hcju/isabelleps huggingface dataset:
ResolveRacing_def: fixes TS :: "('e \<times> (('e, 'f) expr \<times> ('d data \<Rightarrow> bool) \<times> 'f set \<times> 'd pupdate) \<times> 'e) set" shows "ResolveRacing TS \<equiv> Let (Expr.PUpdate (Label TS)) SequentialRacing"
at_within_ball: fixes e :: "real"   and x :: "'b"   and y :: "'b" assumes "0 < e"   and "dist x y < e" shows "at y within ball x e = at y"
has_bochner_integral_erlang_ith_moment: fixes l :: "real"   and X :: "'a \<Rightarrow> real"   and k :: "nat"   and i :: "nat" assumes "0 < l"   and "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))" shows "has_bochner_integral M (\<lambda>x. X x ^ i) (fact (k + i) / (fact k * l ^ i))"
monom_altdef: fixes c :: "'c"   and n :: "nat" shows "monom c n = smult c ([:0::'c, 1::'c:] ^ n)"
content_pos_lt_eq: fixes a :: "'a"   and b :: "'a" shows "(0 < content (cbox a b)) = (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
assms_5: shows "\<xi>1 \<noteq> \<xi>2"
prj_char_1: shows "Maps.PRJ\<^sub>0 \<lbrakk>\<lbrakk>tab\<^sub>0 r\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 s\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>rs.p\<^sub>0\<rbrakk>\<rbrakk>"
freeIn_allzero: fixes var :: "nat"   and p :: "real mpoly" shows "freeIn var (allZero p var)"
g_deriv_on: shows "(g has_vderiv_on g') {a..b}"
midpoint_between: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "(b = midpoint a c) = (between (a, c) b \<and> dist a b = dist b c)"
pred_gpv_Fail: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "pred_gpv P Q Fail"
semilattice_order_axioms: shows "semilattice_order inf (\<le>) (<)"
fresh_eq: fixes x :: "char list"   and xs :: "char list set" assumes "x \<notin> xs" shows "fresh_string xs x = x"
surjective: fixes r :: "('c, 'd, 'e) SES_rec_scheme" shows "r = \<lparr>S_SES = S_SES r, s0_SES = s0_SES r, E_SES = E_SES r, I_SES = I_SES r, O_SES = O_SES r, T_SES = T_SES r, \<dots> = SES_rec.more r\<rparr>"
Q''Chain: shows "Q''[x::=y] \<Longrightarrow>\<^sub>\<tau> Q'"
D3: fixes \<psi> :: "\<nu> \<Rightarrow> \<o>" shows "d\<^sub>\<kappa> (\<^bold>\<iota>x. \<psi> x) = (if \<exists>x. [\<psi> x in w\<^sub>0] \<and> (\<forall>y. [\<psi> y in w\<^sub>0] \<longrightarrow> y = x) then Some (THE x. [\<psi> x in w\<^sub>0]) else None)"
composite_simps_3: shows "local.dom (\<pi> \<star> \<rho>) = local.dom \<pi> \<star> local.dom \<rho>"
dis_rews_5: fixes x :: "Ordering" shows "(is_EQ\<cdot>x = \<bottom>) = (x = \<bottom>)"
remove_fset_cases: fixes S :: "'b fset" shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = insert_fset x (remove_fset x S))"
max_id: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.non_strict_order less_eq"   and "less_eq y x" shows "ord.max less_eq x y = x"
rep_eq: fixes x :: "'f \<Rightarrow> 'd"   and xa :: "'f set!['e]" shows "set_nebset (map_nebset x xa) = x ` set_nebset xa"
new_msg_in_set_implies_occurrence: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and m :: "'c message"   and i :: "nat"   and p :: "'a"   and q :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "m \<notin> set (msgs c i)"   and "m \<in> set (msgs c' i)"   and "channel i = Some (p, q)" shows "occurs_on ev = p"
hyps_6: fixes xs :: "(nat \<times> nat) set" assumes "descendants k i ((p # ps, a) # branch) xs"   and "Nom k at j in branch" shows "A \<turnstile> mapi_branch (bridge k j xs) ((p # ps, a) # branch)"
contour_integral_nearby: fixes S :: "complex set"   and p :: "real \<Rightarrow> complex"   and atends :: "bool" assumes "open S"   and "path p"   and "path_image p \<subseteq> S" shows "\<exists>d>0. \<forall>g h. valid_path g \<and> valid_path h \<and> (\<forall>t\<in>{0..1}. cmod (g t - p t) < d \<and> cmod (h t - p t) < d) \<and> linked_paths atends g h \<longrightarrow> path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and> (\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f)"
