hcju/isabelleps huggingface dataset:
l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_ancestors_di :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_owner_document get_parent remove_child remove_child_locs get_disconnected_nodes set_disconnected_nodes set_disconnected_nodes_locs get_ancestors_di adopt_node adopt_node_locs \<equiv> adopt_node = l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node get_owner_document get_parent remove_child get_disconnected_nodes set_disconnected_nodes get_ancestors_di \<and> adopt_node_locs = l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs remove_child_locs set_disconnected_nodes_locs"
simps_2: fixes n :: "nat" shows "simpnum (Bound n) = CN n 1 (C 0)"
coplanar_perm_14: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet C B A D"
abs_eq: fixes y :: "'b \<Rightarrow> 'c option" shows "Mapping.lookup (mapping.Mapping y) = y"
simps_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.same_level_path_aux kind get_return_edges cs [] = True"
ucast_bl: fixes w :: "'b word" shows "UCAST('b \<rightarrow> 'a) w = of_bl (to_bl w)"
CP_not_subseteq_NUM: shows "\<not> CP \<subseteq> NUM"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide x" shows "Adj.\<epsilon> x = \<psi> x (G x)"
admits_triangular_reduction_1x2: assumes "\<forall>A. A \<in> carrier_mat 1 2 \<longrightarrow> admits_triangular_reduction A" shows "All admits_triangular_reduction"
o: shows "card (V // per_union R x y) = Suc (card (\<Union>x\<in>V - R `` {x} - R `` {y}. {R `` {x}}))"
prime_factors_multiplicity: fixes n :: "'a" shows "prime_factors n = {p. prime p \<and> 0 < multiplicity p n}"
simps_2: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and m :: "'c match_expr"   and chain :: "char list"   and rs :: "'c rule list" shows "process_call \<Gamma> (Rule m (Call chain) # rs) = add_match m (process_ret (the (\<Gamma> chain))) @ process_call \<Gamma> rs"
iffalse: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<iota> :: "nat"   and c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom" assumes "BMap (E b m) = False" shows "\<langle>if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi,m\<rangle> \<rightarrow>\<lhd>[]\<rhd> \<langle>Some c2,m\<rangle>"
is_digitI: fixes n :: "nat" assumes "n = 0 \<or> n = 1 \<or> n = 2 \<or> n = 3 \<or> n = 4 \<or> n = 5 \<or> n = 6 \<or> n = 7 \<or> n = 8 \<or> n = 9" shows "is_digit n"
fun_vars_def: fixes f :: "nat \<Rightarrow> 'a" shows "fun_vars f = {v. f v \<noteq> (0::'a)}"
read_def: fixes c :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and P :: "'e \<Rightarrow> 'd process" shows "read c A P \<equiv> Mprefix (c ` A) (P \<circ> inv c)"
ord_Zp_p_int_pow: fixes n :: "int" assumes "0 \<le> n" shows "ord_Zp ([p] \<cdot> \<one> [^] n) = n"
addfunsetD_range': fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and M :: "'b set"   and x :: "'a" assumes "f \<in> addfunset A M" shows "f x \<in> M"
limsup_infi: shows "limsup (\<lambda>x. ereal (aa (x + 1) / prod aa {0..x} powr (2 + 2 / \<epsilon> + \<delta>) * (1 / bb (x + 1)))) = \<infinity>"
has_bochner_integral_add: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b"   and g :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "has_bochner_integral M f x"   and "has_bochner_integral M g y" shows "has_bochner_integral M (\<lambda>x. f x + g x) (x + y)"
