hcju/isabelleps huggingface dataset:
accepting_pair\<^sub>G\<^sub>R_LTS: fixes w :: "nat \<Rightarrow> 'd"   and \<Sigma> :: "'d set"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e"   and q\<^sub>0 :: "'e"   and F :: "('e \<times> 'd \<times> 'e) set"   and \<I> :: "('e \<times> 'd \<times> 'e) set set" assumes "range w \<subseteq> \<Sigma>" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (F, \<I>) w = accepting_pair\<^sub>G\<^sub>R_LTS (reach\<^sub>t \<Sigma> \<delta> q\<^sub>0) q\<^sub>0 (F, \<I>) w"
zdiv_mono1: fixes a :: "int"   and a' :: "int"   and b :: "int" assumes "a \<le> a'"   and "0 < b" shows "a div b \<le> a' div b"
prems_3: fixes tm :: "taylor_model" assumes "tm \<in> set VS" shows "num_params (tm_poly tm) \<le> length I"
map_spmf_conv_bind_spmf: fixes f :: "'d \<Rightarrow> 'b"   and p :: "'d spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
invar_rderiv_and_add_init: fixes as :: "'a"   and r :: "'b rexp" shows "invar_rderiv_and_add as r (True, [\<guillemotleft>r\<guillemotright>])"
msig_simp: fixes c :: "qtname"   and s :: "sig"   and m :: "methd" shows "msig (c, s, m) = s"
select_convs_2: fixes nothing :: "unit"   and more :: "'m" shows "RObject.more \<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> = more"
less_imp_ordinal_of_hmset_less: fixes k :: "hmultiset"   and l :: "hmultiset" assumes "k < l" shows "ordinal_of_hmset k < ordinal_of_hmset l"
HComplex_eq_0: fixes x :: "real star"   and y :: "real star" shows "(HComplex x y = 0) = (x = 0 \<and> y = 0)"
PO_hoare_defs_1: fixes pre :: "'e set"   and R :: "('e \<times> 'e) set"   and post :: "'e set" shows "{pre} R {> post} \<equiv> R `` pre \<subseteq> post"
pth_a: fixes x :: "'a"   and y :: "'a" shows "0 *\<^sub>R x + y \<equiv> y"
nolb_def: fixes \<phi> :: "atom fm"   and xs :: "'a list"   and l :: "'a"   and x :: "'a" shows "nolb \<phi> xs l x = (\<forall>y\<in>{l<..<x}. y \<notin> LB \<phi> xs)"
compact_imp_closed: fixes "open" :: "'c set \<Rightarrow> bool"   and s :: "'c set" assumes "class.t2_space open"   and "topological_space.compact open s" shows "topological_space.closed open s"
echelon_form_upt_row_condition1_explicit_neg: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c"   and ia :: "'c" assumes "echelon_form_upt_row A k"   and "\<not> is_zero_row i A"   and "ia < i"   and "mod_type_class.to_nat i < k" shows "\<not> is_zero_row ia A"
eval_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" obtains ca :: "('Var, 'AExp, 'BExp) Stmt" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "c = cxt_to_stmt E__ ca__" and "c'a__ = cxt_to_stmt E__ c'b__" and "mds' = mds" and "(ca__, mem) \<leadsto>\<^sub>s (c'b__, mem')"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" where "c = c\<^sub>1 ;; c\<^sub>2" and "c'a__ = c\<^sub>1' ;; c\<^sub>2" and "\<langle>c\<^sub>1, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c\<^sub>1', mds', mem'\<rangle>\<^sub>w"   | ca :: "('Var, 'AExp, 'BExp) Stmt" and mu :: "'Var ModeUpd" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "c = cxt_to_stmt E (ca@[mu])" and "c'a__ = cxt_to_stmt E c'" and "\<langle>ca, update_modes mu mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w"
iso_set_sym: fixes G :: "('d, 'e) monoid_scheme"   and h :: "'d \<Rightarrow> 'f"   and H :: "('f, 'g) monoid_scheme" assumes "Group.group G"   and "h \<in> Group.iso G H" shows "inv_into (carrier G) h \<in> Group.iso H G"
f_Exec_Stream_append: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and ys :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream trans_fun (xs @ ys) c = f_Exec_Comp_Stream trans_fun xs c @ f_Exec_Comp_Stream trans_fun ys (f_Exec_Comp trans_fun xs c)"