HEndPhase2_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "HEndPhase2 s s' p = (EndPhase2 s s' p \<and> HNextPart s s')"
ordered_map_sound: fixes S :: "('a \<times> 'b) set"   and a :: "'a"   and b :: "'b" assumes "is_map S"   and "finite S"   and "(a, b) \<in> set (ordered_map S)" shows "(a, b) \<in> S"
not_sub_single: fixes A :: "'c set"   and a :: "'c" assumes "A \<noteq> {}"   and "A \<noteq> {a}" shows "\<not> A \<subseteq> {a}"
identity_functor_axioms: shows "identity_functor (\<cdot>\<^sub>A)"
max_reg_list_append_singleton: fixes as :: "vname gexp list"   and bs :: "vname gexp" shows "max_reg_list (as @ [bs]) = max (max_reg_list as) (max_reg_list [bs])"
ind: fixes st' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps0 (st, l, r) ap stp = (st', l', r')"   and "0 < st'" shows "steps0 (st, l, r) (adjust0 ap) stp = (st', l', r')"
override_on_bot_2: fixes m :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "m ++\<^bsub>S\<^esub> \<bottom> = m f|` (- S)"
left_right_inverse_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "class.monoid_mult one times"   and "times x y = one" shows "times (power.power one times x n) (power.power one times y n) = one"
ope: fixes i :: "'a" assumes "i \<in> I" shows "openin X (T i)"
partitioning1_disjoint: fixes ss :: "'b set list"   and ts :: "'b set list" assumes "\<Union> (set ss) \<subseteq> \<Union> (set ts)"   and "disjoint_list_rec ts" shows "disjoint_list_rec (partitioning1 ss ts)"
simps_2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "find_indices P (x # xs) = (if P x then 0 # map Suc (find_indices P xs) else map Suc (find_indices P xs))"
Confidentiality_Kas_lemma: fixes authK :: "nat"   and A :: "agent"   and evs :: "event list"   and Ta :: "nat" assumes "authK \<in> symKeys"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets"   and "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs"   and "Key authK \<in> analz (knows Spy evs)" shows "expiredAK Ta evs"
assert_tableau: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom"   and v :: "nat \<Rightarrow> 'a" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "let s' = assert a s in v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> s' \<and> \<triangle> (\<T> s') \<and> \<nabla> s'"
pnet_node_trans: fixes s :: "'c net_state"   and a :: "'d node_action"   and s' :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and i :: "nat"   and R :: "nat set" assumes "(s, a, s') \<in> automaton.trans (pnet np \<langle>i; R\<rangle>)" shows "(s, a, s') \<in> node_sos (automaton.trans (np i))"
qinc: shows "quality_increases (\<sigma> sip) (\<sigma>' sip)"
assms_1: shows "\<guillemotleft>f : a \<cong> b\<guillemotright>"
CE1pre_valid2: shows "valid (! (! (Geq (Differential (Var vid1)) (Differential (Const 0)) && Geq ($' vid1) (Const 0)) && ! (! (Geq (Differential (Var vid1)) (Differential (Const 0))) && ! (Geq ($' vid1) (Const 0)))))"
Dirichlet_kernel_cosine_sum: fixes x :: "real"   and n :: "nat" assumes "\<bar>x\<bar> < 2 * pi" shows "Dirichlet_kernel n x = 1 / 2 + (\<Sum>k = Suc 0..n. cos (real k * x))"
typ_of1_decr_gen: fixes Ts :: "typ list"   and T :: "typ"   and Ts' :: "typ list"   and t :: "Core.term"   and tyo :: "typ option" assumes "typ_of1 (Ts @ [T] @ Ts') t = tyo"   and "\<not> loose_bvar1 t (length Ts)" shows "typ_of1 (Ts @ Ts') (decr (length Ts) t) = tyo"
assms_1: shows "f|\<^bsub>r\<^esub> = g|\<^bsub>r\<^esub>"
