>nd\<in>snd x. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst x) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old (fst x) \<union> new (fst x))) \<and> (\<forall>nd\<in>snd x. \<forall>nd'\<in>snd x. \<mu> R\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd \<or> \<eta> \<in> old nd \<and> \<mu> R\<^sub>r \<eta> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)"
snd_snd_Gauss_Jordan_column_k_PA_eq: fixes P :: "(('b, 'd) vec, 'd) vec"   and i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "snd (snd (Gauss_Jordan_column_k_PA (P, i, A) k)) = snd (Gauss_Jordan_column_k (i, A) k)"
smalloD_tendsto: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> (0::'c)) F"
lower_interval_of: fixes a :: "'b" shows "lower (interval_of a) = a"
comp_eq_dest: fixes a :: "'h \<Rightarrow> 'g"   and b :: "'f \<Rightarrow> 'h"   and c :: "'i \<Rightarrow> 'g"   and d :: "'f \<Rightarrow> 'i"   and v :: "'f" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
rel_fun_conversep: fixes A :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "A\<inverse>\<inverse> ===> B\<inverse>\<inverse> = (A ===> B)\<inverse>\<inverse>"
isOK_E: fixes m :: "'a + 'b" assumes "isOK m" obtains x :: "'b" where "m = Inr x"
rev_rev_ident: fixes xs :: "'d list" shows "rev (rev xs) = xs"
CD_on_idem: fixes ds :: "'b set"   and B :: "'a set" shows "CD_on ds (CD_on ds B) = CD_on ds B"
basis_enum_of_vec_span: fixes Y :: "complex vec set" assumes "length canonical_basis = n"   and "Y \<subseteq> carrier_vec n" shows "basis_enum_of_vec ` local.span Y = cspan (basis_enum_of_vec ` Y)"
less_iff_sym_greater: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and b :: "'b"   and a :: "'b" assumes "comparator cmp" shows "(cmp b a = Less) = (cmp a b = Greater)"
preserves_ide: fixes a :: "'j" assumes "J.ide a" shows "C.ide (D a)"
coeff_a_inv: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "UP_ring R"   and "p \<in> carrier (UP R)" shows "coeff (UP R) (\<ominus>\<^bsub>UP R\<^esub> p) n = \<ominus>\<^bsub>R\<^esub> coeff (UP R) p n"
hyps_2: assumes "\<And>x a\<^sub>1 a\<^sub>2. \<lbrakk>x \<in> S; f x a\<^sub>1 a\<^sub>2\<rbrakk> \<Longrightarrow> r a\<^sub>1 a\<^sub>2" shows "r a\<^sub>0 a\<^sub>1"
set_mod_maps_def: fixes empty :: "'a"   and up :: "'b \<Rightarrow> 'c list \<Rightarrow> 'a \<Rightarrow> 'a"   and map_of :: "'a \<Rightarrow> 'b \<Rightarrow> 'c list"   and M :: "'a \<Rightarrow> bool"   and qle :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and subsumed :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and I :: "'c \<Rightarrow> bool"   and key :: "'c \<Rightarrow> 'b" shows "set_mod_maps empty up map_of M qle subsumed I key \<equiv> maps empty up map_of M \<and> quasi_order qle \<and> set_mod_maps_axioms qle subsumed I key"
conforms_deallocL_aux: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option"   and vn :: "lname"   and T :: "ty" assumes "(x, s)\<Colon>\<preceq>(G, L(vn \<mapsto> T))"   and "L vn = None" shows "G,s\<turnstile>locals s[\<sim>\<Colon>\<preceq>]L"
type_wf_defs: fixes h :: "(_) heap" shows "CharacterDataClass.type_wf h = (ElementClass.type_wf h \<and> (\<forall>character_data_ptr\<in>fset (character_data_ptr_kinds h). get\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr h \<noteq> None))"
complex_cnj_divide: fixes x :: "complex"   and y :: "complex" shows "cnj (x / y) = cnj x / cnj y"
assms_2: shows "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs"
le_listD: fixes xs :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and p :: "nat" assumes "xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys"   and "p < length xs" shows "xs ! p \<sqsubseteq>\<^bsub>r\<^esub> ys ! p"
modusponens: fixes s :: "thm"   and s' :: "thm" assumes "\<turnstile> concl s"   and "\<turnstile> concl s'" shows "\<turnstile> concl (modusponens s s')"
dim_gen_eigenspace_def: fixes A :: "'c mat"   and ev :: "'c"   and k :: "nat" shows "dim_gen_eigenspace A ev k = kernel_dim (char_matrix A ev ^\<^sub>m k)"
l2_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and gny :: "msg" shows "l2_step3 Ra A B gny \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xgnx} \<and> guessed_frame Ra xgny = Some gny \<and> guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx))) \<and> Auth B A \<langle>Number 0, gny, Exp Gen (NonceF (Ra $ nx))\<rangle> \<in> chan s \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xgnx, xgny, xsk, xEnd}), chan := {Auth A B \<langle>Number 1, Exp Gen (NonceF (Ra $ nx)), gny\<rangle>} \<union> chan s, secret := {x. x = Exp gny (NonceF (Ra $ nx)) \<and> Ra = test} \<union> secret s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Commit A B (Exp gny (NonceF (Ra $ nx)))) else signalsInit s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Running A B (Exp gny (NonceF (Ra $ nx)))) else signalsResp s\<rparr>}"
cfe: assumes "f \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "f O e = f"
equivalentFormulaeIffEntailEachOther: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list" shows "equivalentFormulae formula1 formula2 = (formulaEntailsFormula formula1 formula2 \<and> formulaEntailsFormula formula2 formula1)"
simplify_forest_components_f: fixes p :: "'a"   and e :: "'a"   and f :: "'a" assumes "regular p"   and "regular e"   and "injective (f \<sqinter> - e\<^sup>T \<sqinter> - p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e)"   and "injective f" shows "forest_components (f \<sqinter> - e\<^sup>T \<sqinter> - p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e) = (f \<squnion> f\<^sup>T \<squnion> e \<squnion> e\<^sup>T)\<^sup>\<star>"
excess_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and v :: "nat" shows "excess f v \<equiv> sum f (incoming v) - sum f (outgoing v)"
g_def: fixes x :: "'a"   and y :: "'b" shows "g x y = \<langle>x, G y\<rangle>"
cases: fixes \<I> :: "('c, 'd) \<I>"   and a :: "('c, 'd) resource" assumes "lossless_resource \<I> a" obtains res :: "('c, 'd) resource" where "a = res" and "\<And>a. a \<in> outs_\<I> \<I> \<Longrightarrow> lossless_spmf (run_resource res a)" and "\<And>a b res'. \<lbrakk>a \<in> outs_\<I> \<I>; (b, res') \<in> set_spmf (run_resource res a)\<rbrakk> \<Longrightarrow> lossless_resource \<I> res'"
upto_enum_step_def: fixes a :: "'b word"   and b :: "'b word"   and c :: "'b word" shows "[a , b .e. c] \<equiv> if c < a then [] else map (\<lambda>x. a + x * (b - a)) [0 .e. (c - a) div (b - a)]"
lnull_lunstream: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lnull (lunstream g s) = (case g s of Done \<Rightarrow> True | Skip s' \<Rightarrow> lnull (lunstream g s') | Yield x xa \<Rightarrow> False)"
load_word_mem_8_unchanged: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addra :: "word32"   and addr :: "word36"   and val :: "word8" assumes "low_equal s1 s2 \<and> load_word_mem s1 addra 8 = load_word_mem s2 addra 8" shows "load_word_mem (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) addra 8 = load_word_mem (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) addra 8"