hcju/isabelleps huggingface dataset:
sum4sq_int_nat_eq: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" shows "int (a\<^sup>2 + b\<^sup>2 + c\<^sup>2 + d\<^sup>2) = sum4sq_int (int a, int b, int c, int d)"
weight_spmf'_def: fixes p :: "'g spmf" shows "weight_spmf' p = weight_spmf p"
union_code: fixes xs :: "'k list"   and ys :: "'k list" shows "mset xs + mset ys = mset (xs @ ys)"
ord_term_strict_canc: fixes t :: "'a"   and v :: "'t"   and w :: "'t" assumes "t \<oplus> v \<prec>\<^sub>t t \<oplus> w" shows "v \<prec>\<^sub>t w"
in_spanE: fixes v :: "'a vec"   and W :: "'a vec set" assumes "v \<in> local.span W" shows "\<exists>a A. v = lincomb a A \<and> finite A \<and> A \<subseteq> W"
llist_all2_LNil2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist" shows "llist_all2 P xs LNil = (xs = LNil)"
finsum_infinite: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "infinite A" shows "sum f A = (0::'a)"
bl_word_and: fixes v :: "'b word"   and w :: "'b word" shows "to_bl (v AND w) = map2 (\<and>) (to_bl v) (to_bl w)"
Suc_1: fixes i :: "nat" assumes "k = j - i"   and "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i < j"   and "j < length l"   and "x \<in> rvars (\<T> (l ! i))"   and "x \<in> lvars (\<T> (l ! j))" shows "\<exists>k\<ge>i. k < j \<and> k + 1 < length l \<and> x \<in> rvars (\<T> (l ! k)) \<and> x \<in> lvars (\<T> (l ! (k + 1)))"
split: fixes P :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and nondetT :: "('f, 'e) nondetT" shows "P (case nondetT of NondetT x \<Rightarrow> f x) = (\<forall>x. nondetT = NondetT x \<longrightarrow> P (f x))"
nat_of_agent_def: shows "nat_of_agent \<equiv> case_agent (curry prod_encode 0) (curry prod_encode 1) (curry prod_encode 2) (curry prod_encode 3) (prod_encode (4, 0))"
assms_1: shows "reparam \<gamma>1 \<gamma>2"
iIN_add_neg: fixes n :: "nat"   and d :: "nat"   and k :: "nat" shows "[n\<dots>,d] \<oplus>- k = (if k \<le> n then [n - k\<dots>,d] else if k \<le> n + d then [\<dots>n + d - k] else {})"
bound_def: shows "bound = Suc (Max (snd ` as))"
induct: fixes P :: "'b hp option \<Rightarrow> bool"   and a0 :: "'b hp option" assumes "P None"   and "\<And>x hs. P (Some (Hp x hs))" shows "P a0"
union_inter_neutral: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "\<forall>x\<in>A \<inter> B. g x = (0::'d)" shows "sum g (A \<union> B) = sum g A + sum g B"
int_pow_distrib: fixes x :: "'a"   and y :: "'a"   and i :: "int" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes> y) [^] i = x [^] i \<otimes> y [^] i"
the_equality: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(THE x. P x) = a"
has_field_derivative_Arccos: fixes z :: "complex" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "(Arccos has_field_derivative - inverse (sin (Arccos z))) (at z)"
add_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b"   and "less_eq c d" shows "less_eq (plus a c) (plus b d)"
counit_in_hom_2: shows "\<guillemotleft>\<epsilon>' : f \<star> g' \<Rightarrow> trg f\<guillemotright>"
mset_le_add_mset: fixes x :: "'a"   and B :: "'a multiset"   and C :: "'a multiset" assumes "add_mset x B \<subseteq># C" shows "{#x#} \<subseteq># C \<and> B \<subseteq># C"
zero_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<equiv> Matrix.mat nr nc (\<lambda>ij. 0::'a)"
B2b: shows "\<^bold>\<forall>x. dom cod x \<cong> cod x"
prems_2: shows "b \<in>\<^sub>\<circ> mk_doubleton_image X Y"