rel_fundef_arities: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and gd1 :: "'c fundef"   and gd2 :: "'d fundef" assumes "rel_fundef r gd1 gd2" shows "arity gd1 = arity gd2"
centre_in_ball: fixes x :: "'a"   and e :: "real" shows "(x \<in> ball x e) = (0 < e)"
assms_5: shows "msg (Suc 0) req \<and> msg (Suc 0) a \<and> msg (Suc 0) stop \<and> ts lose \<longrightarrow> (\<forall>t. (ack t = [init_state] \<and> req (Suc t) = [init] \<and> lose (Suc t) = [False] \<and> lose (Suc (Suc t)) = [False] \<longrightarrow> ack (Suc (Suc t)) = [connection_ok]) \<and> (ack t = [connection_ok] \<and> req (Suc t) = [send] \<and> (\<forall>k\<le>Suc d. lose (t + k) = [False]) \<longrightarrow> i (Suc (t + d)) = inf_last_ti dt t \<and> ack (Suc t) = [sending_data]) \<and> (ack (t + d) = [sending_data] \<and> a (Suc t) = [sc_ack] \<and> (\<forall>k\<le>Suc d. lose (t + k) = [False]) \<longrightarrow> vc (Suc (t + d)) = [vc_com]))"
fmdom_fmmap_keys: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and m :: "('a, 'c) fmap" shows "fmdom (fmmap_keys f m) = fmdom m"
star_decom: fixes x :: "'a list"   and A :: "'a list set" assumes "x \<in> A\<star>"   and "x \<noteq> []" shows "\<exists>a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> A \<and> b \<in> A\<star>"
rank_take_cols: fixes inds :: "nat list" assumes "distinct inds" shows "rank (take_cols (1\<^sub>m n) inds) = length (filter (\<lambda>y. y < n) inds)"
2: shows "(\<exists>x. linear_orderable_1 x) = (\<exists>x. linear_orderable_3 x)"
subscene_antisym: fixes Y :: "'a scene"   and X :: "'a scene" assumes "idem_scene Y"   and "X \<subseteq>\<^sub>S Y"   and "Y \<subseteq>\<^sub>S X" shows "X = Y"
finfun_to_list_update: fixes f :: "('a, 'b) finfun"   and a :: "'a"   and b :: "'b" shows "finfun_to_list (finfun_update f a b) = (if b = finfun_default f then remove1 a (finfun_to_list f) else insort_insert a (finfun_to_list f))"
circ_mult: fixes x :: "'a"   and y :: "'a" shows "(x * y)\<^sup>\<circ> = Rf (x * (y * x)\<^sup>\<circ>) y"
law_of_cosines: fixes B :: "complex"   and C :: "complex"   and A :: "complex" shows "(cdist B C)\<^sup>2 = (cdist A C)\<^sup>2 + (cdist A B)\<^sup>2 - 2 * cdist A C * cdist A B * cos (\<angle> (C - A) (B - A))"
hyps_2: shows "TP \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> TQ"
trans_invert: fixes c :: "('test, 'atom) com"   and s :: "'state" obtains c' :: "('test, 'atom) com" and s' :: "'state" where "(c, s) \<rightarrow>c (c', s'__)"   | s' :: "'state" where "(c, s) \<rightarrow>t s'"
subdivision_gen_contr: fixes V :: "'a set" assumes "is_slim V" shows "subdivision_pair (gen_contr_graph (with_proj G) V) G"
of_int_0_eq_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(zero = ring_1.of_int one plus zero minus z) = (z = 0)"
filter_b_def: shows "filter_b = filter (\<lambda>x. x = b)"
eq: shows "hd xs = e_hd (e2xs (encode_config c))"
map_replicate_const: fixes k :: "'b"   and lst :: "'c list" shows "map (\<lambda>x. k) lst = replicate (length lst) k"
cut_less_not_empty_imp: fixes I :: "'a set"   and t :: "'a" assumes "I \<down>< t \<noteq> {}" shows "I \<noteq> {}"
agree_refl: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and A :: "('a + 'a) set" shows "Vagree \<nu> \<nu> A"
pp_inf_sup_p: fixes x :: "'a" shows "- - x \<sqinter> (x \<squnion> - x) = x"
collapse_2: fixes resumption :: "('f, 'g, 'h) Resumption.resumption" assumes "\<not> Resumption.resumption.is_Done resumption" shows "Resumption.resumption.Pause (Resumption.resumption.output resumption) (Resumption.resumption.resume resumption) = resumption"