rep_eq: fixes x :: "'g \<Rightarrow> 'f"   and xa :: "('g, 'e) generator" shows "generator (map_trans x xa) = map_raw x (generator xa)"
parallel_handler_right: fixes left :: "'r \<Rightarrow> 'u \<Rightarrow> 'r spmf"   and right :: "'s \<Rightarrow> 't \<Rightarrow> 's spmf" shows "map_fun id (map_fun Inr id) (parallel_handler left right) = (\<lambda>(s_l, s_r) q. map_spmf (Pair s_l) (right s_r q))"
arr_cod: fixes f :: "'a" assumes "L.arr f" shows "L.arr (L.cod f)"
K_def: shows "K = (\<lambda>p n. {x \<in> (T ^^ n) --` A. local_time A n x < p})"
RedStaticUpCast: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list"   and E :: "char list \<Rightarrow> ty option"   and a :: "nat"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>ref (a, Cs),s\<rangle> \<rightarrow> \<langle>ref (a, Ds),s\<rangle>"
isSame_equi_dev: fixes n :: "nat"   and V :: "nat set"   and A :: "nat set"   and t1 :: "'b option list"   and B :: "nat set"   and t2 :: "'b option list"   and s :: "nat set" assumes "wf_set n V"   and "wf_tuple n A t1"   and "wf_tuple n B t2"   and "s \<subseteq> A"   and "s \<subseteq> B"   and "A \<subseteq> V"   and "B \<subseteq> V" shows "New_max.isSameIntersection t1 s t2 = (restrict s t1 = restrict s t2)"
symcl_def: fixes Rel :: "('a \<times> 'a) set" shows "symcl Rel = Rel \<union> Rel\<inverse>"
nonneg_bounded: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_linear f" shows "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
list_before_irrefl_eq_distinct: shows "irrefl (list_before_rel l) = distinct l"
UINF_insert: fixes x :: "'c"   and xs :: "'c set"   and P :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Sqinter> i \<in> insert x xs \<bullet> P i) = P x \<sqinter> (\<Sqinter> i \<in> xs \<bullet> P i)"
slp_to_intra_path_with_slice_edges: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and S :: "'node SDG_node set"   and cs :: "'edge list" assumes "n -as\<rightarrow>\<^bsub>sl\<^esub>* n'"   and "slice_edges S cs as = []" obtains as' :: "'edge list" where "n -as'\<rightarrow>\<^sub>\<iota>* n'" and "slice_edges S cs as' = []"
join_binary: fixes x :: "'b"   and y :: "'b" shows "x \<squnion> y = \<Squnion>{x, y}"
circ_sub_dist: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> \<le> (x \<squnion> y)\<^sup>\<star>"
Ide_HcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Ide t"   and "Ide u"   and "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Ide (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u)"
butlast_noteq: fixes Cs :: "'a list" assumes "Cs \<noteq> []" shows "butlast Cs \<noteq> Cs"
acyclic_converse: fixes r :: "('a \<times> 'a) set" shows "acyclic (r\<inverse>) = acyclic r"
mem_Ch_CH: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and h :: "'f"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch"   and "x \<in> Ch h X" shows "x \<in> Contracts.CH Ch X"
Grover_def: shows "Grover = hadamard_n n;; While_P vars2 M0 M1 D;; Measure_P vars1 N testN (replicate N SKIP)"
aff_dim_affine_hull: fixes S :: "'c set" shows "aff_dim (affine hull S) = aff_dim S"
submodule_mOp_closed: fixes H :: "'a set"   and h :: "'a" assumes "submodule R M H"   and "h \<in> H" shows "-\<^sub>a h \<in> H"
mcont_if_bot: fixes lub' :: "'a set \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bound :: "'b"   and bot :: "'a"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.ccpo lub' ord (mk_less ord)"   and "\<And>x. \<not> bound \<le> x \<Longrightarrow> ord bot (f x)"   and "\<And>x. g x = (if bound \<le> x then bot else f x)"   and "\<And>x y. \<lbrakk>y \<le> x; \<not> bound \<le> x\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>Y. \<lbrakk>Complete_Partial_Order.chain (\<lambda>x y. y \<le> x) Y; Y \<noteq> {}; \<And>x. x \<in> Y \<Longrightarrow> \<not> bound \<le> x\<rbrakk> \<Longrightarrow> f (\<Sqinter> Y) = lub' (f ` Y)" shows "mcont Inf (\<lambda>x y. y \<le> x) lub' ord g"