bisims1List2: fixes P :: "'b prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and es' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e :: "'addr expr1"   and v :: "'addr val" assumes "P,es,h \<turnstile> (es', xs) [\<leftrightarrow>] (stk, loc, pc, xcp)" shows "P,e # es,h \<turnstile> (Val v # es', xs) [\<leftrightarrow>] (stk @ [v], loc, length (compE2 e) + pc, xcp)"
fun_cong: fixes f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g"   and x :: "'f" assumes "f = g" shows "f x = g x"
INF_lower: fixes i :: "'a::type"   and A :: "'a::type set"   and f :: "'a::type \<Rightarrow> 'b::complete_lattice" assumes "(i::'a::type) \<in> (A::'a::type set)" shows "\<Sqinter> ((f::'a::type \<Rightarrow> 'b::complete_lattice) ` (A::'a::type set)) \<le> f (i::'a::type)"
hyps: fixes vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "g \<in> set gs" shows "monom_mult (1::'b) (pp_of_term v - lp g) g \<in> set (snd (sym_preproc_addnew gs vs fs v))"
Star_val: fixes ss :: "'b list list"   and r :: "'b rexp" assumes "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r" shows "\<exists>vs. concat (map flat vs) = concat ss \<and> (\<forall>v\<in>set vs. \<turnstile> v : r)"
repc_def: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" shows "repc cfg = repcs (reps (state cfg)) cfg"
iso_verts_inv_iso: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and u :: "'e" assumes "pre_digraph.digraph_isomorphism G hom"   and "u \<in> verts G" shows "iso_verts (pre_digraph.inv_iso G hom) (iso_verts hom u) = u"
not_is_Pure_conv: fixes r :: "('g, 'h, 'i) generat" shows "(\<not> is_Pure r) = (\<exists>out c. r = IO out c)"
iso_set_trans: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme"   and i :: "'b \<Rightarrow> 'e"   and I :: "('e, 'f) monoid_scheme" assumes "h \<in> Group.iso G H"   and "i \<in> Group.iso H I" shows "i \<circ> h \<in> Group.iso G I"
sum_nonzero_representation_eq: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and v :: "'d" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis" shows "(\<Sum>b | module.representation scale basis v b \<noteq> (0::'c). scale (module.representation scale basis v b) b) = v"
sum_prog_def: fixes \<Phi> :: "nat \<Rightarrow> bool"   and l :: "nat"   and u :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "sum_prog \<Phi> l u f \<equiv> nfoldli [l..<u] (\<lambda>_. True) (\<lambda>i s. ASSERT (\<Phi> i) \<bind> (\<lambda>_. RETURN (s + f i))) (0::'b)"
integral_distr: fixes g :: "'e \<Rightarrow> 'f"   and M :: "'e measure"   and N :: "'f measure"   and f :: "'f \<Rightarrow> 'g" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integral\<^sup>L (distr M N g) f = \<integral> x. f (g x) \<partial>M"
f0_is_0: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and N :: "('h, 'l, 'm) module_scheme"   and f :: "'j \<Rightarrow> 'l" assumes "mod_hom R M N f" shows "f \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>N\<^esub>"
ce_rel_lm_11: fixes x :: "nat"   and y :: "nat"   and r :: "(nat \<times> nat) set" assumes "c_pair x y \<in> ce_rel_to_set r" shows "(x, y) \<in> r"
surjective_var: fixes x :: "'a" shows "(top \<sqinter> x = top) = (top \<le> id x \<sqinter> x)"
list_quotient3: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b" assumes "Quotient3 R Abs Rep" shows "Quotient3 (list_all2 R) (map Abs) (map Rep)"
