hcju/isabelleps huggingface dataset:
some_src_in_sources: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_src \<mu> \<in> sources \<mu>"
i2: assumes "a \<sharp> s1" shows "a \<sharp> s2"
pure_state_self_outer_prod_is_partial_density_operator: fixes v :: "complex vec"   and n :: "nat" assumes "v \<in> carrier_vec n"   and "vec_norm v = 1" shows "partial_density_operator (outer_prod v v)"
cinfinite_csum: fixes r1 :: "('p \<times> 'p) set"   and r2 :: "('q \<times> 'q) set" assumes "cinfinite r1 \<or> cinfinite r2" shows "cinfinite (r1 +c r2)"
sortnMTQ: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "p \<noteq> []" shows "qsort p l \<noteq> []"
obsp_contradict: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat" assumes "cs\<^bsup>path \<sigma>\<^esup> k = cs\<^bsup>path \<sigma>'\<^esup> k'"   and "obsp \<sigma> k \<noteq> obsp \<sigma>' k'" shows "(\<sigma>', k') \<cc> (\<sigma>, k)"
observable_move_intra: fixes f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a :: "'edge"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and ms :: "'node list"   and S :: "'node SDG_node set"   and ms' :: "'node list" assumes "pred (f a) s"   and "transfer (f a) s = s'"   and "valid_edge a"   and "intra_kind (kind a)"   and "\<forall>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "hd ms \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "length s' = length s"   and "length ms = length s"   and "hd ms = sourcenode a"   and "ms' = targetnode a # tl ms" shows "S,f \<turnstile> (ms,s) -a\<rightarrow> (ms',s')"
IH: assumes "suffix vs' (v # vs)"   and "vs' \<noteq> []" shows "liveVal' g vs'"
add: shows "\<forall>v w. v \<in> carrier M \<and> w \<in> carrier M \<longrightarrow> v \<oplus>\<^bsub>M\<^esub> w \<in> carrier M"
int_pow_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and i :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> i \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
eq_add_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.ab_group_add plus zero minus uminus" shows "(x = plus x y) = (y = zero)"
iso_cancel_left: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "local.iso f"   and "f \<cdot> g = f \<cdot> g'"   and "seq f g" shows "g = g'"
DERIV_mirror: fixes f :: "real \<Rightarrow> real"   and y :: "real"   and x :: "real" shows "(f has_real_derivative y) (at (- x)) = ((\<lambda>x. f (- x)) has_real_derivative - y) (at x)"
q_rep_t: shows "q = replicate (num_args (fun t)) dir.Left"
l9_38: 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 P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TSP Bet A B C P Q" shows "Tarski_neutral_dimensionless.TSP Bet A B C Q P"
if_True: fixes x :: "'a"   and y :: "'a" shows "(if True then x else y) = x"
t_s: shows "t \<succeq>[\<R>] mix_pmf \<gamma> t u \<and> (mix_pmf \<gamma> t u, t) \<notin> \<R>"
mu_help_2: fixes u :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and h :: "'b \<Rightarrow> real"   and n :: "nat" assumes "\<And>n. (\<lambda>m. u m n)\<up>f n"   and "f\<up>h" shows "mu u n \<le> f n"
Suc_llength: fixes n :: "nat"   and xs :: "'a llist" assumes "enat (Suc n) < llength xs" shows "enat n < llength xs"
set_of_vector_def: fixes A :: "('b, 'c) vec" shows "set_of_vector A = {A $ i |i. i \<in> UNIV}"
IH: fixes xs :: "'a list"   and zs :: "'a list" assumes "filter P xs = ys @ zs" shows "\<exists>us vs. xs = us @ vs \<and> filter P us = ys \<and> filter P vs = zs"