trms\<^sub>s\<^sub>s\<^sub>t_subst: fixes S :: "('b, 'a) stateful_strand_step list"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "bvars\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}" shows "trms\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>) = trms\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
2: shows "real (card G) - real (card (pseudoAllocation aa \<inter> pseudoAllocation a)) = real (sum (pseudoAllocation a <| (N \<times> finestpart G)) (pseudoAllocation a)) - real (sum (summedBidVector (pseudoAllocation a <| (N \<times> finestpart G)) N G) aa)"
Assum: fixes a :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "a \<in> set G" shows "G \<turnstile> a"
vec_to_iarray_nth: fixes i :: "nat"   and A :: "('b, 'a) vec" assumes "i < CARD('a)" shows "vec_to_iarray A !! i = A $ mod_type_class.from_nat i"
assms_2: shows "vsv (T\<lparr>ArrVal\<rparr>)"
COMB_def: fixes h :: "nat list" shows "COMB h = (COMB_init h, COMB_step)"
sees_methods_Object: fixes P :: "'b prog"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "Mm = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms" shows "P \<turnstile> Object sees_methods Mm"
bex_one_point2: fixes A :: "'b set"   and a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>x\<in>A. a = x \<and> P x) = (a \<in> A \<and> P a)"
diamond_y_def: shows "diamond_y \<equiv> \<lambda>t. d / 2 - \<bar>t\<bar>"
Myhill_Nerode: fixes A :: "'a list set" shows "regular A = finite (UNIV // \<approx>A)"
ortho_involution: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.orthocomplemented_lattice minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
Gramian_determinant_1: fixes k :: "nat" assumes "k \<le> length (map of_int_hom.vec_hom fs')" shows "gs2.d k = (\<Prod>j<k. \<parallel>gs2.gso j\<parallel>\<^sup>2)"
normal_subgroup_set_mult_closed: fixes M :: "'a set"   and N :: "'a set" assumes "M \<lhd> G"   and "N \<lhd> G" shows "M <#> N \<lhd> G"
bliftM2_strict1: fixes f :: "'c \<rightarrow> 'a \<rightarrow> 'b" shows "bliftM2 f\<cdot>\<bottom> = \<bottom>"
meet_join_distrib_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "(y \<squnion> z) \<sqinter> x = y \<sqinter> x \<squnion> z \<sqinter> x"
distPath_def: fixes h :: "'b \<Rightarrow> 'b ref"   and x :: "'b ref"   and as :: "'b list"   and y :: "'b ref" shows "distPath h x as y = (Path h x as y \<and> distinct as)"
set: shows "Q \<in> sets (M1 \<Otimes>\<^sub>M M)"
disj: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm"   and q :: "fm" shows "Ifm bbs bs (Cooper.disj p q) = Ifm bbs bs (Or p q)"
LAssReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>V:=e,s\<rangle> \<rightarrow>* \<langle>V:=e',s'\<rangle>"
wfp: fixes AP' :: "'aprop set"   and \<pi> :: "('state \<times> 'aprop set) stream" shows "wfp AP' \<pi> = (\<forall>i. fst (\<pi> !! i) \<in> S \<and> snd (\<pi> !! i) \<subseteq> AP' \<and> snd (\<pi> !! i) \<inter> AP = L (fst (\<pi> !! i)) \<and> fst (\<pi> !! Suc i) \<in> \<delta> (fst (\<pi> !! i)))"
cba_bres2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> - y \<squnion> z) = (x \<le> proto_near_quantale.bres Sup (\<le>) (\<sqinter>) y z)"
intro: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "comm_semiring_ow U plus times"   and "cancel_comm_monoid_add_ow U plus minus zero" shows "comm_semiring_0_cancel_ow U plus minus zero times"
sum_val_process: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market" assumes "portfolio pf1"   and "portfolio pf2" shows "\<forall>n w. val_process Mkt (qty_sum pf1 pf2) n w = val_process Mkt pf1 n w + val_process Mkt pf2 n w"
hyps_3: fixes b :: "binding list" assumes "\<turnstile> b ok"   and "T\<^sub>1 closed_in b" shows "b\<turnstile>T\<^sub>1<:T\<^sub>1"
