hcju/isabelleps huggingface dataset:
Can_red_1: fixes u :: "'a local.term" assumes "Ide u" shows "Can (u\<^bold>\<down>)"
d_indistinguishable_def: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and e1 :: "'a"   and e2 :: "'a" shows "WHATWHERE_Secure_Programs.d_indistinguishable E DA d e1 e2 \<equiv> \<forall>m m'. WHATWHERE.d_equal DA d m m' \<longrightarrow> E e1 m = E e2 m'"
is_pref_profile_update: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent" assumes "is_pref_profile R"   and "total_preorder_on alts Ri'"   and "i \<in> agents" shows "is_pref_profile (R(i := Ri'))"
ord_isom_minimum: fixes E :: "('c, 'd) Order_scheme"   and f :: "'a \<Rightarrow> 'c"   and S :: "'a set"   and a :: "'a" assumes "Order E"   and "ord_isom D E f"   and "S \<subseteq> carrier D"   and "a \<in> carrier D"   and "minimum_elem D S a" shows "minimum_elem E (f ` S) (f a)"
vimage_vsingleton_nin: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<notin>\<^sub>\<circ> A" shows "set {\<langle>a, b\<rangle>} `\<^sub>\<circ> A = 0"
cyclic: fixes ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and p :: "'a"   and q :: "'a"   and r :: "'a" assumes "ccw_system12 ccw S"   and "ccw p q r" shows "ccw q r p"
cond_a_conv: fixes ls :: "'b lock_actions"   and nts :: "('a, 'c, 'd) new_thread_action list"   and js :: "'a conditional_action list"   and wws :: "('a, 'e) wait_set_action list \<times> 'a interrupt_action list \<times> 'f list" shows "\<lbrace>(ls, nts, js, wws)\<rbrace>\<^bsub>c\<^esub> = js"
dvd_gcdD2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd normalization_euclidean_semiring_class.gcd b c" shows "a dvd c"
ND_F_dir2': fixes s :: "'c event list"   and P :: "'c process"   and S :: "'c process"   and Q :: "'c process" assumes "s \<notin> D P"   and "s \<in> T P"   and "P \<sqsubseteq> S"   and "Q \<sqsubseteq> S" shows "s \<in> T Q"
weakly_preferred_by_men_refl: fixes \<mu> :: "('m \<times> 'w) set" assumes "match \<mu>" shows "weakly_preferred_by_men \<mu> \<mu>"
halt_least_step: fixes l :: "nat"   and lm :: "'b"   and tp :: "(action \<times> nat) list"   and stp :: "nat"   and m :: "nat"   and rs :: "nat"   and n :: "nat" assumes "steps0 (Suc 0, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"   and "tm_wf0 tp"   and "0 < rs" shows "\<exists>stp. nonstop (code tp) (bl2wc (<lm>)) stp = 0 \<and> (\<forall>stp'. nonstop (code tp) (bl2wc (<lm>)) stp' = 0 \<longrightarrow> stp \<le> stp')"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" shows "trimming \<Gamma> f g = (current \<Gamma> g \<and> wave \<Gamma> g \<and> g \<le> f \<and> (\<forall>x. x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<longrightarrow> x \<notin> A \<Gamma> \<longrightarrow> KIR g x) \<and> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g) - A \<Gamma> = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) - A \<Gamma>)"
vconverse_iff: fixes b :: "V"   and a :: "V"   and r :: "V" shows "(\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
body_rewrite_fundef_body: fixes fd :: "'b fundef"   and n :: "nat"   and x :: "'b" shows "body (rewrite_fundef_body fd n x) = rewrite (body fd) n x"
spr_tLast: fixes a :: "'a"   and t :: "'s Trace"   and t' :: "'s Trace" assumes "spr_jview a t = spr_jview a t'" shows "envObs a (tLast t) = envObs a (tLast t')"
linepath_1': fixes a :: "'a"   and b :: "'a" shows "linepath a b 1 = b"
parPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto><Rel> Q"   and "(P, Q) \<in> Rel"   and "\<And>P Q R. (P, Q) \<in> Rel \<Longrightarrow> (P \<parallel> R, Q \<parallel> R) \<in> Rel'"   and "\<And>P Q a. (P, Q) \<in> Rel' \<Longrightarrow> (<\<nu>a>P, <\<nu>a>Q) \<in> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "P \<parallel> R \<leadsto><Rel'> Q \<parallel> R"
sum_mono2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> less_eq zero (f b)" shows "less_eq (comm_monoid_add.sum plus zero f A) (comm_monoid_add.sum plus zero f B)"
join_ti_hint2: fixes s :: "nat \<Rightarrow> 'a list"   and x :: "nat"   and i :: "nat" assumes "join_ti s x (Suc i) = []" shows "s (x + Suc i) = []"
sum_representation_eq: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set"   and v :: "'c"   and B :: "'c set" assumes "module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. scale (module.representation scale basis v b) b) = v"
a_shadow_root_valid_def: fixes h :: "(_) heap" shows "local.a_shadow_root_valid h = (\<forall>shadow_root_ptr\<in>fset (shadow_root_ptr_kinds h). \<exists>host\<in>fset (element_ptr_kinds h). |h \<turnstile> get_tag_name host|\<^sub>r \<in> safe_shadow_root_element_types \<and> |h \<turnstile> get_shadow_root host|\<^sub>r = Some shadow_root_ptr)"
assms: shows "G\<turnstile>C\<preceq>\<^sub>C D"
toplevel_summands_nonempty: fixes r :: "'c rexp" shows "toplevel_summands r \<noteq> {}"
that: fixes a :: "'a" assumes "a \<noteq> (0::'a)"   and "\<And>y. y \<in> S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"   and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y" shows "thesis"
return_ok: fixes h :: "'x"   and x :: "'z" shows "h \<turnstile> ok return x"
N_minus_commute: fixes x :: "int"   and y :: "int" shows "N (x - y) = N (y - x)"
SN_one_mono_ordered_semiring_1_axioms: shows "SN_one_mono_ordered_semiring_1 d (\<succ>)"
nonzero: fixes h :: "'a \<Rightarrow> real" assumes "h \<in> G" shows "\<forall>\<^sub>F x in F. h x \<noteq> 0"
G_def: shows "G = sample_uniform (order \<G>) \<bind> (\<lambda>w. return_spmf (\<^bold>g [^] w, w))"
that: fixes y :: "'b" assumes "y \<in> carrier"   and "x \<prec>[Pareto R] y" shows "thesis"
tendsto_ident_at: fixes "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and s :: "'a set" assumes "class.topological_space open" shows "LIM x topological_space.at_within open a s. x :> topological_space.nhds open a"
reachable_reaches: fixes a :: "'a"   and b :: "'a" assumes "reachable a"   and "reaches a b" shows "reachable b"
interior_real_atLeast: fixes a :: "real" shows "interior {a..} = {a<..}"
mrconcNone: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and x :: "'c"   and p :: "'b list"   and b :: "'b" assumes "applied_rule_rev C x p = \<bottom>" shows "applied_rule_rev C x (b # p) = applied_rule_rev C x [b]"
assms_1: shows "\<^bold>|r\<^bold>| < \<^bold>|w\<^bold>|"
jumpF_polyL_def: fixes q :: "real poly"   and p :: "real poly"   and a :: "real" shows "jumpF_polyL q p a = jumpF (\<lambda>x. poly q x / poly p x) (at_left a)"
lfp_fusion3_aux: fixes g :: "'d \<rightarrow> 'e"   and f :: "'d \<rightarrow> 'd"   and h :: "'e \<rightarrow> 'e"   and i :: "nat" assumes "g oo f = h oo g"   and "g\<cdot>\<bottom> = \<bottom>" shows "g\<cdot>(iterate i\<cdot>f\<cdot>\<bottom>) = iterate i\<cdot>h\<cdot>\<bottom>"
components_are_iso: fixes a :: "'a" assumes "hom'.ide a" shows "hom'.iso (\<psi>.map a)"
