hcju/isabelleps huggingface dataset:
wf_ruleset_singleton: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and m :: "'b match_expr"   and a :: "action" shows "wf_ruleset \<gamma> p [Rule m a] = (\<not> matches \<gamma> m a p \<or> (\<nexists>chain. a = Call chain) \<and> a \<noteq> Return \<and> (\<nexists>chain. a = Goto chain) \<and> a \<noteq> Unknown)"
list_ex_append: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "list_ex P (xs @ ys) = (list_ex P xs \<or> list_ex P ys)"
indiff_set_def: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "indiff_set S A = (A \<noteq> {} \<and> (\<forall>x\<in>A. \<forall>y\<in>A. S x y))"
blocks_nempty: shows "\<B> \<noteq> {#} \<or> \<B>' \<noteq> {#}"
distinct_13: fixes x8 :: "'a \<Rightarrow> ('a, 'b, 'c) com" shows "Skip \<noteq> DynCom x8"
effect__strips_iii_b: fixes s' :: "'a \<Rightarrow> bool option"   and s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and v :: "'a" assumes "s' = s \<then> op"   and "v \<in> set (delete_effects_of op) \<and> v \<notin> set (add_effects_of op)" shows "s' v = Some False"
conjugate_exponent_ennreal_1_2_PInf_1: shows "conjugate_exponent 1 = \<infinity>"
prime_aprimedivisor: fixes p :: "'a"   and n :: "'a" assumes "prime p"   and "p dvd n" shows "prime (aprimedivisor n)"
open_prod_elim: fixes S :: "('d \<times> 'e) set"   and x :: "'d \<times> 'e" assumes "open S"   and "x \<in> S" obtains A :: "'d set" and B :: "'e set" where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
fun_rel_def: fixes A :: "('c \<times> 'e) set"   and B :: "('d \<times> 'f) set" shows "A \<rightarrow> B \<equiv> {(f, f'). \<forall>(a, a')\<in>A. (f a, f' a') \<in> B}"
collapse_2: fixes strand_step :: "('c, 'd) strand_step" assumes "is_Receive strand_step" shows "receive\<langle>un_Receive strand_step\<rangle>\<^sub>s\<^sub>t = strand_step"
mat2matofpoly_def: fixes A :: "(('c, 'd) vec, 'e) vec" shows "mat2matofpoly A = (\<chi>i j. [:A $ i $ j:])"
cong_dvd_modulus: fixes x :: "'a"   and y :: "'a"   and m :: "'a"   and n :: "'a" assumes "[x = y] (mod m)"   and "n dvd m" shows "[x = y] (mod n)"
pw_abs_inres: fixes R :: "('c \<times> 'b) set"   and M :: "'c nres"   and a :: "'b" shows "inres (\<Up> R M) a = (nofail (\<Up> R M) \<longrightarrow> (\<exists>c. inres M c \<and> (c, a) \<in> R))"
itypeI': fixes c :: "'a" shows "c ::\<^sub>i TYPE('b)"
wtSB_wt: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and x :: "Preliminaries.var" assumes "wtSB \<pi>" shows "wt (\<pi> x)"
p: shows "p = smult c (\<Prod>(a, i)\<in>set pis. a ^ Suc i)"
lrev_final_def: shows "lrev_final \<equiv> \<Lambda> xs. lrev_work_final\<cdot>xs\<cdot>lnil"
antisym: fixes y :: "'a"   and x :: "'a" assumes "y \<sqsubseteq> x"   and "x \<sqsubseteq> y"   and "x \<in> X"   and "y \<in> X" shows "x = y"
zcf_zero_degree_zero: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "deg R f = 0"   and "zcf f = \<zero>" shows "f = \<zero>\<^bsub>P\<^esub>"
simp_imp: fixes P :: "bool"   and Q :: "bool"   and Q' :: "bool"   and PP :: "bool" assumes "P \<Longrightarrow> CNV Q Q'"   and "PP \<longrightarrow> Q'" shows "PP \<longrightarrow> P \<longrightarrow> Q"
to_vec_matrix_matrix_mult: fixes A :: "('c, 'd) sq_matrix"   and B :: "('c, 'd) sq_matrix" shows "to_vec (A * B) = to_vec A ** to_vec B"
K: shows "\<forall>k'\<le>n. \<one> \<le> fw m n k n n k' k'"
Kbase_def: fixes K :: "('c, 'd) Ring_scheme"   and n :: "nat"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set" shows "Kb\<^bsub>K n P\<^esub> = (SOME f. KbaseP K P n f)"