pat_vars_ptyping: fixes p :: "pat"   and T :: "ty"   and \<Delta> :: "(name \<times> ty) list" assumes "\<turnstile> p : T \<Rightarrow> \<Delta>" shows "pat_vars p = map fst \<Delta>"
wcl1: shows "steps0 (Suc 0, [], <code p # args>) t_wcode stp = (0, [Bk], Oc \<up> Suc (code p) @ Bk # Oc \<up> Suc (bl_bin (<args>)) @ Bk \<up> rn)"
chain: shows "\<forall>i. (f i, f (Suc i)) \<in> r\<^sup>+"
assms_2: shows "\<exists>x\<in>I. (0::'b) < f x"
is_glb_def: fixes \<delta> :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'a"   and s :: "'a"   and s1 :: "'a"   and s2 :: "'a" shows "pre_RDR.is_glb \<delta> s s1 s2 \<equiv> pre_RDR.is_lb \<delta> s s1 s2 \<and> (\<forall>s'. pre_RDR.is_lb \<delta> s' s1 s2 \<longrightarrow> pre_RDR.less_eq \<delta> s' s)"
simps_2: fixes A :: "agent"   and ev :: "event"   and evs :: "event list" shows "knows A (ev # evs) = (if A = Spy then case ev of Says A' B X \<Rightarrow> insert X (knows Spy evs) | Gets A' X \<Rightarrow> knows Spy evs | Notes A' X \<Rightarrow> if A' \<in> bad then insert X (knows Spy evs) else knows Spy evs else case ev of Says A' B X \<Rightarrow> if A' = A then insert X (knows A evs) else knows A evs | Gets A' X \<Rightarrow> if A' = A then insert X (knows A evs) else knows A evs | Notes A' X \<Rightarrow> if A' = A then insert X (knows A evs) else knows A evs)"
Ana_invar_substD: fixes \<M> :: "('fun, 'var) Term.term set"   and f :: "'fun"   and T :: "('fun, 'var) Term.term list"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "Ana_invar_subst \<M>"   and "Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<M>"   and "Ana (Fun f T) = (K, M)" shows "Ana (Fun f T \<cdot> \<I>) = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<I>, M \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<I>)"
eq: shows "(if \<exists>y. snd (rep_nat_term y) = i then i else 0) = 0"
nontrans2: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and u :: "'a"   and r :: "'a" assumes "class.interval meets \<I>"   and "meets u r" shows "\<nexists>t. meets r t \<and> meets t u"
bot_apply: fixes x :: "'d" shows "\<bottom> x = \<bottom>"
simps_5: fixes R :: "('f, 'g, 'h) Resumption.resumption set" assumes "\<not> (\<forall>r\<in>R. Resumption.resumption.is_Done r)" shows "Resumption.resumption.resume (resumption_lub R) = (\<lambda>inp. resumption_lub ((\<lambda>c. c inp) ` Resumption.resumption.resume ` (R \<inter> {r. \<not> Resumption.resumption.is_Done r})))"
IH2_3: assumes "\<not> M \<Turnstile> G" shows "\<not> M(n := True) \<Turnstile> G"
subresultant_prs_def: fixes div_exp :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> 'b"   and f :: "'b poly"   and g :: "'b poly" shows "subresultant_prs div_exp f g = (let h = pseudo_mod f g; \<delta> = degree f - degree g; d = lead_coeff g ^ \<delta> in if h = 0 then (g, d) else subresultant_prs_main div_exp g ((- 1) ^ (\<delta> + 1) * h) d)"
gt_diff: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t"   and "wt t = wt s"   and "head t >\<^sub>h\<^sub>d head s" shows "t >\<^sub>t s"
ODnums_def: shows "ODnums = {X. \<exists>S. Worder S \<and> X = ordinal_number S}"
fresh_star_finite_insert: fixes S :: "'a set"   and a :: "atom set"   and x :: "'a" assumes "finite S" shows "a \<sharp>* insert x S = (a \<sharp>* x \<and> a \<sharp>* S)"
simps_1: fixes v :: "(('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> 'a" shows "\<Gamma> (Var v) = \<Gamma>\<^sub>v v"
add_le_cancel_right: fixes a :: "'b zmultiset"   and c :: "'b zmultiset"   and b :: "'b zmultiset" shows "(a + c \<subseteq>#\<^sub>z b + c) = (a \<subseteq>#\<^sub>z b)"