gfp_F_stable_on: fixes ds :: "'b set" shows "stable_on ds (match (gfp_F ds))"
simps_1: fixes v0 :: "(char list \<times> v) list" shows "is_cupcake_ns (Bind v0 []) = list_all (is_cupcake_value \<circ> snd) v0"
real_root_less_mono: fixes n :: "nat"   and x :: "real"   and y :: "real" assumes "0 < n"   and "x < y" shows "root n x < root n y"
edom_env_delete: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd" shows "edom (env_delete x m) = edom m - {x}"
WTrtFAssNT: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty"   and F :: "char list"   and D :: "char list" assumes "P,E,h,sh \<turnstile> e\<^sub>1 : NT"   and "P,E,h,sh \<turnstile> e\<^sub>2 : T\<^sub>2" shows "P,E,h,sh \<turnstile> e\<^sub>1\<bullet>F{D} := e\<^sub>2 : Void"
Abs_def: fixes xs :: "'varSort"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" shows "Abs xs x X \<equiv> asAbs (qAbs xs x (pick X))"
inv_all_uppertD: fixes n :: "nat"   and A :: "'b mat"   and j :: "nat"   and i :: "nat" assumes "??.Jordan_Normal_Form_Existence.inv_all n ??.Jordan_Normal_Form_Existence.uppert A"   and "j < i"   and "i < n" shows "A $$ (i, j) = (0::'b)"
R28_bc_1: shows "pmf (sds R28) b = 0"
1: shows "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} = lessThan ` {x<..} \<union> greaterThan ` {..<x}"
completion_def: fixes M :: "'b measure" shows "completion M = measure_of (space M) {uu_. \<exists>S N N'. uu_ = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'} (emeasure M \<circ> main_part M)"
Compl_eq: fixes A :: "'c set" shows "- A = {x. x \<notin> A}"
callee: fixes x :: "'a"   and s :: "'d" assumes "x \<in> outs_\<I> \<I>" shows "lossless_spmf (callee s x)"
the_\<nu>_props_2: shows "local.iso \<omega>.the_\<nu>"
1: shows "f ` set xs \<inter> f ` set (y # ys) = {}"
that: fixes G :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_Groebner_basis G"   and "ideal G = ideal F"   and "G \<subseteq> P[X]"   and "\<And>g. g \<in> G \<Longrightarrow> \<exists>q. g = (\<Sum>f\<in>F. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> Dube (Suc (card X)) (maxdeg F) \<and> (f \<notin> F \<longrightarrow> q f = 0))" shows "thesis"
vk: shows "v \<in> mat_kernel (char_matrix A lam)"
karatsuba_lower_bound_def: shows "karatsuba_lower_bound = 7"
llexord_LNil_right: fixes ys :: "'a llist"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a llist" assumes "lnull ys" shows "llexord r xs ys = lnull xs"
l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs \<equiv> get_disconnected_document = l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_document get_disconnected_nodes \<and> get_disconnected_document_locs = l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_document_locs get_disconnected_nodes_locs"
elims_3: fixes x :: "label"   and xa :: "label" assumes "\<not> x \<sqsubset>l xa" obtains l :: "label" where "x = Old" and "xa = l__" and "\<not> l__ \<noteq> Old"   | l :: "label" where "x = Processed" and "xa = la__" and "la__ \<noteq> New"   | l :: "label" where "x = New" and "xa = l"
mkChannels_def: fixes g :: "'b gState_scheme"   and p :: "pState"   and cs :: "channel list" shows "mkChannels g p cs = (if cs = [] then (g, p) else let l = length (gState.channels g) in if max_channels < l + length cs then abort STR ''Too much channels'' (\<lambda>_. (g, p)) else let cs\<^sub>p = map integer_of_nat [l..<l + length cs]; g' = g\<lparr>gState.channels := gState.channels g @ cs\<rparr> in Let (p\<lparr>pState.channels := pState.channels p @ cs\<^sub>p\<rparr>) (Pair g'))"