remdups_hd_notin_tl: fixes dl :: "'b list"   and h :: "'b"   and t :: "'b list" assumes "remdups dl = h # t" shows "h \<notin> set t"
dg_Par_def: fixes \<alpha> :: "V" shows "dg_Par \<alpha> = [Vset \<alpha>, ZFC_in_HOL.set {T. arr_Par \<alpha> T}, \<lambda>T\<in>\<^sub>\<circ>ZFC_in_HOL.set {T. arr_Par \<alpha> T}. T\<lparr>ArrDom\<rparr>, \<lambda>T\<in>\<^sub>\<circ>ZFC_in_HOL.set {T. arr_Par \<alpha> T}. T\<lparr>ArrCod\<rparr>]\<^sub>\<circ>"
bind_FAIL: fixes f :: "'d \<Rightarrow> 'c nres" shows "FAIL \<bind> f = FAIL"
at_within_order: fixes x :: "'a"   and s :: "'a set" assumes "UNIV \<noteq> {x}" shows "at x within s = inf (INF a\<in>{x<..}. principal ({..<a} \<inter> s - {x})) (INF a\<in>{..<x}. principal ({a<..} \<inter> s - {x}))"
simps: fixes a1 :: "nat"   and a2 :: "((nat \<times> nat) \<times> transition) fset"   and a3 :: "nat"   and a4 :: "nat \<Rightarrow>f value option"   and a5 :: "(String.literal \<times> value list) list" shows "visits a1 a2 a3 a4 a5 = ((\<exists>s e r. a1 = s \<and> a2 = e \<and> a3 = s \<and> a4 = r \<and> a5 = []) \<or> (\<exists>e s r l i target t. a1 = target \<and> a2 = e \<and> a3 = s \<and> a4 = r \<and> a5 = (l, i) # t \<and> (\<exists>(s', T)|\<in>|possible_steps e s r l i. visits target e s' (evaluate_updates T i r) t)))"
mat_seq_minus_def: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat" shows "mat_seq_minus X A = (\<lambda>n. X n - A)"
permStarFresh: fixes xvec :: "name list"   and p :: "(name \<times> name) list"   and T :: "'a" assumes "xvec \<sharp>* p" shows "xvec \<sharp>* (p \<bullet> T) = xvec \<sharp>* T"
f_nonneg: shows "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
open_fun_def: fixes U :: "('c \<Rightarrow> 'd) set" shows "open U = openin (product_topology (\<lambda>i. euclidean) UNIV) U"
eq_inf_concat_end: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w :: "'item stream" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "w\<^sub>1 @- w =\<^sub>I w\<^sub>2 @- w"
iff_comm: shows "(A \<and> B) = (B \<and> A)"
exhaust: fixes y :: "'a tree" obtains x1 :: "'a" and x2 :: "'a tree list" where "y = NODE x1 x2"
HPhase0Read_HInv4b_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase0Read s s' p d" shows "HInv4b s' p"
prv_LLq_Num_imp_Suc: fixes t :: "'trm"   and n :: "nat" assumes "t \<in> atrm" shows "prv (imp (LLq t (Num n)) (LLq t (suc (Num n))))"
weight_outside: fixes x :: "'v" assumes "x \<notin> \<^bold>V" shows "weight \<Gamma> x = 0"
4: shows "fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q (x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<delta>) = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) - set X"
rsp: shows "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'c)}) (\<lambda>k. 0::'c) (\<lambda>k. 0::'c)"
Jordan_inside_outside_R2: fixes c :: "real \<Rightarrow> 'a" assumes "simple_path c"   and "pathfinish c = pathstart c" shows "inside (path_image c) \<noteq> {} \<and> open (inside (path_image c)) \<and> connected (inside (path_image c)) \<and> outside (path_image c) \<noteq> {} \<and> open (outside (path_image c)) \<and> connected (outside (path_image c)) \<and> bounded (inside (path_image c)) \<and> \<not> bounded (outside (path_image c)) \<and> inside (path_image c) \<inter> outside (path_image c) = {} \<and> inside (path_image c) \<union> outside (path_image c) = - path_image c \<and> frontier (inside (path_image c)) = path_image c \<and> frontier (outside (path_image c)) = path_image c"