less_asym: fixes x :: "'c"   and y :: "'c"   and P :: "bool" assumes "x < y"   and "\<not> P \<Longrightarrow> y < x" shows "P"
simps_1: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and more :: "'f" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
MVOKBAnotRF: fixes x :: "Vals" assumes "MVOKBA x" shows "x \<noteq> RPCFailure"
cod_dom: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "cod (local.dom f) = local.dom f"
propagate_all_def: fixes c0 :: "('loc, 't) Propagate.configuration" shows "propagate_all c0 = while_option (\<lambda>c. \<exists>loc. c_work c loc \<noteq> {#}\<^sub>z) (\<lambda>c. SOME c'. next_propagate c c') c0"
finY2: fixes X :: "'v set"   and n :: "nat" assumes "X \<subseteq> A_n n" shows "integral\<^sup>N (count_space (Y2 n X)) (weight \<Gamma>) \<noteq> \<top>"
pe_cond2: fixes m :: "nat"   and i :: "nat"   and l :: "nat"   and k :: "nat"   and \<delta>S :: "real" assumes "te m i \<le> te l i"   and "correct k (te k (i + 1))"   and "correct l (te k (i + 1))"   and "correct m (te k (i + 1))"   and "\<bar>IC l i (te l i) - IC m i (te l i)\<bar> \<le> \<delta>S" shows "\<bar>\<theta> k (i + 1) l - \<theta> k (i + 1) m\<bar> \<le> 2 * \<Lambda> + \<delta>S + 2 * \<rho> * (rmax + \<beta>)"
eout_def: fixes sP :: "specID"   and E :: "Expression" shows "eout sP E \<equiv> \<exists>ch. ch \<in> out sP \<and> exprChannel ch E"
takeWhile_replicate: fixes P :: "'a \<Rightarrow> bool"   and n :: "nat"   and x :: "'a" shows "takeWhile P (x \<up> n) = (if P x then x \<up> n else [])"
isAugmentingPath_def: fixes p :: "(nat \<times> nat) list" shows "isAugmentingPath p \<equiv> cf.isSimplePath s p t"
subsumes_accessibleI: fixes r\<^sub>2 :: "'r"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r\<^sub>1 :: "'r" assumes "r\<^sub>2 \<in> RID\<^sub>L (the (s r\<^sub>1)) \<Longrightarrow> LID\<^sub>S (the (s r\<^sub>2)\<^sub>\<sigma>) \<subseteq> doms (the (s r\<^sub>1))" shows "\<A> r\<^sub>1 r\<^sub>2 s"
stable_on_range': fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "stable_on ds X"   and "x \<in> X" shows "x \<in> Field (Pd (Xd x))"
fv\<^sub>s\<^sub>t_subset_vars\<^sub>s\<^sub>t: fixes S :: "('d, 'c) strand_step list" shows "fv\<^sub>s\<^sub>t S \<subseteq> vars\<^sub>s\<^sub>t S"
floor_unique: fixes z :: "int"   and x :: "'a" assumes "of_int z \<le> x"   and "x < of_int z + (1::'a)" shows "\<lfloor>x\<rfloor> = z"
simps_2: fixes lev :: "nat"   and T :: "typ"   and t :: "Core.term" shows "decr lev (Abs T t) = Abs T (decr (lev + 1) t)"
I: fixes X :: "'a set"   and x :: "'a"   and y :: "'a"   and a :: "'a"   and b :: "real" assumes "X \<in> \<G>"   and "x \<in> X"   and "y \<in> X"   and "(a, b) \<in> I" shows "a \<bullet> x \<le> b \<and> a \<bullet> y \<le> b \<or> b \<le> a \<bullet> x \<and> b \<le> a \<bullet> y"
prems_2: shows "ops = xs @ [(oid, Some ref)] @ ys @ [x]"
Worder_ord_isom_mem: fixes S :: "('c, 'd) Order_scheme"   and T :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c" assumes "Worder S"   and "Worder T"   and "ord_isom S T f"   and "a \<in> carrier S" shows "f a \<in> carrier T"
finprod_singleton_set: fixes f :: "'a \<Rightarrow> int vec"   and a :: "'a" assumes "f a \<in> carrier_vec n" shows "finsum V f {a} = f a"
pos_Max_imp_neg_notin: fixes C :: "'b literal multiset"   and A :: "'b" assumes "Max_mset C = Pos A" shows "Neg A \<notin># C"
punit_component_of_term: shows "MPoly_Type_Class.punit.component_of_term = (\<lambda>_. ())"
inverse_divide: fixes a :: "'a"   and b :: "'a" shows "inverse (a / b) = b / a"