induced_imp_subgraph: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "induced_subgraph H G" shows "subgraph H G"
simps: fixes xs :: "'a list"   and n :: "nat"   and m :: "nat"   and a :: "'a array" shows "dyn_array_raw (xs, n) (Dyn_Array m a) = a \<mapsto>\<^sub>a xs * \<up> (m = n)"
ereal_Inf_finite_Min: fixes S :: "ereal set" assumes "finite S"   and "S \<noteq> {}" shows "Inf S = Min S"
atLeastLessThan_empty_iff: fixes a :: "'a"   and b :: "'a" shows "({a..<b} = {}) = (\<not> a < b)"
gt_arg: fixes s :: "('s, 'v) tm" assumes "wary s"   and "is_App s" shows "s >\<^sub>t arg s"
D: fixes \<nu> :: "'a" assumes "\<guillemotleft>\<nu> : v \<Rightarrow> g \<star> u\<guillemotright>" shows "trnl\<^sub>\<eta> v (trnl\<^sub>\<epsilon> u \<nu>) = \<nu>"
that: fixes r :: "V"   and A :: "V" assumes "x = \<langle>\<langle>r, A\<rangle>, r `\<^sub>\<circ> A\<rangle>"   and "r \<in>\<^sub>\<circ> R"   and "A \<in>\<^sub>\<circ> X" shows "thesis"
Inf_pres_radj: fixes g :: "'a \<Rightarrow> 'b" assumes "Inf_pres g" shows "\<exists>f. f \<stileturn> g"
tag_in_interval: fixes p :: "('a \<times> 'a set) set"   and i :: "'a set"   and x :: "'a"   and k :: "'a set" assumes "p tagged_division_of i"   and "(x, k) \<in> p" shows "x \<in> i"
rep_abs_b: fixes y :: "'bl" assumes "y \<in> \<UU>B" shows "y = Rep_b (Abs_b y)"
that: fixes w :: "complex" assumes "w \<in> frontier S"   and "\<And>z. z \<in> closure S \<Longrightarrow> cmod (f z) \<le> cmod (f w)" shows "thesis"
a_all_ptrs_in_heap_def: fixes get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_all_ptrs_in_heap get_shadow_root h = (\<forall>host shadow_root_ptr. h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr \<longrightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h)"
pell_solutions_def: fixes D :: "nat" shows "pell_solutions D = (let z = find_fund_sol D in siterate (pell_mul_nat D z) (1, 0))"
sim': shows "(ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b') \<sim> (ts', m', \<S>')"
chk_s't_st: assumes "s' >\<^sub>t s" shows "chksubs (>\<^sub>t) (App s' t) (App s t)"
series_and_derivative_comparison_complex: fixes S :: "complex set"   and f :: "nat \<Rightarrow> complex \<Rightarrow> complex"   and f' :: "nat \<Rightarrow> complex \<Rightarrow> complex" assumes "open S"   and "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod (f n y) \<le> cmod (h n))" shows "\<exists>g g'. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (\<lambda>n. f' n x) sums g' x \<and> (g has_field_derivative g' x) (at x)"
tailrec_implT_def: shows "tailrec_implT \<equiv> gds_init gds \<bind> FOREACH\<^sub>C\<^bsup>\<lambda>it s. gen_rwof s \<and> (\<not> gds_is_break gds s \<longrightarrow> gds_is_empty_stack gds s) \<and> V0 - it \<subseteq> gen_discovered s\<^esup> V0 ((Not \<circ>\<circ> gds_is_break) gds) (\<lambda>v0 s. let s0 = s in if gds_is_discovered gds v0 s then RETURN s else gds_new_root gds v0 s \<bind> WHILE\<^sub>T\<^bsup>\<lambda>s. gen_rwof s \<and> insert v0 (gen_discovered s0) \<subseteq> gen_discovered s\<^esup> (\<lambda>s. \<not> gds_is_break gds s \<and> \<not> gds_is_empty_stack gds s) tr_impl_while_body)"
assms_3: shows "[a ^ (n - 1) = 1] (mod n)"
simps_5: fixes b :: "bool" shows "isVal (Bool b) = True"