br_if_false: fixes n :: "i32"   and i :: "nat" assumes "int_eq n 0" shows "\<lparr>[$C ConstInt32 n, $Br_if i]\<rparr> \<leadsto> \<lparr>[]\<rparr>"
lcm_least: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c"   and "b dvd c" shows "lcm a b dvd c"
cases: fixes x :: "(val \<times> val) list" obtains "x = []"   | v1 :: "val" and v2 :: "val" and t :: "(val \<times> val) list" where "x = (v1, v2) # t"
parallel_converter2_id_id: fixes \<I>1 :: "('y, 'aa) \<I>"   and \<I>2 :: "('z, 'ab) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2,\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<turnstile>\<^sub>C 1\<^sub>C |\<^sub>= 1\<^sub>C \<sim> 1\<^sub>C"
kas: shows "init \<in> {[x, y], [y, x]}"
res': fixes R_ab :: "'b \<times> 'd \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool" assumes "\<And>k v. m k = Some v \<Longrightarrow> set_iterator_genord (it_b v) (S_b v) (\<lambda>_ _. True)"   and "\<And>k v u k' v' u'. \<lbrakk>m k = Some v; u \<in> S_b v; m k' = Some v'; u' \<in> S_b v'; if k = k' then True else True\<rbrakk> \<Longrightarrow> R_ab (k, u) (k', u')" shows "set_iterator_genord (map_iterator_product it_a it_b) {(k, e). \<exists>v. m k = Some v \<and> e \<in> S_b v} R_ab"
reachable_target: fixes q :: "'d"   and execute :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and enabled :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'e list" assumes "q \<in> transition_system.reachable execute enabled p"   and "transition_system.path execute enabled r q" shows "fold execute r q \<in> transition_system.reachable execute enabled p"
Invariants_func_charac: fixes f :: "'a \<Rightarrow> 'b"   and N :: "'b measure" assumes "random_variable N f"   and "\<And>x. x \<in> space M \<Longrightarrow> f (T x) = f x" shows "f \<in> Invariants \<rightarrow>\<^sub>M N"
composePermInduct: fixes xvec :: "name list"   and yvec :: "name list"   and P :: "name list \<Rightarrow> name list \<Rightarrow> bool" assumes "length xvec = length yvec"   and "P [] []"   and "\<And>x xvec y yvec. \<lbrakk>length xvec = length yvec; P xvec yvec\<rbrakk> \<Longrightarrow> P (x # xvec) (y # yvec)" shows "P xvec yvec"
sinits_sscanl: fixes f :: "'s \<rightarrow> 'r \<rightarrow> 's"   and z :: "'s" shows "smap\<cdot>(sfoldl\<cdot>f\<cdot>z) oo sinits = sscanl\<cdot>f\<cdot>z"
inverse_unit: shows "comm.additive.inverse \<zero> = \<zero>"
Lq_def2: fixes z :: "'var" assumes "z \<in> var"   and "z \<noteq> yy"   and "z \<noteq> xx" shows "Lq = exi z (eql (Var yy) (pls (Var z) (Var xx)))"
wd_def: shows "wd TYPE('c) = (length ode_e = DIM('c))"
RedT_preserves_wset_thread_ok: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ttas :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -\<triangleright>ttas\<rightarrow>* s'"   and "wset_thread_ok (wset s) (thr s)" shows "wset_thread_ok (wset s') (thr s')"
eq_iff_diff_eq_0: fixes minus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and plus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and zero :: "'i"   and uminus :: "'i \<Rightarrow> 'i"   and a :: "'i"   and b :: "'i" assumes "class.group_add minus plus zero uminus" shows "(a = b) = (minus a b = zero)"
last_ConsL: fixes xs :: "'a list"   and x :: "'a" assumes "xs = []" shows "List.last (x # xs) = x"
conductor_le_iff: fixes a :: "nat" shows "(conductor \<le> a) = (\<exists>d\<le>a. induced_modulus d)"
mono: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset"   and d :: "'b multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<inter># b \<subseteq># c \<inter># d"