nat_term_comp_nat_term_compare: fixes to :: "'b nat_term_order" shows "nat_term_comp (nat_term_compare to)"
keys_vectorize_poly: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "keys (vectorize_poly p) = component_of_term ` keys p"
tendsto_exp: fixes f :: "'e \<Rightarrow> 'f"   and a :: "'f"   and F :: "'e filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. exp (f x)) \<longlongrightarrow> exp a) F"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_diff a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> wt t \<ge>\<^sub>p wt s \<and> head t >\<^sub>h\<^sub>d head s)"
assms: shows "(f (lead_coeff p) = (0::'a)) = (p = 0)"
widen_trans: fixes P :: "'b prog"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "P \<turnstile> S \<le> U"   and "P \<turnstile> U \<le> T" shows "P \<turnstile> S \<le> T"
module_hom_zero: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "module_pair s1 s2" shows "module_hom s1 s2 (\<lambda>x. 0::'c)"
imap_def: fixes history :: "nat \<Rightarrow> ('c \<times> ('c, 'd) operation) event list" shows "imap history \<equiv> network_with_constrained_ops history interpret_op (\<lambda>x. ({}, {})) valid_behaviours"
rel_\<I>I: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I>1 :: "('a, 'c) \<I>"   and \<I>2 :: "('b, 'd) \<I>"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_set C (outs_\<I> \<I>1) (outs_\<I> \<I>2)"   and "\<And>x y. C x y \<Longrightarrow> rel_set R (responses_\<I> \<I>1 x) (responses_\<I> \<I>2 y)" shows "rel_\<I> C R \<I>1 \<I>2"
map_add_find_right: fixes n :: "'e \<Rightarrow> 'd option"   and k :: "'e"   and xx :: "'d"   and m :: "'e \<Rightarrow> 'd option" assumes "n k = Some xx" shows "(m ++ n) k = Some xx"
map_cong: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "map f xs = map g ys"
fds_nth_norm: fixes f :: "'a fds"   and n :: "nat" shows "fds_nth (fds_norm f) n = norm (fds_nth f n)"
const_subterm_obtain: fixes t :: "('b, 'a) Term.term" assumes "fv t = {}" obtains c :: "'b" where "Fun c [] \<sqsubseteq> t"
l_get_owner_document_wf_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_owner_document_wf heap_is_wellformed type_wf known_ptr known_ptrs get_disconnected_nodes get_owner_document get_parent \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_get_owner_document_wf_axioms heap_is_wellformed type_wf known_ptrs get_disconnected_nodes get_owner_document get_parent"
less_asym: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat"   and P :: "bool" assumes "x \<prec> y"   and "\<not> P \<Longrightarrow> y \<prec> x" shows "P"
I1: shows "blinfun_scaleR_left (inverse (blinfun_apply (Ds (flow0 x (return_time {x \<in> S. s x = 0} x))) (f (flow0 x (return_time {x \<in> S. s x = 0} x))))) o\<^sub>L (Ds (flow0 (fst (x, return_time {x \<in> S. s x = 0} x)) (snd (x, return_time {x \<in> S. s x = 0} x))) o\<^sub>L flowderiv (fst (x, return_time {x \<in> S. s x = 0} x)) (snd (x, return_time {x \<in> S. s x = 0} x)) o\<^sub>L embed2_blinfun) = 1\<^sub>L"
invertible_matrices_mult_rank: fixes P :: "(('a, 'b) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'c) vec, 'b) vec" assumes "invertible P"   and "invertible Q" shows "rank (P ** A ** Q) = rank A"
hom_one: shows "inv hom (1::'b) = (1::'a)"
qAFresh_qSwap_id: fixes zs :: "'d"   and z1 :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and z2 :: "'e" assumes "qAFresh zs z1 X"   and "qAFresh zs z2 X" shows "X #[[z1 \<and> z2]]_zs = X"
exhaust: fixes y :: "Vertices" obtains "y = u0"   | "y = v0"   | "y = w0"