residue_of_diff: fixes b :: "nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and k :: "nat" assumes "b \<in> carrier Zp" shows "(a \<ominus> b) k = a k \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> b k"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "Left_a.in_hom f a b"   and "Left_a.in_hom g b c" shows "Left_a.seq g f"
small_these_tdghms: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V" shows "small {\<NN>. \<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>}"
nsuf_rev_pref_iff: fixes s :: "'b list"   and w :: "'b list" shows "s \<le>ns w = (rev s \<le>np rev w)"
ex_bij_betw_strict_mono_card: fixes M :: "V set" assumes "finite M"   and "M \<subseteq> ON" obtains h :: "nat \<Rightarrow> V" where "bij_betw h {..<card M} M" and "strict_mono_on h {..<card M}"
assms_5: shows "(ir, Delete is e2) \<in> set (node_deliver_messages xs)"
bij_betw_imp_funcset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "f \<in> A \<rightarrow> B"
ex_closed: fixes p :: "('a, 'b) form" shows "\<exists>m. closed m p"
iIN_conv: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] = [\<dots>d] \<oplus> n"
itrm_brackets_dist: fixes vs :: "nat list"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_brackets vs x = Some x'" shows "opaque_dist vs x' \<simeq>\<^sup>+ x"
mset_set_Diff: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "B \<subseteq> A" shows "mset_set (A - B) = mset_set A - mset_set B"
fold_remove1_split: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'b" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"   and "x \<in> set xs" shows "fold f xs = fold f (remove1 x xs) \<circ> f x"
obj_char: fixes a :: "'a arrow_of_spans_data" shows "obj a = (ide a \<and> a = mkObj (Chn a))"
mult_pCons_left: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "pCons a p * q = Polynomial.smult a q + pCons (0::'a) (p * q)"
consec_assoc_mult: fixes i2 :: "nat_int"   and i1 :: "nat_int"   and i3 :: "nat_int"   and i4 :: "nat_int" shows "(i2 = bot \<or> consec i1 i2) \<and> (i3 = bot \<or> consec i3 i4) \<and> consec (i1 \<squnion> i2) (i3 \<squnion> i4) \<longrightarrow> (i1 \<squnion> i2) \<squnion> (i3 \<squnion> i4) = (i1 \<squnion> (i2 \<squnion> i3)) \<squnion> i4"
case_distrib: fixes h :: "'d \<Rightarrow> 'c"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and f3 :: "'f \<Rightarrow> 'e \<Rightarrow> 'd"   and step :: "('f, 'e) step" shows "h (case step of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = (case step of Done \<Rightarrow> h f1 | Skip x \<Rightarrow> h (f2 x) | Yield x1 x2 \<Rightarrow> h (f3 x1 x2))"
Power: fixes y :: "ZF"   and x :: "ZF" shows "Elem y (Power x) = HOLZF.subset y x"
composite_functor_axioms: shows "composite_functor B.VV.comp (\<cdot>\<^sub>B) (\<cdot>) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>) UP"
maxsimpchainD_adj: fixes X :: "'b set set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimpchain X xs" shows "adjacentchain xs"
relto_fun_trans: fixes A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set"   and n1 :: "nat"   and as1 :: "nat \<Rightarrow> 'a"   and sel1 :: "nat \<Rightarrow> bool"   and m1 :: "nat"   and a :: "'a"   and b :: "'a"   and n2 :: "nat"   and as2 :: "nat \<Rightarrow> 'a"   and sel2 :: "nat \<Rightarrow> bool"   and m2 :: "nat"   and c :: "'a" assumes "relto_fun A B n1 as1 sel1 m1 (a, b)"   and "relto_fun A B n2 as2 sel2 m2 (b, c)" shows "\<exists>as sel. relto_fun A B (n1 + n2) as sel (m1 + m2) (a, c)"
multrow_i: shows "((=) ===> R ===> mat_rel R ===> mat_rel R) (multrow_i ops) multrow"