zero_reducibility_implies_lt_divisibility: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and f :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* f 0"   and "f \<noteq> 0" obtains h :: "'a \<Rightarrow>\<^sub>0 'c" where "h \<in> F" and "h \<noteq> 0" and "local.punit.adds_term (punit.lt h) (punit.lt f)"
B: shows "\<forall>k<length ns. Suc i \<le> k \<longrightarrow> 0 < offs_num (length ns) xs index key mi ma k \<longrightarrow> ns ! k + offs_num (length ns) xs index key mi ma k \<le> ub"
leq_refl: fixes x :: "'c spmf" shows "ord_spmf (=) x x"
isFDERIV_aform_def: fixes prec :: "nat"   and N :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and AS :: "(real \<times> real pdevs) list" shows "isFDERIV_aform prec N xs fas AS = isFDERIV_approx prec N xs fas (ivls_of_aforms prec AS)"
lfp_dual_gfp: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "lfp f = undual (gfp (dual \<circ> f \<circ> undual))"
CD_on_card: fixes ds :: "'d set"   and X :: "'x set" shows "card (CD_on ds X) = (\<Sum>d\<in>ds. card (Cd d X))"
assms: shows "\<forall>PROB as s. P PROB \<and> finite PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)"
linear_inj_imp_surj: fixes f :: "'a \<Rightarrow> 'a" assumes "linear f"   and "inj f" shows "surj f"
distinct_1: fixes x21 :: "'a"   and x22 :: "'a llist" shows "[]\<^sub>l \<noteq> x21 #\<^sub>l x22"
size_Diff_submset: fixes M :: "'c multiset"   and M' :: "'c multiset" assumes "M \<subseteq># M'" shows "size (M' - M) = size M' - size M"
simps: fixes guard :: "'a \<Rightarrow> bool"   and body :: "'a \<Rightarrow> 'a spmf"   and s :: "'a" shows "loop_spmf.while guard body s = (if guard s then body s \<bind> loop_spmf.while guard body else return_spmf s)"
AbGroup_subset_inner_dirsum: fixes Gs :: "'b set list"   and H :: "'b set" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "H \<in> set Gs" shows "H \<subseteq> (\<Oplus>G\<leftarrow>Gs. G)"
components_def: fixes S :: "'b set" shows "components S \<equiv> connected_component_set S ` S"
le_dbm_le: fixes a :: "'d"   and b :: "'d" assumes "Le a \<preceq> Le b" shows "a \<le> b"
BigOr_is_not_cnf'': fixes ls :: "'b formula list" assumes "is_cnf \<^bold>\<Or>ls" shows "\<forall>l\<in>set ls. is_lit_plus l"
better_LOr1_intro: fixes y :: "name"   and x1 :: "name"   and N1 :: "trm"   and x2 :: "name"   and N2 :: "trm"   and b :: "coname"   and a :: "coname"   and M :: "trm" assumes "y \<sharp> ([x1].N1, [x2].N2)"   and "b \<sharp> [a].M" shows "Cut <b>.OrR1 <a>.M b y.OrL x1.N1 x2.N2 y \<longrightarrow>\<^sub>l Cut <a>.M x1.N1"
size_list_estimation: fixes x :: "'c"   and xs :: "'c list"   and y :: "nat"   and f :: "'c \<Rightarrow> nat" assumes "x \<in> set xs"   and "y < f x" shows "y < size_list f xs"
assms_3: shows "set xvec \<subseteq> supp N"
case_1: fixes f1 :: "'a"   and f2 :: "'a" shows "(case \<zero> of \<zero> \<Rightarrow> f1 | \<one> \<Rightarrow> f2) = f1"
sup_subset_mset_count: fixes A :: "'k multiset"   and B :: "'k multiset"   and x :: "'k" shows "count (A \<union># B) x = max (count A x) (count B x)"
preserves_arr: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "B.arr (G f)"
proj_mcont2mcont: fixes lub :: "'a set \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> cnf llist"   and c :: "'id" assumes "mcont lub ord lSup lprefix f" shows "mcont lub ord lSup lprefix (\<lambda>x. \<pi>\<^bsub>c\<^esub>f x)"
verts3_in_slim_G: fixes x :: "'a" assumes "x \<in> verts3 (with_proj H)" shows "x \<in> pverts H.slim"