prems_2: shows "\<forall>ia. ia \<in> insert i s \<longrightarrow> 0 \<le> a ia"
is_zero_row_upt_k_suc: fixes i :: "'b"   and k :: "nat"   and A :: "(('c, 'd) vec, 'b) vec" assumes "is_zero_row_upt_k i k A"   and "A $ i $ mod_type_class.from_nat k = (0::'c)" shows "is_zero_row_upt_k i (Suc k) A"
inf_subs: fixes T :: "bool list set"   and ds :: "bool list" assumes "infinite (subtree T ds)" shows "infinite (subtree T (ds @ [Left])) \<or> infinite (subtree T (ds @ [Right]))"
assms_2: fixes x :: "'v" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)"
compatibleI: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "class.cpo below"   and "below x z"   and "below y z"   and "\<And>a. \<lbrakk>below x a; below y a\<rbrakk> \<Longrightarrow> below z a" shows "cpo.compatible below x y"
cases: fixes G :: "('a, 'b) graph_scheme"   and A :: "'a set"   and B :: "'a set"   and S :: "'a set" assumes "separating_gen G A B S" obtains "\<And>x y p. \<lbrakk>x \<in> A; y \<in> B; path G x p y\<rbrakk> \<Longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S"
g_def: shows "g \<equiv> \<^bold>\<tau>x. \<Phi> x"
post_weak_hoare_r: fixes p :: "'b upred"   and C :: "'b hrel"   and q\<^sub>2 :: "'b upred"   and q\<^sub>1 :: "'b upred" assumes "\<lbrace>p\<rbrace> C \<lbrace>q\<^sub>2\<rbrace>\<^sub>u"   and "`q\<^sub>2 \<Rightarrow> q\<^sub>1`" shows "\<lbrace>p\<rbrace> C \<lbrace>q\<^sub>1\<rbrace>\<^sub>u"
fmrel_on_fset_fmrel_restrict: fixes S :: "'c fset"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'c \<rightharpoonup>\<^sub>f 'd"   and n :: "'c \<rightharpoonup>\<^sub>f 'e" shows "fmrel_on_fset S P m n = fmrel P (fmrestrict_fset S m) (fmrestrict_fset S n)"
extremed_iff_UNIV_complete: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "extremed A less_eq = {A}-complete A less_eq"
rel_simps_44: shows "(0::'a) \<le> (1::'a)"
prj_simps_6: shows "Maps.dom SPN_fgh.Prj\<^sub>0 = Chn ((SPN f \<circ> SPN g) \<circ> SPN h)"
out_one_side: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "\<not> Col A B X \<or> \<not> Col A B Y"   and "A Out X Y" shows "A B OS X Y"
simps_2: fixes v :: "'r"   and vs :: "[:'r:]"   and pat :: "[:'r:]"   and us :: "[:'r:]" assumes "v \<noteq> \<bottom>"   and "vs \<noteq> \<bottom>" shows "right2\<cdot>pat\<cdot>(us, v :# vs) = rep2\<cdot>pat\<cdot>(us :@ [:v:], vs)"
\<psi>_img_\<phi>_img: fixes A :: "'s set" assumes "A \<subseteq> S.Univ" shows "\<psi> ` \<phi> ` A = A"
unfair: shows "\<not> fair_state_seq (lmap state_of_wstate Sts)"
lesub_def: fixes x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> y) = r x y"
isconstant_polybound0: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "isconstant p = polybound0 p"
smultfunsetI: fixes G :: "'g set"   and FH :: "('r, 'g) aezfun set"   and f :: "('r, 'g) aezfun \<Rightarrow> 'v" assumes "\<forall>a. \<forall>g\<in>G. \<forall>x\<in>FH. f (a \<delta>\<delta> g * x) = (a \<delta>\<delta> g) \<cdot> f x" shows "f \<in> smultfunset G FH"
arith_extra_simps_18: fixes a :: "'a" shows "(0::'a) \<cdot> a = (0::'a)"
wellformed_items_def: fixes X :: "item set" shows "wellformed_items X = Ball X wellformed_item"
Derives1_implies_derives1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derives1 \<NN> \<TT> \<RR> u i r v" shows "CFG.derives1 \<NN> \<TT> \<RR> u v"
UN_bool_eq: fixes A :: "bool \<Rightarrow> 'c set" shows "\<Union> (range A) = A True \<union> A False"
le_numeral_extra_4: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one one"
gen_lossless_gpvI: fixes gpv :: "('a, 'out, 'in) gpv" assumes "lossless_spmf (the_gpv gpv)"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> local.gen_lossless_gpv (c input)" shows "local.gen_lossless_gpv gpv"