split_sel: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b"   and llist :: "'c llist" shows "P (case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = ((llist = LNil \<longrightarrow> P f1) \<and> (llist = LCons (lhd llist) (ltl llist) \<longrightarrow> P (f2 (lhd llist) (ltl llist))))"
tauChainAppend: fixes P :: "ccs"   and P' :: "ccs"   and P'' :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "P' \<Longrightarrow>\<^sub>\<tau> P''" shows "P \<Longrightarrow>\<^sub>\<tau> P''"
in_snapshotI_1: fixes r :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "black r s" shows "in_snapshot r s"
left_unit: fixes a :: "'a" assumes "a \<in> R" shows "\<one> \<cdot> a = a"
submatrix_split2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and A :: "'c mat"   and I :: "nat set"   and J :: "nat set" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "submatrix A I J = submatrix (submatrix A I UNIV) UNIV J"
MClkReply_enabled: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState" assumes "basevars (rtrner send!p, caller rcv!p, cst!p)" shows "\<turnstile> Calling send p \<and> \<not> Calling rcv p \<and> cst!p = #clkB \<longrightarrow> Enabled (<MClkReply send rcv cst p>_(cst!p, rtrner send!p, caller rcv!p))"
vfsequence_at_last: fixes i :: "V"   and x :: "V" assumes "i = vcard xs\<^sub>2" shows "(xs\<^sub>2 #\<^sub>\<circ> x)\<lparr>i\<rparr> = x"
im_set_un2: fixes A :: "'e set"   and A1 :: "'e set"   and A2 :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "A = A1 \<union> A2" shows "f ` A = f ` A1 \<union> f ` A2"
norm_cmul_rule_thm: fixes x :: "'a"   and b :: "real"   and c :: "real" assumes "norm x \<le> b" shows "norm (c *\<^sub>R x) \<le> \<bar>c\<bar> * b"
edge_obs_intra_slice_eq: fixes a :: "'edge"   and S :: "'node SDG_node set" assumes "valid_edge a"   and "intra_kind (kind a)"   and "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "obs_intra (targetnode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
disjnt_ba: fixes i :: "nat"   and j :: "nat"   and k :: "nat"   and r :: "nat" assumes "i < j"   and "j \<le> k" shows "disjnt (b k (j, i)) (a r)"
bchoice_iff: fixes S :: "'c set"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<forall>x\<in>S. \<exists>y. Q x y) = (\<exists>f. \<forall>x\<in>S. Q x (f x))"
delete_edges_def: fixes G :: "'a graph"   and es :: "('a \<times> 'a) set" shows "delete_edges G es \<equiv> \<lparr>nodes = nodes G, edges = {(e1, e2). (e1, e2) \<in> edges G \<and> (e1, e2) \<notin> es}\<rparr>"
lcm_dvdD1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "lcm a b dvd c" shows "a dvd c"
exec_to_exec_sequence_flatten: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>sequence Seq (flatten c),s\<rangle> \<Rightarrow> t"
ord_option_map1: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and x :: "'e option"   and y :: "'d option" shows "ord_option ord (map_option f x) y = ord_option (\<lambda>x. ord (f x)) x y"
empty_neutral_2: fixes x :: "'a multiset" shows "x + \<Empt> = x"
axioms_3: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "equivalence A equiv" shows "transitive A equiv"
I_nnf: fixes \<phi> :: "'a fm"   and xs :: "'b list" shows "I (nnf \<phi>) xs = I \<phi> xs"
is_maxsplitter_def: fixes P :: "'b list \<Rightarrow> bool"   and f :: "'b list \<Rightarrow> 'b list \<times> 'b list" shows "is_maxsplitter P f = (\<forall>xs ps qs. (f xs = (ps, qs)) = (xs = ps @ qs \<and> is_maxpref P ps xs))"