square_free_factorization_def: fixes p :: "'c poly"   and cbs :: "'c \<times> ('c poly \<times> nat) list" shows "square_free_factorization p cbs \<equiv> case cbs of (c, bs) \<Rightarrow> p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i) \<and> (p = 0 \<longrightarrow> c = (0::'c) \<and> bs = []) \<and> (\<forall>a i. (a, i) \<in> set bs \<longrightarrow> square_free a \<and> 0 < degree a) \<and> (\<forall>a i b j. (a, i) \<in> set bs \<longrightarrow> (b, j) \<in> set bs \<longrightarrow> (a, i) \<noteq> (b, j) \<longrightarrow> coprime a b) \<and> distinct bs"
map_bind_pmf: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'c pmf"   and g :: "'c \<Rightarrow> 'b pmf" shows "map_pmf f (M \<bind> g) = M \<bind> (\<lambda>x. map_pmf f (g x))"
Card_Union: fixes A :: "V set" assumes "\<And>x. x \<in> A \<Longrightarrow> Card x" shows "Card (\<Squnion> A)"
dom_cod: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.dom (hom'.cod f) = hom'.cod f"
homp: shows "homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S (p \<circ> (\<lambda>z. Arg2pi z / (2 * pi))) (\<lambda>x. a)"
reduce_nested_mpoly_sum: fixes p1 :: "'b mpoly mpoly"   and p2 :: "'b mpoly mpoly" shows "reduce_nested_mpoly (p1 + p2) = reduce_nested_mpoly p1 + reduce_nested_mpoly p2"
mdl_def: fixes M :: "('h, 'i, 'j) Module_scheme"   and H :: "'h set" shows "mdl M H = \<lparr>carrier = H, pop = (\<plusminus>\<^bsub>M\<^esub>), mop = mop M, zero = \<zero>\<^bsub>M\<^esub>, sprod = \<lambda>a. restrict ((\<cdot>\<^sub>s\<^bsub>M\<^esub>) a) H\<rparr>"
lookup_make_fmap: fixes k :: "'a"   and ks :: "'a fset"   and v :: "'b" assumes "k \<in> fset ks" shows "lookup [ ks |=> v ] k = v"
TI_trancl: shows "\<forall>(a, b)\<in>(set TI)\<^sup>+. a \<noteq> b \<longrightarrow> (a, b) \<in> set TI"
freshstuff_sema_def: fixes fresh :: "'a set \<Rightarrow> 'a" shows "freshstuff_sema fresh \<equiv> freshstuff fresh"
fold_stores_def: fixes ws :: "('a, 'c, 'd) mem_store_action list" shows "fold_stores ws = fold (\<lambda>w. (\<circ>) (do_store_action w)) ws id"
minus_of_real_eq_of_real_iff: fixes x :: "real"   and y :: "real" shows "(- of_real x = of_real y) = (- x = y)"
HAInitState_def: fixes HA :: "('a, 'b, 'c) hierauto" shows "HAInitState HA \<equiv> InitState (HARoot HA)"
conj_var_subst: fixes x :: "'a \<Longrightarrow> 'b"   and P :: "'b upred"   and v :: "('a, 'b) uexpr" assumes "vwb_lens x" shows "(P \<and> x =\<^sub>u v) = (P\<lbrakk>v/x\<rbrakk> \<and> x =\<^sub>u v)"
neg_intro: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "\<phi> \<pi>l \<Longrightarrow> False" shows "neg \<phi> \<pi>l"
S_def: shows "S = (\<lambda>x. ln (real (nat \<lfloor>sqrt x\<rfloor>)) - ln x / 2)"
comp_ide_self: fixes a :: "'a1" assumes "C1.ide a" shows "a \<cdot>\<^sub>1 a = a"
uniqueness_of_extended_map_to_freeword_hom': fixes k :: "('a \<times> bool) list \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and xs :: "('a \<times> bool) list" assumes "\<And>s. k [pairtrue s] = f s"   and "\<And>xs ys. \<lbrakk>proper_signed_list xs; proper_signed_list ys\<rbrakk> \<Longrightarrow> k (prappend_signed_list xs ys) = k xs + k ys"   and "proper_signed_list xs" shows "k xs = (\<Sum>(s, b)\<leftarrow>xs. if b then f s else - f s)"
sumE: fixes x :: "'b"   and U :: "'b set"   and V :: "'b set" assumes "x \<in> U + V" obtains u :: "'b" and v :: "'b" where "x = u + v" and "u \<in> U" and "v \<in> V"