finite_reach_af_opt: fixes \<Sigma> :: "'b set set"   and \<phi> :: "'b ltl" shows "finite (reach \<Sigma> \<up>af\<^sub>\<UU> (Abs \<phi>))"
integral_pbernpoly_1: fixes s :: "complex" assumes "s \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "(\<lambda>N. integral {0..real N} (\<lambda>x. complex_of_real (pbernpoly 1 x) / (complex_of_real x + s))) \<longlonglongrightarrow> - ln_Gamma s - s + (s - 1 / 2) * Ln s + complex_of_real (ln (2 * pi) / 2)"
simps_1: fixes ks :: "int list"   and l :: "int"   and xs :: "nat list" shows "eval_pf (Eq ks l) xs = (eval_dioph ks xs = l)"
same_length_different: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> ys"   and "length xs = length ys" shows "\<exists>pre x xs' y ys'. x \<noteq> y \<and> xs = pre @ [x] @ xs' \<and> ys = pre @ [y] @ ys'"
lcoeff_nonzero_deg: fixes p :: "nat \<Rightarrow> 'a" assumes "deg R p \<noteq> 0"   and "p \<in> carrier P" shows "lcoeff p \<noteq> \<zero>"
combine_is_t_des: shows "t_design \<V>\<^sup>+ \<B>\<^sup>+ \<k> \<t> (\<Lambda>\<^sub>t + \<Lambda>\<^sub>t')"
Labels_Call: fixes p :: "char list"   and es :: "expr list"   and rets :: "char list list" shows "labels (Call p es rets) 1 Skip"
getFr_FvarsT: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and t :: "'trm" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla"   and "t \<in> set ts" shows "getFr xs ts \<phi>s \<notin> FvarsT t"
join_pmf_def: fixes M :: "'b pmf pmf" shows "join_pmf M = M \<bind> (\<lambda>x. x)"
00: shows "rawpsubst (rawpsubst \<phi> (zip (map Var uus) (map snd (tx # txs)))) (zip (map fst (tx # txs)) uus) = rawpsubst (subst (rawpsubst \<phi> (zip (map Var us) (map snd txs) @ [(Var u, x)])) t u) (zip (map fst txs) us)"
inf_sup_aci_2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice inf less_eq less sup" shows "inf (inf x y) z = inf x (inf y z)"
cset_count_def: fixes A :: "'c cset" shows "cset_count A = (if finite (rcset A) then SOME f. inj_on f (rcset A) else SOME f. bij_betw f (rcset A) UNIV)"
chart_homeomorphism: fixes c :: "('a, 'b) chart" shows "homeomorphism (domain c) (codomain c) (apply_chart c) (inv_chart c)"
order_l: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a l\<rightarrow> b = (1::'a))"
exp_series_add_commuting: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x * y = y * x" shows "(x + y) ^ n /\<^sub>R fact n = (\<Sum>i\<le>n. x ^ i /\<^sub>R fact i * (y ^ (n - i) /\<^sub>R fact (n - i)))"
Pxy: shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"
leI: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "\<not> x \<prec> y" shows "y \<preceq> x"
from_hma_to_hma\<^sub>v: fixes v :: "'d Matrix.vec" assumes "v \<in> carrier_vec CARD('e)" shows "Mod_Type_Connect.from_hma\<^sub>v (Mod_Type_Connect.to_hma\<^sub>v v) = v"
one_in_kernel: shows "\<one>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> \<in> kernel (G\<lparr>carrier := generate G {a}\<rparr>) (H\<lparr>carrier := generate H {b}\<rparr>) x"
finite_extension_is_subfield: fixes K :: "'a list set"   and xs :: "'a list list" assumes "subfield K (A [X])"   and "set xs \<subseteq> carrier (A [X])"   and "\<And>x. x \<in> set xs \<Longrightarrow> (UP_A.algebraic over K) x" shows "subfield (UP_A.finite_extension K xs) (A [X])"
lookup_RBT: fixes t :: "('b, 'c) RBT_Impl.rbt" assumes "is_rbt t" shows "RBT.lookup (RBT t) = rbt_lookup t"
int_msb_not: fixes x :: "int" shows "msb (NOT x) = (\<not> msb x)"