outstanding_volatile_writes_unowned_by_others_tl: fixes t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "outstanding_volatile_writes_unowned_by_others (t # ts)" shows "outstanding_volatile_writes_unowned_by_others ts"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "abel_semigroup f"   and "abel_semigroup_set_axioms f z" shows "abel_semigroup_set f z"
path_entry_cons_conv: fixes G :: "nat graph_rec"   and E :: "(nat \<times> nat) set"   and u' :: "nat"   and l :: "nat list"   and w :: "nat" assumes "cfg_doms G" shows "path_entry E (u' # l) w = ((u', w) \<in> E \<and> path_entry E l u')"
p2r_subid: fixes P :: "'b \<Rightarrow> bool" shows "p2r P \<subseteq> Id"
real_of_float_max: fixes x :: "float"   and y :: "float" shows "real_of_float (max x y) = max (real_of_float x) (real_of_float y)"
unifiers_insert_Var_left: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and x :: "'a"   and t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" assumes "\<sigma> \<in> unifiers (insert (Var x, t) E)" shows "\<sigma> \<in> unifiers (subst_set (subst x t) E)"
simps_3: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> (atom list \<times> atom fm list \<times> nat) list \<Rightarrow> atom fm"   and \<phi> :: "atom fm" shows "QE_dnf' opt step (Neg \<phi>) = neg (QE_dnf' opt step \<phi>)"
induct: fixes x1 :: "BExpr"   and x2 :: "TP"   and P :: "BExpr \<Rightarrow> TP \<Rightarrow> bool" assumes "(x1, x2) \<in> VS_Bexpr"   and "\<And>e1 t e2 f. \<lbrakk>(e1, t) \<in> VS_expr; (e2, t) \<in> VS_expr\<rbrakk> \<Longrightarrow> P (compB f e1 e2) t"   and "\<And>e. P e high" shows "P x1 x2"
bisimulation_upto_bisimulation: fixes f :: "('s \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> bool"   and R :: "'s \<Rightarrow> 's \<Rightarrow> bool" assumes "compatible f"   and "bisimulation_upto R f" obtains S :: "'s \<Rightarrow> 's \<Rightarrow> bool" where "bisimulation S" and "R \<le> S"
comp_arr_inv': fixes f :: "'b" assumes "D.iso f" shows "f \<cdot>\<^sub>D D.inv f = D.cod f"
remove_vertex_V': fixes G :: "('c, 'd) Graph_scheme"   and x :: "'c" assumes "Digraph G" shows "V\<^bsub>Digraph.remove_vertex G x\<^esub> \<subseteq> V\<^bsub>G\<^esub>"
\<phi>: shows "bij_betw \<phi> {0..<n} (Collect arr)"
deltafun_is_aezfun: fixes b :: "'b"   and a :: "'a" shows "b \<delta> a \<in> aezfun_set"
imply_completeness: fixes X :: "('a, 'b) fm set"   and p :: "('a, 'b) fm" assumes "\<forall>E F G. Ball X \<lbrakk>E, F, G\<rbrakk> \<longrightarrow> \<lbrakk>E, F, G\<rbrakk> p"   and "finite (params X)"   and "infinite UNIV" shows "\<exists>ps. set ps \<subseteq> X \<and> (ps \<turnstile> p)"
lm129: fixes f :: "('c \<times> 'd) set"   and x :: "'c" assumes "runiq f"   and "x \<in> Domain f" shows "(f Else (0::'d)) x = toFunction f x"
MJ_def: shows "MJ = distr M borel (\<lambda>x n. U ((T ^^ n) x))"
not_negative_int_iff: fixes k :: "int" shows "(NOT k < 0) = (0 \<le> k)"
Nml_Trg: fixes t :: "'c local.term" assumes "Nml t" shows "Nml (Trg t)"
psubsetE: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subset> B" obtains "A \<subseteq> B" and "\<not> B \<subseteq> A"
set_child_nodes_get_shadow_root: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>Shadow_DOM.set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
OneState_HPT_Target: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and T :: "'d"   and SA :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<in> Target TS"   and "T \<in> Target TS"   and "S \<in> States SA"   and "T \<in> States SA"   and "SA \<in> SAs (HA ST)" shows "S = T"