solves_ode_on_subset: fixes x :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "real set"   and Y :: "'b set"   and T :: "real set"   and X :: "'b set" assumes "(x solves_ode f) S Y"   and "T \<subseteq> S"   and "Y \<subseteq> X" shows "(x solves_ode f) T X"
sup_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "sup f g x = sup (f x) (g x)"
1: shows "B.ide f \<and> B.ide g \<and> B.ide h \<and> src\<^sub>B f = trg\<^sub>B g \<and> src\<^sub>B g = trg\<^sub>B h"
reaches_step_1: fixes x :: "'e"   and y :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and z :: "'e" assumes "(x reaches y) s"   and "(y points_to z) s" shows "(x reaches z) s"
intros_22: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es :: "char list exp list"   and vs :: "val list"   and ex :: "char list exp"   and es' :: "char list exp list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs @ throw ex # es',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw ex,s\<^sub>2\<rangle>"
norm_eq_if_face_cong: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs \<cong> ys"   and "distinct xs"   and "xs \<noteq> []" shows "rotate_min xs = rotate_min ys"
simps: fixes M :: "('d, 'e, 'f) FSM"   and V :: "'d list set" shows "is_det_state_cover M V = (\<exists>f. is_det_state_cover_ass M f \<and> V = f ` d_reachable M (initial M))"
Sup_bot_conv_1: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup A = bot) = (\<forall>x\<in>A. x = bot)"
non_volatile_conflict_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat" shows "P,E \<turnstile>a\<dagger>a' = ((a \<in> read_actions E \<and> a' \<in> write_actions E \<or> a \<in> write_actions E \<and> a' \<in> read_actions E \<or> a \<in> write_actions E \<and> a' \<in> write_actions E) \<and> (\<exists>ad al. (ad, al) \<in> action_loc P E a \<inter> action_loc P E a' \<and> \<not> is_volatile P al))"
3: shows "pderiv f = [2] \<cdot> \<one> \<odot>\<^bsub>UP Zp\<^esub> X_poly Zp"
distinct_11: fixes x3 :: "real \<times> real \<times> real"   and x4 :: "real \<times> real \<times> real" shows "LeqUni x3 \<noteq> NeqUni x4"
a_2N_sqrt: shows "\<forall>\<^sub>F N in sequentially. real N * (sqrt (real N) / 2 - 1) < real_of_int (a (2 * N))"
rb_aux_inv2_E: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and j :: "nat"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rb_aux_inv2 (bs, ss, ps)"   and "j < length fs"   and "Inr j \<notin> set ps"   and "b \<in> set bs"   and "component_of_term (lt b) < j" obtains s :: "'t" where "s \<in> set ss" and "s adds\<^sub>t term_of_pair (punit.lt (rep_list b), j)"
sup_bot_left: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "t_zero \<union> x = x"
adds_pp_splus: fixes t :: "'a"   and v :: "'t"   and s :: "'a" assumes "t adds\<^sub>p v" shows "t adds\<^sub>p s \<oplus> v"
BT_lemma_1_13: fixes F :: "'b poly"   and B :: "'b poly"   and G :: "'b poly"   and H :: "'b poly" assumes "F + B * G = H"   and "degree G \<le> degree F"   and "degree H < degree G \<or> H = 0 \<and> F \<noteq> 0 \<and> G \<noteq> 0"   and "degree H < degree G \<or> H \<noteq> 0" shows "subresultant (degree H) F G = Polynomial.smult ((- (1::'b)) ^ ((degree F - degree H) * (degree G - degree H)) * lead_coeff G ^ (degree F - degree H) * lead_coeff H ^ (degree G - degree H - 1)) H"
ltl_Release_StrongRelease_con_1: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> R\<^sub>c \<psi> = \<xi> \<Turnstile>\<^sub>c \<phi> M\<^sub>c \<psi> or\<^sub>c G\<^sub>c \<psi>"
segment_1: fixes a :: "'a"   and b :: "'a" shows "open_segment a b \<equiv> closed_segment a b - {a, b}"
116: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (x \<sqinter> - y \<squnion> z) = x \<squnion> z"