pgalleryD_distinct: fixes xs :: "'b set list" assumes "codomain.pgallery xs" shows "distinct xs"
elims_3: fixes x :: "assn_raw"   and xa :: "pheap" assumes "\<not> aseval x xa" obtains f :: "pheap \<Rightarrow> bool" and h :: "pheap" where "x = Assn f" and "xa = h" and "\<not> f h"
induct: fixes prog :: "cmd"   and procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and x1 :: "char list \<times> label"   and x2 :: "(char list, val, char list \<times> label, char list) edge_kind"   and x3 :: "char list \<times> label"   and P :: "char list \<times> label \<Rightarrow> (char list, val, char list \<times> label, char list) edge_kind \<Rightarrow> char list \<times> label \<Rightarrow> bool" assumes "prog,procs \<turnstile> x1 -x2\<rightarrow> x3"   and "\<And>n et n'. prog \<turnstile> n -IEdge et\<rightarrow>\<^sub>p n' \<Longrightarrow> P (Main, n) et (Main, n')"   and "\<And>p ins outs c n et n' ps. \<lbrakk>(p, ins, outs, c) \<in> set procs; c \<turnstile> n -IEdge et\<rightarrow>\<^sub>p n'; containsCall procs prog ps p\<rbrakk> \<Longrightarrow> P (p, n) et (p, n')"   and "\<And>l p es rets n' ins outs c. \<lbrakk>prog \<turnstile> Label l -CEdge (p, es, rets)\<rightarrow>\<^sub>p n'; (p, ins, outs, c) \<in> set procs\<rbrakk> \<Longrightarrow> P (Main, Label l) (\<lambda>s. True:(Main, n')\<hookrightarrow>\<^bsub>p\<^esub>map interpret es) (p, Entry)"   and "\<And>p ins outs c l p' es' rets' l' ins' outs' c' ps. \<lbrakk>(p, ins, outs, c) \<in> set procs; c \<turnstile> Label l -CEdge (p', es', rets')\<rightarrow>\<^sub>p Label l'; (p', ins', outs', c') \<in> set procs; containsCall procs prog ps p\<rbrakk> \<Longrightarrow> P (p, Label l) (\<lambda>s. True:(p, Label l')\<hookrightarrow>\<^bsub>p'\<^esub>map interpret es') (p', Entry)"   and "\<And>l p es rets l' ins outs c. \<lbrakk>prog \<turnstile> Label l -CEdge (p, es, rets)\<rightarrow>\<^sub>p Label l'; (p, ins, outs, c) \<in> set procs\<rbrakk> \<Longrightarrow> P (p, Exit) (\<lambda>cf. snd cf = (Main, Label l')\<hookleftarrow>\<^bsub>p\<^esub>\<lambda>cf cf'. cf'(rets [:=] map cf outs)) (Main, Label l')"   and "\<And>p ins outs c l p' es' rets' l' ins' outs' c' ps. \<lbrakk>(p, ins, outs, c) \<in> set procs; c \<turnstile> Label l -CEdge (p', es', rets')\<rightarrow>\<^sub>p Label l'; (p', ins', outs', c') \<in> set procs; containsCall procs prog ps p\<rbrakk> \<Longrightarrow> P (p', Exit) (\<lambda>cf. snd cf = (p, Label l')\<hookleftarrow>\<^bsub>p'\<^esub>\<lambda>cf cf'. cf'(rets' [:=] map cf outs')) (p, Label l')"   and "\<And>n p es rets n'. prog \<turnstile> n -CEdge (p, es, rets)\<rightarrow>\<^sub>p n' \<Longrightarrow> P (Main, n) (\<lambda>s. False)\<^sub>\<surd> (Main, n')"   and "\<And>p ins outs c n p' es' rets' n' ps. \<lbrakk>(p, ins, outs, c) \<in> set procs; c \<turnstile> n -CEdge (p', es', rets')\<rightarrow>\<^sub>p n'; containsCall procs prog ps p\<rbrakk> \<Longrightarrow> P (p, n) (\<lambda>s. False)\<^sub>\<surd> (p, n')" shows "P x1 x2 x3"
assms_2: shows "thr s t = \<lfloor>(x, ln)\<rfloor>"
lm097: fixes y :: "'a"   and f :: "('b \<times> 'a) set"   and x :: "'b" assumes "y \<in> f `` {x}"   and "runiq f" shows "f ,, x = y"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b \<cdot> (a \<cdot> c) = a \<cdot> (b \<cdot> c)"
swapVal_involutive: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'g" shows "val ^[x \<and> y]_zs ^[x \<and> y]_zs = val"
assms_2: shows "round_robin_invar \<sigma> (dom (thr s))"