mono: fixes "consider" :: "'e \<Rightarrow> bool"   and x :: "('d, 'e, 'f) gpv" shows "lfp.mono_body (\<lambda>interaction_bound. \<Squnion>generat\<in>set_spmf (the_gpv x). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound (c input)) else \<Squnion>input. interaction_bound (c input))"
ag_pOp_commute: fixes A :: "('e, 'f) aGroup_scheme"   and x :: "'e"   and y :: "'e" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y = y \<plusminus>\<^bsub>A\<^esub> x"
stermsl_ctermsl_choice2: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp"   and p1 :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> stermsl p2" shows "q \<in> ctermsl (p1 \<oplus> p2)"
Dirichlet_L_0_nonprincipal: assumes "\<chi> \<noteq> \<chi>\<^sub>0" shows "Dirichlet_L n \<chi> 0 = - (\<Sum>k = 1..<n. of_nat k * \<chi> k) / of_nat n"
restrict_fun_upd: fixes m :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b option"   and D :: "'a set" shows "m(x := y) |` D = (if x \<in> D then (m |` (D - {x}))(x := y) else m |` D)"
homomorphism: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "ap_spmf (return_spmf f) (return_spmf x) = return_spmf (f x)"
Rep_formula_var: fixes i :: "'a" shows "Rep_formula (var i) = {A. i \<in> A}"
antipar_2: shows "src\<^sub>D (map\<^sub>0 a) = trg\<^sub>D (\<tau>\<^sub>0 a)"
mat_mul_finsum_alt: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * B = mat\<^sub>r nr nc (\<lambda>i. finsum_vec TYPE('a) nc (\<lambda>k. A $$ (i, k) \<cdot>\<^sub>v row B k) {0..<n})"
IH_1: fixes x :: "perm" assumes "x \<bullet> t1 =\<^sub>\<alpha> t2" shows "S_transform_Tree (x \<bullet> t1) = S_transform_Tree t2"
gbinomial_asymptotic': fixes z :: "real"   and k :: "nat" assumes "z \<notin> \<nat>" shows "(\<lambda>n. z gchoose (n + k)) \<sim>[sequentially] (\<lambda>n. (- 1) ^ (n + k) / (Gamma (- z) * real n powr (z + 1)))"
elims: fixes x :: "op"   and xa :: "dynamic list"   and y :: "opinl option" assumes "inl x xa = y" obtains uu_ :: "nat" and uv_ :: "nat" where "x = OpAdd" and "xa = [DNum uu_, DNum uv_]" and "y = Some OpAddNum"   | uw_ :: "nat" and ux_ :: "nat" where "x = OpMul" and "xa = [DNum uw_, DNum ux_]" and "y = Some OpMulNum"   | uz_ :: "dynamic list" where "x = OpNeg" and "xa = uz_" and "y = None"   | "x = OpMul" and "xa = []" and "y = None"   | va :: "dynamic list" where "x = OpMul" and "xa = DNil # va__" and "y = None"   | vb :: "bool" and va :: "dynamic list" where "x = OpMul" and "xa = DBool vb__ # vaa__" and "y = None"   | v :: "dynamic" where "x = OpMul" and "xa = [v__]" and "y = None"   | v :: "dynamic" and vc :: "dynamic list" where "x = OpMul" and "xa = vc__ # DNil # vca__" and "y = None"   | v :: "dynamic" and vd :: "bool" and vc :: "dynamic list" where "x = OpMul" and "xa = vd__ # DBool vda__ # vcb__" and "y = None"   | v :: "dynamic" and vb :: "dynamic" and vd :: "dynamic" and ve :: "dynamic list" where "x = OpMul" and "xa = ve__ # vba__ # vdb__ # vea__" and "y = None"   | uy_ :: "op" where "x = uy___" and "xa = []" and "y = None"   | uy_ :: "op" and va :: "dynamic list" where "x = uya___" and "xa = DNil # vab__" and "y = None"   | uy_ :: "op" and vb :: "bool" and va :: "dynamic list" where "x = uyb___" and "xa = DBool vbb__ # va" and "y = None"   | uy_ :: "op" and v :: "dynamic" where "x = uyc___" and "xa = [vf__]" and "y = None"   | uy_ :: "op" and v :: "dynamic" and vc :: "dynamic list" where "x = uyd___" and "xa = vg__ # DNil # vcc__" and "y = None"   | uy_ :: "op" and v :: "dynamic" and vd :: "bool" and vc :: "dynamic list" where "x = uye___" and "xa = vh__ # DBool vdc__ # vc" and "y = None"   | uy_ :: "op" and v :: "dynamic" and vb :: "dynamic" and vd :: "dynamic" and ve :: "dynamic list" where "x = uy_" and "xa = v # vb # vd # ve" and "y = None"