inv_part_def: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and m_i :: "nat"   and m_j :: "nat" shows "local.inv_part p A m_i m_j \<equiv> \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> j < m_j \<or> j = m_j \<and> m_i \<le> i \<longrightarrow> p A i j"
emeasure_distr: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and A :: "'b set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "A \<in> sets N" shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)"
dens_ctxt: shows "density_context V V' \<Gamma> \<delta>"
Schottky_lemma0: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and a :: "complex" assumes "f holomorphic_on S"   and "contractible S"   and "a \<in> S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 1 \<and> f z \<noteq> - 1" obtains g :: "complex \<Rightarrow> complex" where "g holomorphic_on S" and "cmod (g a) \<le> 1 + cmod (f a) / 3" and "\<And>z. z \<in> S \<Longrightarrow> f z = cos (complex_of_real pi * g z)"
points_index_0_left_imp: fixes B :: "'b set multiset"   and ps :: "'b set"   and b :: "'b set" assumes "B index ps = 0"   and "b \<in># B" shows "\<not> ps \<subseteq> b"
sums_iff: fixes x :: "'a" shows "f sums x = (summable f \<and> suminf f = x)"
assms_3: shows "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
thetaSeqZO_def: shows "thetaSeqZO \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. c1 \<approx>01T d1 \<and> c2 \<approx>01 d2}"
abs_zero: fixes x :: "int" assumes "Preliminaries.abs x = 0" shows "x = 0"
assms_2: shows "set (Abs_State.dom S1) \<subseteq> X"
sel_1: fixes x1 :: "'f option" shows "Resumption.resumption.result (Resumption.resumption.Done x1) = x1"
iMODb_icard: fixes m :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < m" shows "icard [ r, mod m, c ] = enat (Suc c)"
hyps_2: assumes "list_all stateful_strand_step.is_Send (unlabel A)" shows "list_all stateful_strand_step.is_Receive (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
mk_next_pow_def: fixes n :: "nat"   and x :: "'a ltln" shows "X\<^sub>n' n x \<equiv> case x of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> (Next_ltln ^^ n) x"
raise1: fixes ck :: "bool"   and s1 :: "v sem_env"   and env :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v" assumes "evaluate ck s1 env e (s2, Rval v1)" shows "evaluate ck s1 env (Raise e) (s2, Rerr (Rraise v1))"
tMap_tFirst: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b Trace" shows "tFirst (tMap f t) = f (tFirst t)"
simps_3: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and p :: "real \<times> real \<times> real" shows "substInfinitesimalQuadraticUni a b c d (LeqUni p) = OrUni (map_atomUni (quadraticSubUni a b c d) (convertDerivativeUni p)) (allZero' p)"
assms: fixes r :: "real" assumes "0 < r" shows "f holomorphic_on ball c r"
cases: fixes x :: "'a \<times> 'b \<times> 'c \<times> 'd" obtains A :: "'a" and B :: "'b" and C :: "'c" and D :: "'d" where "x = (A, B, C, D)"
pow_integer_mod_group: fixes n :: "nat"   and x :: "int"   and m :: "nat" shows "x [^]\<^bsub>integer_mod_group n\<^esub> m = int m * x mod int n"
plus_absorb1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "L F (\<lambda>x. f x + g x) = L F g"
lunit\<^sub>F\<^sub>M\<^sub>C_def: fixes a :: "'c local.term set" shows "\<l>[a] = \<F>C.mkarr \<^bold>\<l>\<^bold>[\<F>C.rep a\<^bold>]"
closure_approachable: fixes x :: "'b"   and S :: "'b set" shows "(x \<in> closure S) = (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"