downset_set_prop_var2: fixes x :: "'a"   and X :: "'a set"   and y :: "'a" assumes "x \<in> \<Down> X"   and "y \<le> x" shows "y \<in> \<Down> X"
OINV: shows "cscc_outer_invar it (l, D0)"
prefix_match_semantics_wordset: fixes pfx :: "'b prefix_match"   and a :: "'b word" assumes "valid_prefix pfx" shows "prefix_match_semantics pfx a = (a \<in> prefix_to_wordset pfx)"
leD: fixes y :: "'b zmultiset"   and x :: "'b zmultiset" assumes "y \<subseteq>#\<^sub>z x" shows "\<not> x \<subset>#\<^sub>z y"
subset_\<Q>\<P>Suc: fixes k :: "nat"   and u :: "nat" shows "\<Q> k \<subseteq> \<P> (Suc k) u"
simps: fixes ly :: "nat list"   and x :: "nat" shows "start_of ly x = Suc (sum_list (take x ly))"
ipcidr_tuple_to_wordinterval_def: fixes iprng :: "'a word \<times> nat" shows "ipcidr_tuple_to_wordinterval iprng \<equiv> iprange_interval (ipcidr_to_interval iprng)"
select_result_E: fixes P :: "'aq \<Rightarrow> bool"   and h :: "'as"   and f :: "('as, 'ar, 'aq) prog" assumes "P |h \<turnstile> f|\<^sub>r"   and "h \<turnstile> ok f" obtains x :: "'aq" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "P x"
eSuc_mono: fixes n :: "enat"   and m :: "enat" shows "(eSuc n < eSuc m) = (n < m)"
parallel_def: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs \<parallel> ys) = (\<not> prefix xs ys \<and> \<not> prefix ys xs)"
path_connected_def: fixes S :: "'b set" shows "path_connected S = (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>g. path g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y)"
wfp_to_wffp: fixes sl :: "('a, 'b, 'c) state list"   and i :: "nat"   and \<pi> :: "(('a, 'b, 'c) state \<times> ('b, 'c, 'd) Noninterference.aprop set) stream"   and do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and out :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "sl = map fst (stake i \<pi>)"   and "0 < i"   and "Kripke.wfp UNIV (GM_sec_model.\<delta> do) UNIV (GM_sec_model.L out) UNIV \<pi>" shows "GM_sec_model.wffp do sl \<and> (\<forall>j<length sl. fst (\<pi> !! j) = sl ! j) \<and> stateOf \<pi> = hd sl"
pred_inject_2: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and aa :: "'a ttree list tree" shows "pred_ttree P (ttree.TNode a aa) = (P a \<and> pred_tree (list_all (pred_ttree P)) aa)"
0: shows "(0::'a) \<notin> set (diag_mat A)"
axioms_2: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "ideal I R" shows "ring R"
term_p_subid: fixes x :: "('a \<times> 'a set) set" shows "(x \<cdot> 1\<^sub>\<pi> = x) = (x \<subseteq> 1\<^sub>\<pi>)"
update_convs_2: fixes IntMapK' :: "rat option \<Rightarrow> rat option"   and NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a" shows "IntMapK_update IntMapK' \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = \<lparr>NonceS = NonceS, IntMapK = IntMapK' IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr>"
set_iterator_foldli_conv: fixes iti :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b set" shows "set_iterator iti S = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> iti = foldli l0)"
simps_5: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp"   and e2 :: "exp" shows "FV (EPrim f e1 e2) = FV e1 \<union> FV e2"
map_ident: fixes t :: "'e \<Rightarrow> 'f" shows "(\<lambda>x. x) \<circ> t = t"
less_imp_Suc_add: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "\<exists>k. n = Suc (m + k)"
gen_rwof_eq_assert: assumes "nofail gen_dfs" shows "gen_rwof = gen_rwof_assert"
funion_fempty_right: fixes A :: "'m fset" shows "A |\<union>| {||} = A"