intros_2: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma>\<^sub>\<sigma> :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and state_update :: "'b \<Rightarrow> final_decision \<Rightarrow> 'c \<Rightarrow> 'b"   and \<sigma>\<^sub>0 :: "'b"   and built_in_chain :: "char list"   and default_policy :: "action"   and p :: "'c"   and ps :: "'c list"   and ps_processed :: "('c \<times> final_decision) list"   and \<sigma>' :: "'b"   and X :: "final_decision" assumes "semantics_stateful \<Gamma> \<gamma>\<^sub>\<sigma> state_update \<sigma>\<^sub>0 (built_in_chain, default_policy) (p # ps) ps_processed \<sigma>'"   and "\<Gamma>,\<gamma>\<^sub>\<sigma> \<sigma>',p\<turnstile> \<langle>[Rule MatchAny (Call built_in_chain), Rule MatchAny default_policy], Undecided\<rangle> \<Rightarrow> Decision X" shows "semantics_stateful \<Gamma> \<gamma>\<^sub>\<sigma> state_update \<sigma>\<^sub>0 (built_in_chain, default_policy) ps (ps_processed @ [(p, X)]) (state_update \<sigma>' X p)"
Int_insert_left: fixes a :: "'b"   and B :: "'b set"   and C :: "'b set" shows "insert a B \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
emeasure_count_space: fixes X :: "'a set"   and A :: "'a set" assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"
finite_Collect_le_nat: fixes k :: "nat" shows "finite {n. n \<le> k}"
IS_LEFT_TOTAL_alt: fixes R :: "('a \<times> 'b) set" shows "IS_LEFT_TOTAL R = (\<forall>x. \<exists>y. (x, y) \<in> R)"
less_imp_le: fixes x :: "'b list"   and y :: "'b list" assumes "x <s y" shows "x \<le>s y"
simps_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and u :: "nat"   and v :: "nat"   and w :: "nat"   and ws :: "nat list" assumes "class.linordered_ab_monoid_add plus zero less_eq less" shows "linordered_ab_monoid_add.len plus m u v (w # ws) = plus (m u w) (linordered_ab_monoid_add.len plus m w v ws)"
reindex: fixes h :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'d \<Rightarrow> 'a" assumes "inj_on h A" shows "F g (h ` A) = F (g \<circ> h) A"
lessaliveI: fixes s :: "Store"   and t :: "Store" assumes "\<And>x. alive x s \<Longrightarrow> s \<equiv>[x] t"   and "\<And>f. s@@staticLoc f = t@@staticLoc f" shows "s \<lless> t"
islimpt_subset: fixes x :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "x islimpt S"   and "S \<subseteq> T" shows "x islimpt T"
PAR_IMAGE_nres_def: fixes fi :: "'c \<Rightarrow> 'd nres"   and Xi :: "'c list" shows "local.PAR_IMAGE_nres fi Xi \<equiv> RETURN Xi \<bind> (\<lambda>xa. nres_of_nress_impl (Parallel.map (\<lambda>xb. fi xb \<bind> (\<lambda>xc. RETURN (xb, xc))) xa) \<bind> RETURN)"
spoly_alt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "spoly p q = fst (crit_pair p q) - snd (crit_pair p q)"
pell_valuation_solution_gt1D: fixes D :: "nat"   and z :: "int \<times> int" assumes "pell D"   and "pell.solution D z"   and "1 < pell.pell_valuation D z" shows "0 < fst z \<and> 0 < snd z"
yun_monic_factorization_int_yun_rel: fixes f :: "rat poly"   and res :: "(rat poly \<times> nat) list"   and F :: "int poly"   and Res :: "(int poly \<times> nat) list"   and c :: "rat" assumes "yun_gcd.yun_monic_factorization gcd f = res"   and "yun_gcd.yun_monic_factorization gcd F = Res"   and "yun_rel F c f" shows "list_all2 (rel_prod yun_erel (=)) Res res"
isCont_eq_Ub: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x" shows "\<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x\<ge>a. x \<le> b \<and> f x = M)"
uint_div: fixes x :: "'b word"   and y :: "'b word" shows "uint (x div y) = uint x div uint y"
find_first_first: fixes i :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "i < length xs"   and "x \<notin> set (take i xs)"   and "xs ! i = x" shows "find_first x xs = Some i"
empty_not_insert: fixes a :: "'c"   and A :: "'c set" shows "{} \<noteq> insert a A"