dim_eq_card_independent: fixes B :: "'d set" assumes "cindependent B" shows "cdim B = card B"
distinct_168: fixes x151 :: "floatarith"   and x152 :: "nat"   and x6 :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Arctan x6"
absorb_iff2: fixes b :: "'a Word.word"   and a :: "'a Word.word" shows "(b \<le>s a) = (signed.min a b = b)"
dy_fake_msg_extr: fixes bad :: "agent set"   and ik :: "msg set"   and chan :: "chan set"   and X :: "msg set" assumes "extr bad ik chan \<subseteq> synth (analz X)" shows "dy_fake_msg bad ik chan \<subseteq> synth (analz X)"
map_fst_enumerate: fixes n :: "nat"   and xs :: "'c list" shows "map fst (List.enumerate n xs) = [n..<n + length xs]"
args_Nil_iff_is_Hd: fixes s :: "('a, 'b) tm" shows "(args s = []) = is_Hd s"
conjugate_vec_sprod_comm: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "v \<bullet>c w = conjugate w \<bullet> v"
encode_0: fixes base :: "nat"   and e :: "nat" assumes "2 \<le> base" shows "encode base e 0 = 0"
BSPTaxonomyDifferentCorrections_def: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec" shows "BSPTaxonomyDifferentCorrections ES \<V> \<equiv> ES_valid ES \<and> isViewOn \<V> E\<^bsub>ES\<^esub>"
assms_2: fixes n :: "nat" assumes "0 < n"   and "n < \<^bold>|w\<^bold>|" shows "rotate n w \<noteq> w"
prems_2: shows "\<S>\<bullet>\<C> \<turnstile> xs @ [x] : ts _> ts'"
pos: shows "if pos then \<phi>''' = \<phi>'' else \<phi>''' = formula.Neg \<phi>''"
example_OFSM_1: shows "OFSM M\<^sub>S"
scope_split: fixes v :: "'a"   and outPorts :: "'b \<Rightarrow> 'c fset"   and vertices :: "'a fset"   and nodeOf :: "'a \<Rightarrow> 'b"   and edges :: "(('a \<times> 'c) \<times> 'a \<times> 'd) set"   and ps :: "'a \<times> 'd"   and v' :: "'a"   and pth :: "(('a \<times> 'c) \<times> 'a \<times> 'd) list" assumes "v \<in> Pre_Port_Graph.scope outPorts vertices nodeOf edges ps"   and "Pre_Port_Graph.path edges v v' pth"   and "Vertex_Graph.terminal_vertex outPorts vertices nodeOf v'" obtains pth1 :: "(('a \<times> 'c) \<times> 'a \<times> 'd) list" and e :: "('a \<times> 'c) \<times> 'a \<times> 'd" and pth2 :: "(('a \<times> 'c) \<times> 'a \<times> 'd) list" where "pth = (pth1 @ [e]) @ pth2" and "Pre_Port_Graph.path edges v (fst ps) (pth1 @ [e])" and "Pre_Port_Graph.path edges (fst ps) v' pth2" and "snd e = ps" and "ps \<notin> snd ` set pth1"
fold_A_min_gallery_from1_map: fixes F :: "'a set"   and Fs :: "'a set list" assumes "chamber F"   and "F \<in> fold_A \<turnstile> A"   and "min_gallery (C # Fs @ [F])" shows "min_gallery (C # fold_A \<Turnstile> Fs @ [F])"
cSUP_upper: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c multiset" assumes "x \<in> A"   and "subset_mset.bdd_above (f ` A)" shows "f x \<subseteq># \<Squnion> (f ` A)"
one: shows "(1::'a) \<le> eval_poly \<alpha> q"
nn_integral_count_space_finite: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "finite A" shows "integral\<^sup>N (count_space A) f = sum f A"
exp_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "exp_aform_err p X = (let ivl = ivl_of_aform_err p X in min_range_mono p Exp Exp (lower ivl) (upper ivl) X)"
normalize_def: fixes x :: "'d" shows "Projections.normalize x = (1 / norm x) *\<^sub>R x"
two_in_unreach: fixes Q :: "'a set"   and b :: "'a" assumes "Q \<in> \<P>"   and "b \<in> \<E>"   and "b \<notin> Q" shows "\<exists>x\<in>\<emptyset> Q b. \<exists>y\<in>\<emptyset> Q b. x \<noteq> y"
is_monoid: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"