ex_sym1: fixes A :: "'p"   and B :: "'p"   and X :: "'p" assumes "A \<noteq> B" shows "\<exists>Y. (A B Perp X Y \<or> X = Y) \<and> (\<exists>M. Col A B M \<and> M Midpoint X Y \<and> X Y Reflect A B)"
WHILEIT_eq_WHILEI: fixes I :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres"   and s0 :: "'b" assumes "WHILE\<^sub>T\<^bsup>I\<^esup> b f s0 \<noteq> FAIL" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f s0 = WHILE\<^bsup>I\<^esup> b f s0"
converse_converse: fixes r :: "('c \<times> 'd) set" shows "(r\<inverse>)\<inverse> = r"
c_tr_append: fixes step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s :: "'c"   and xs :: "'a list"   and ys :: "'a list" shows "c_tr step out s (xs @ ys) = c_tr step out s xs @ c_tr step out (foldl step s xs) ys"
simps_7: fixes bs :: "real list"   and c :: "int"   and a :: "MIR.num" shows "Inum bs (Mul c a) = real_of_int c * Inum bs a"
r: shows "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in>{N<..r (Suc n)}. u (r (Suc n)) \<le> u i)"
hyps_26: shows "A\<^sub>Q \<sharp>* Q"
discriminant_negative: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" assumes "a \<noteq> 0"   and "discrim a b c < 0" shows "a * x\<^sup>2 + b * x + c \<noteq> 0"
assms: shows "T \<in> set\<^sub>v (tensors_from_net m)"
pseudominimal_map: fixes x :: "'a" assumes "domain.has_bottom P"   and "domain.pseudominimal_in P x" shows "codomain.pseudominimal_in (f ` P) (f x)"
le_cSup_finite: fixes X :: "'a multiset set"   and x :: "'a multiset" assumes "finite X"   and "x \<in> X" shows "x \<subseteq># Sup X"
complete: fixes B :: "'a set"   and A :: "'a set"   and M :: "'a measure" assumes "B \<subseteq> A"   and "A \<in> null_sets (completion M)" shows "B \<in> sets (completion M)"
scale_scale: fixes a :: "'c"   and b :: "'c"   and x :: "'c" shows "a * (b * x) = a * b * x"
visits_deadend: fixes p :: "Player" assumes "lset P \<inter> deadends p \<noteq> {}" shows "winning_path p** P"
l10_4_spec: fixes P :: "'p"   and P' :: "'p"   and A :: "'p"   and B :: "'p" assumes "P P' ReflectL A B" shows "P' P ReflectL A B"
fold_Mapping_filter_None: fixes ts :: "('b, 'a) mapping"   and as :: "'b"   and ds :: "('a \<times> 'b set) list" assumes "Mapping.lookup ts as = None" shows "Mapping.lookup (fold (\<lambda>(t, X) ts. Mapping.filter (filter_cond X ts t) ts) ds ts) as = None"
permute_fun_def: fixes p :: "'b perm"   and f :: "'b \<Rightarrow> 'b" shows "atom_fun_permute.permute_fun p f = (\<lambda>x. permute_atom p (f (permute_atom (- p) x)))"
assert_bound_tableau_id: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and a :: "'a \<times> 'b atom" assumes "AssertBound assert_bound"   and "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<T> (assert_bound a s) = \<T> s"
simps_1: fixes \<mu> :: "'a" shows "\<^bold>\<lfloor>\<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0\<^bold>\<rfloor> = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
cis_inj_on: shows "inj_on cis {0..<2 * pi}"
finfun_dom_finfunI: fixes f :: "('a, 'b) finfun" shows "(\<lambda>a. finfun_apply f a \<noteq> finfun_default f) \<in> finfun"
poly_prod: fixes p :: "'c \<Rightarrow> 'b poly"   and A :: "'c set"   and x :: "'b" shows "poly (prod p A) x = (\<Prod>k\<in>A. poly (p k) x)"
assms: shows "\<forall>c\<in>set r. u c = d"
distinct_30: fixes x51 :: "trm"   and x52 :: "POPLmarkRecord.type"   and x31 :: "POPLmarkRecord.type"   and x32 :: "trm" shows "x51 \<bullet>\<^sub>\<tau> x52 \<noteq> (\<lambda><:x31. x32)"