sup_zhmultiset_def: fixes A :: "zhmultiset"   and B :: "zhmultiset" shows "sup A B = (if A < B then B else A)"
simps_4: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int option" shows "pbval' (Less a\<^sub>1 a\<^sub>2) s = optless (paval' a\<^sub>1 s) (paval' a\<^sub>2 s)"
tMap_eq_imp_tLength_eq: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c Trace"   and f' :: "'d \<Rightarrow> 'b"   and t' :: "'d Trace" assumes "tMap f t = tMap f' t'" shows "tLength t = tLength t'"
expand_superset: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "{a. g a \<noteq> (1::'d)} \<subseteq> A" shows "Prod_any g = prod g A"
counit_simps_2: shows "local.dom \<epsilon>' = f \<star> g"
that: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* x z" shows "\<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>*\<^sup>* z u"
try_spmf_cong: fixes p :: "'h spmf"   and p' :: "'h spmf"   and q :: "'h spmf"   and q' :: "'h spmf" assumes "p = p'"   and "\<not> lossless_spmf p' \<Longrightarrow> q = q'" shows "TRY p ELSE q = TRY p' ELSE q'"
is_renaming_list_inv_renaming_cancel_atm_mset_lists: fixes AAs :: "'a multiset list"   and \<rho>s :: "'s list" assumes "length AAs = length \<rho>s"   and "is_renaming_list \<rho>s" shows "AAs \<cdot>\<cdot>aml \<rho>s \<cdot>\<cdot>aml map inv_renaming \<rho>s = AAs"
extr_mono: fixes b :: "agent set"   and b' :: "agent set"   and I :: "msg set"   and I' :: "msg set"   and C :: "chan set"   and C' :: "chan set" assumes "b \<subseteq> b'"   and "I \<subseteq> I'"   and "C \<subseteq> C'" shows "extr b I C \<subseteq> extr b' I' C'"
INFM_iff_infinite: fixes P :: "'a \<Rightarrow> bool" shows "(\<exists>\<^sub>\<infinity>x. P x) = infinite {x. P x}"
indRelRPOp_indRelRPO_eq: fixes Enc :: "'a \<Rightarrow> 'b" shows "encoding.indRelRPOp Enc = (\<lambda>x xa. (x, xa) \<in> encoding.indRelRPO Enc)"
designI: assumes "\<And>b. b \<in># del_block bl \<Longrightarrow> b \<noteq> {}"   and "del_block bl \<noteq> {#}"   and "\<V> \<noteq> {}" shows "design \<V> (del_block bl)"
eval_fm_SentP: fixes e :: "(name, hf) finfun"   and x :: "tm" shows "eval_fm e (SentP x) = (\<lbrakk>x\<rbrakk>e \<in> Sent)"
HInfiniteI: fixes x :: "'a star" assumes "\<And>r. r \<in> \<real> \<Longrightarrow> r < hnorm x" shows "x \<in> HInfinite"
fresh_star_Un: fixes as :: "atom set"   and bs :: "atom set"   and x :: "'a" shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
assms_2: shows "Undirected_Graph.connected g\<^sub>2"
fair_implies_Liminf_saturated: fixes Ns :: "'a set llist" assumes "chain (\<rhd>) Ns"   and "fair Ns" shows "saturated (Liminf_llist Ns)"
sigma_minr_lemma: fixes rf :: "recf"   and xs :: "nat list"   and w :: "nat" assumes "primerec rf (Suc (length xs))" shows "UF.Sigma (rec_exec (rec_all (recf.id (Suc (length xs)) (length xs)) (Cn (Suc (Suc (length xs))) rec_not [Cn (Suc (Suc (length xs))) rf (get_fstn_args (Suc (Suc (length xs))) (length xs) @ [recf.id (Suc (Suc (length xs))) (Suc (length xs))])]))) (xs @ [w]) = Minr (\<lambda>args. 0 < rec_exec rf args) xs w"
max_in_chain_def: fixes i :: "nat"   and C :: "nat \<Rightarrow> 'c" shows "max_in_chain i C = (\<forall>j\<ge>i. C i = C j)"
BV_correct: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "jvm_prog"   and \<sigma> :: "jvm_state"   and \<sigma>' :: "jvm_state" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>'"   and "P,\<Phi> |- \<sigma> [ok]" shows "P,\<Phi> |- \<sigma>' [ok]"