converse_rtrancl_into_rtrancl: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and c :: "'d" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>*"
IH: assumes "wf\<^sub>e\<^sub>s\<^sub>t V A" shows "wf\<^sub>e\<^sub>s\<^sub>t V (A @ D)"
tp_minus_keys': shows "tp - len + 1 \<in> Mapping.keys a2_map"
dom\<sigma>: shows "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems j"
x: shows "x = fst (split_n_m_x (dim_col A\<^sup>T) (dim_row A\<^sup>T) X)"
inf_exists: fixes A :: "'a set" assumes "A \<subseteq> carrier L"   and "A \<noteq> {}" shows "\<exists>i. is_glb L i A"
sigma_finite: fixes M :: "'b measure" assumes "sigma_finite_measure M" obtains A :: "nat \<Rightarrow> 'b set" where "range A \<subseteq> sets M" and "\<Union> (range A) = space M" and "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
trunc_simps_2: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "f \<ominus>\<^bsub>UP R\<^esub> Cring_Poly.truncate R f = monom (UP R) (f (deg R f)) (deg R f)"
minus_of_int: fixes x :: "real"   and n :: "int" shows "perzeta (x - real_of_int n) = perzeta x"
intro: fixes G_invar :: "'a \<Rightarrow> bool"   and g :: "'a" assumes "G_invar g" shows "Prim_Impl_axioms G_invar g"
cryptsp_crypts_eq: fixes H :: "msg set" shows "cryptsp (\<lambda>x. x \<in> H) = (\<lambda>x. x \<in> crypts H)"
pmf_eqI: fixes M :: "'h pmf"   and N :: "'h pmf" assumes "\<And>i. pmf M i = pmf N i" shows "M = N"
vunion_vinsert_commutativity_right: fixes A :: "V"   and a :: "V"   and B :: "V" shows "A \<union>\<^sub>\<circ> vinsert a B = vinsert a A \<union>\<^sub>\<circ> B"
gen_dfs_pre_initial: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and V0 :: "'a set"   and it :: "'a set"   and V :: "'a set"   and v0 :: "'a" assumes "gen_dfs_outer E U V0 it V False"   and "v0 \<in> U - V" shows "gen_dfs_pre E U {} V v0"
last_appendR: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys \<noteq> []" shows "last (xs @ ys) = last ys"
union: fixes M :: "'b set set"   and B :: "'b set set" assumes "\<And>x. x \<in> M \<Longrightarrow> x \<in> topo B" shows "\<Union> M \<in> topo B"
c: fixes A :: "'b matrix"   and n :: "nat"   and B :: "'c matrix"   and fadd :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and fmul :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "ncols A \<le> n"   and "nrows B \<le> n"   and "fadd (0::'d) (0::'d) = (0::'d)"   and "fmul (0::'b) (0::'c) = (0::'d)" shows "mult_matrix fmul fadd A B = mult_matrix_n n fmul fadd A B"
TS: fixes x :: "nat" assumes "x \<in> T - S" shows "n \<le> x"
toprefixmatch_def: fixes m :: "'b word \<times> nat" shows "toprefixmatch m \<equiv> let pm = PrefixMatch (fst m) (snd m) in if pm = PrefixMatch 0 0 then None else Some pm"
affine_hull_halfspace_lt: fixes a :: "'a"   and r :: "real" shows "affine hull {x. a \<bullet> x < r} = (if a = (0::'a) \<and> r \<le> 0 then {} else UNIV)"
openin_Int: fixes U :: "'d topology"   and S :: "'d set"   and T :: "'d set" assumes "openin U S"   and "openin U T" shows "openin U (S \<inter> T)"
n2'_def: shows "n2' = nfa_steps (nfa_of_dfa M) n1 bss"
inj_sur_semi_swap: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_sur z"   and "is_inj x"   and "z \<le> y ; x" shows "x \<le> y\<^sup>T ; z"
member_def: shows "member \<equiv> (rep_dlist ---> id) List.member"
simps_4: shows "cospan_shape.Cod cospan_shape.AT = cospan_shape.TT"
cases: fixes x :: "('b rexp set \<times> 'b rexp set) list \<times> ('b rexp set \<times> 'b rexp set) list" obtains ws :: "('b rexp set \<times> 'b rexp set) list" and ps :: "('b rexp set \<times> 'b rexp set) list" where "x = (ws, ps)"