leg0_simps_3: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c" assumes "tabulation_data V H \<a> \<i> src trg r \<rho> f g" shows "trg f = src r"
gen_lossless_gpv_mono: fixes \<I> :: "('e, 'f) \<I>" shows "mono (\<lambda>gen_lossless_gpv gpv. \<exists>pa. gpv = GPV pa \<and> lossless_spmf pa \<and> (\<forall>out c input. IO out c \<in> set_spmf pa \<longrightarrow> input \<in> responses_\<I> \<I> out \<longrightarrow> gen_lossless_gpv (c input)))"
SumIdemp: fixes P :: "pi" shows "P \<oplus> P \<equiv>\<^sub>e P"
compet_def: fixes A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and c :: "real"   and S :: "'b list set" shows "compet A c S = (\<forall>s\<in>S. \<exists>b\<ge>0. \<forall>rs. static s rs \<longrightarrow> real (T_on A s rs) \<le> c * real (T_opt s rs) + b)"
UP_domain: fixes R :: "('a, 'b) ring_scheme" assumes "UP_domain R" shows "domain (UP R)"
add_independentS_double_iff: fixes A :: "'a set"   and B :: "'a set" shows "add_independentS [A, B] = (\<forall>b\<in>B. \<forall>a\<in>A. a + b = (0::'a) \<longrightarrow> a = (0::'a))"
Conjunctive_def: fixes H :: "'d upred \<Rightarrow> 'd upred" shows "Conjunctive H = (\<exists>Q. \<forall>P. H P = (P \<and> Q))"
equiv_proj: fixes A :: "'d set"   and R :: "('d \<times> 'd) set"   and z :: "'d \<times> 'd" assumes "equiv A R"   and "z \<in> R" shows "(Equiv_Relations.proj R \<circ> fst) z = (Equiv_Relations.proj R \<circ> snd) z"
E_def: shows "E = uedge ` edges T - {{x, y}}"
apex_def: shows "Dom_\<rho>.apex \<equiv> C.dom Dom_\<rho>.leg0"
upt_eq_Cons_conv: fixes i :: "nat"   and j :: "nat"   and x :: "nat"   and xs :: "nat list" shows "([i..<j] = x # xs) = (i < j \<and> i = x \<and> [i + 1..<j] = xs)"
lb_sgn: shows "\<forall>x\<le>lb. \<forall>p\<in>set ps. sgn (poly p x) = sgn_neg_inf p"
sol: shows "(sol solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. mk_v I (OVar vid1) (sol 0, bb) x \<in> Contexts I pid1 UNIV}"
counit_simps_4: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>C (\<epsilon>'.counit a) = I\<^sub>C.map\<^sub>0 a"
Abs_fin_string_induct: fixes P :: "fin_string \<Rightarrow> bool"   and x :: "fin_string" assumes "\<And>y. y \<in> {s. length s \<le> MAX_STR} \<Longrightarrow> P (Abs_fin_string y)" shows "P x"
scalar_prod_def: fixes v :: "'a Matrix.vec"   and w :: "'a Matrix.vec" shows "v \<bullet> w \<equiv> \<Sum>i = 0..<dim_vec w. v $ i * w $ i"
stepreq: shows "SRabbr (V'' ! i) m1' \<alpha>' p' m2' \<and> stepResultsinR p p' R' \<and> (\<alpha>, \<alpha>') \<in> R' \<and> dhequality_alternative d PP (pp (V' ! i)) m2 m2'"
prems_1: shows "I \<in> {..Suc n} \<rightarrow> {J. ideal J R}"
distinct_1: fixes x1 :: "Keys"   and x2 :: "Secrets" shows "kE x1 \<noteq> sE x2"
always_def: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<box>F \<equiv> %s. \<forall>n. F (s |\<^sub>s n)"
Range_set_zip: fixes A :: "'d list"   and B :: "'e list" assumes "length A = length B" shows "Range (set (zip A B)) = set B"
mod_div_trivial: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and b :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "divide (modulo a b) b = zero"
is_quantum_predicate_def: fixes dims :: "nat list"   and P :: "complex mat" shows "state_sig.is_quantum_predicate dims P = (P \<in> carrier_mat (state_sig.d dims) (state_sig.d dims) \<and> positive P \<and> P \<le>\<^sub>L 1\<^sub>m (state_sig.d dims))"