lim_proc_is_lub1: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "\<forall>u. range S <| u \<longrightarrow> D u \<subseteq> D (lim_proc (range S))"
mangoldt_prime: fixes p :: "nat" assumes "prime p" shows "mangoldt p = of_real (ln (real p))"
iter_hcomp_insert: fixes v :: "'a"   and ws :: "'a fset"   and p :: "'a \<Rightarrow> interface" assumes "v |\<notin>| ws" shows "iter_hcomp (finsert v ws) p = (p v \<otimes> iter_hcomp ws p)"
Throw_normal_steps_end: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "'d"   and c :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate" assumes "\<Gamma> \<turnstile> (Throw, Normal s) \<rightarrow>\<^sup>* (c, s')" shows "c = Throw \<and> s' = Normal s"
determ_def: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "determ t \<equiv> Transformers.additive t \<and> maximal t"
is_natural: fixes f :: "'a" assumes "arr f" shows "\<r>[cod f] \<cdot> R f = local.map f \<cdot> \<r>[local.dom f]"
cross3_simps_10: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
less_iff_add1_le_zhmset: fixes m :: "zhmultiset"   and n :: "zhmultiset" shows "(m < n) = (m + 1 \<le> n)"
Units_pow_closed: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and d :: "nat" assumes "Group.monoid G"   and "x \<in> Units G" shows "x [^]\<^bsub>G\<^esub> d \<in> Units G"
update_def: fixes a :: "'a array"   and i :: "nat"   and x :: "'a"   and h :: "heap" shows "Array.update a i x h = Array.set a ((Array.get h a)[i := x]) h"
bst_inter: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.bst (local.inter t1 t2)"
echelon_form_condition2: fixes A :: "(('d, 'e) vec, 'f) vec"   and j :: "'f" assumes "echelon_form A" shows "\<forall>i. i < j \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row j A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ j $ n \<noteq> (0::'d))"
simps_1: fixes s :: "'b set" shows "partList4 s [] = []"
coinduct: fixes X :: "('a, 'b, 'c) gpv \<Rightarrow> ('d, 'b, 'c) gpv \<Rightarrow> bool"   and x1 :: "('a, 'b, 'c) gpv"   and x2 :: "('d, 'b, 'c) gpv"   and A :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>gpv gpv'. x1 = gpv \<and> x2 = gpv' \<and> rel_spmf (eq_\<I>_generat A \<I> (\<lambda>uu_ uua_. X uu_ uua_ \<or> eq_\<I>_gpv A \<I> uu_ uua_)) (the_gpv gpv) (the_gpv gpv')" shows "eq_\<I>_gpv A \<I> x1 x2"
assms_2: shows "\<I> \<turnstile>res res1 \<surd>"
connected_component_def: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" shows "connected_component S x y \<equiv> \<exists>T. connected T \<and> T \<subseteq> S \<and> x \<in> T \<and> y \<in> T"
simps_2: fixes a :: "'b routing_rule"   and as :: "'b routing_rule list"   and lo :: "'b wordinterval" shows "routing_port_ranges (a # as) lo = (let rpm = range_prefix_match (routing_match a) lo; m = fst rpm; nm = snd rpm in (routing_oiface a, m) # routing_port_ranges as nm)"
envTrans_def: fixes eact :: "unit"   and aact :: "Agent \<Rightarrow> ChildAct"   and s :: "(Agent, (bool \<times> bool \<times> bool) \<times> ChildAct \<times> ChildAct \<times> ChildAct, bool \<times> bool \<times> bool) BEState" shows "envTrans eact aact s \<equiv> s\<lparr>es := envTransES eact aact (es s)\<rparr>"
assert_assume: fixes p :: "'b Assertion" shows "{\<cdot> p } * [\<cdot> p ] = {\<cdot> p }"
that_2: shows "q \<in> outs_\<I> \<I>"
eq_tid: shows "vpeq (Step.partition tid) s1 s2"
less_le_trans: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" assumes "x \<subset># y"   and "y \<subseteq># z" shows "x \<subset># z"