enum_inj: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(enum i = enum j) = (i = j)"
d_OUT_Edge: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a antiparallel_edges.vertex \<times> 'a antiparallel_edges.vertex \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" assumes "antiparallel_edges \<Delta>"   and "flow (antiparallel_edges.\<Delta>'' \<Delta>) f" shows "d_OUT f (antiparallel_edges.Edge x y) = f (antiparallel_edges.Edge x y, antiparallel_edges.Vertex y)"
Form_imp_wf_dbfm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "Form x" obtains A :: "dbfm" where "wf_dbfm A" and "x = \<lbrakk>quot_dbfm A\<rbrakk>e"
assms_1: fixes d :: "nat" assumes "d < length p" shows "lv p d = lv p' d"
order_refl: fixes x :: "'a" shows "domain_semigroup.ds_ord (domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top>) (\<cdot>) x x"
deriv_psubst: fixes z :: "('a, 'b) form list"   and p :: "('a, 'b) form"   and f :: "'a \<Rightarrow> 'a" assumes "infinite UNIV"   and "z \<turnstile> p" shows "map (psubst f) z \<turnstile> psubst f p"
monad_state_prob_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_prob return bind get put sample" shows "monad_state_prob (return_option return) (bind_option return bind) (get_option get) (put_option put) (sample_option sample)"
findr_ex_1: fixes e :: "nat" assumes "\<exists>i<e_length e. e_nth e i \<noteq> 0" shows "the (findr e) < e_length e"
fmpredD: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('b, 'c) fmap"   and x :: "'b"   and y :: "'c" assumes "fmpred P m"   and "fmlookup m x = Some y" shows "P x y"
density_context_def: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" shows "density_context V V' \<Gamma> \<delta> \<equiv> ((\<forall>\<rho>. \<rho> \<in> space (state_measure V' \<Gamma>) \<longrightarrow> subprob_space (dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho>)) \<and> finite V) \<and> finite V' \<and> \<delta> \<in> borel_measurable (state_measure (V \<union> V') \<Gamma>) \<and> V \<inter> V' = {}"
lsumr_rsuml_right_gpv_right_gpv: fixes gpv :: "('h, 'k, 'n) gpv" shows "map_gpv' id lsumr rsuml (right_gpv (right_gpv gpv)) = right_gpv gpv"
intros_1: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[], t\<rangle> \<Rightarrow> t"
idem: fixes a :: "'k multiset" shows "a \<union># a = a"
final_3: shows "\<forall>c. Marker \<notin> set (msgs final c)"
left_diff_repeat_mset_distrib': fixes i :: "nat"   and j :: "nat"   and u :: "'k multiset" shows "repeat_mset (i - j) u = repeat_mset i u - repeat_mset j u"
simps_5: fixes t1 :: "('a, 'b) trm"   and t2 :: "('a, 'b) trm" shows "SIGT (Times t1 t2) = SIGT t1 \<union> SIGT t2"
filterlim_at_top_imp_at_infinity: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "filterlim f at_top F" shows "filterlim f at_infinity F"
beta_l_Suc_l_derive: fixes l :: "nat" shows "2 * (alpha_l l * complex_of_real (cos (\<theta> / 2)) - beta_l l * complex_of_real (sin (\<theta> / 2))) * complex_of_real (sin (\<theta> / 2)) + beta_l l = beta_l (l + 1)"
joinI_left: fixes a :: "'a"   and b :: "'a"   and A :: "('a \<times> 'a) set" assumes "(a, b) \<in> A\<^sup>*" shows "(a, b) \<in> A\<^sup>\<down>"
nth_default_coeffs_eq: fixes p :: "'a poly" shows "nth_default (0::'a) (coeffs p) = poly.coeff p"