insert_Un: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "insert x (A \<union> B) = insert x A \<union> insert x B"
dropWhile_cong: fixes l :: "'b list"   and k :: "'b list"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> set l \<Longrightarrow> P x = Q x" shows "dropWhile P l = dropWhile Q k"
lang_REV: fixes n :: "nat"   and r :: "'b rexp" shows "lang n (REV r) = rev ` lang n r"
extend_guarantees_eq: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program"   and X :: "'c program set"   and Y :: "'c program set" assumes "Extend h" shows "(extend h F \<in> extend h ` X guarantees extend h ` Y) = (F \<in> X guarantees Y)"
homeomorphic_ball01_UNIV: shows "ball (0::'a) 1 homeomorphic UNIV"
nres_more_simps_4: fixes X :: "'c set"   and Y :: "'c set" shows "(RES X = RES Y) = (X = Y)"
exhaust: fixes y :: "event" obtains "y = a"   | "y = b"
space_restrict_space2: fixes \<Omega> :: "'c set"   and M :: "'c measure" assumes "\<Omega> \<in> sets M" shows "space (restrict_space M \<Omega>) = \<Omega>"
min_absorb2: fixes y :: "'d"   and x :: "'d" assumes "y \<le> x" shows "min x y = y"
assms_1: shows "chain (\<leadsto>LGC) Ns"
fset_induct_stronger: fixes P :: "'b fset \<Rightarrow> bool"   and S :: "'b fset" assumes "P {||}"   and "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (insert_fset x S)" shows "P S"
s_eq_aux: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "\<not> (\<forall>k\<le>n. \<phi> (the (eval t [f \<triangleright> n])) k = f k)"   and "f \<in> U" shows "eval s [f \<triangleright> n] = eval r_auxhyp [f \<triangleright> n]"
select_convs_1: fixes access :: "acc_modi"   and cfields :: "fdecl list"   and methods :: "mdecl list"   and init :: "stmt"   and more :: "'b" shows "cfields \<lparr>access = access, cfields = cfields, methods = methods, init = init, \<dots> = more\<rparr> = cfields"
dist_fps_sym: fixes a :: "'a fps"   and b :: "'a fps" shows "dist a b = dist b a"
f1: shows "x \<cdot> d y \<cdot> y = x \<cdot> y"
is_max: fixes \<aa> :: "'a set" assumes "ideal \<aa> R (+) (\<cdot>) \<zero> \<one>"   and "\<aa> \<noteq> R"   and "I \<subseteq> \<aa>" shows "I = \<aa>"
next_recv_marker: fixes c :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and c' :: "('a, 'b, 'c) configuration" shows "c \<turnstile> RecvMarker i p q \<mapsto> c' = (can_occur (RecvMarker i p q) c \<and> (\<forall>r. states c r = states c' r) \<and> (\<forall>r. r \<noteq> p \<longrightarrow> ps c r = ps c' r) \<and> Marker # msgs c' i = msgs c i \<and> cs c' i = (fst (cs c i), Done) \<and> (if ps c p \<noteq> None then ps c p = ps c' p \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> cs c i' = cs c' i') else ps c' p = Some (states c p) \<and> (\<forall>i'. i' \<noteq> i \<and> (\<exists>r. channel i' = Some (p, r)) \<longrightarrow> msgs c' i' = msgs c i' @ [Marker]) \<and> (\<forall>i'. i' \<noteq> i \<and> (\<exists>r. channel i' = Some (r, p)) \<longrightarrow> cs c' i' = (fst (cs c i'), Recording)) \<and> (\<forall>i'. i' \<noteq> i \<and> (\<nexists>r. channel i' = Some (p, r)) \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>i'. i' \<noteq> i \<and> (\<nexists>r. channel i' = Some (r, p)) \<longrightarrow> cs c' i' = cs c i')))"
expand: fixes tree :: "'b tree"   and tree' :: "'b tree" assumes "(tree = Leaf) = (tree' = Leaf)"   and "\<lbrakk>tree \<noteq> Leaf; tree' \<noteq> Leaf\<rbrakk> \<Longrightarrow> l tree = l tree' \<and> r tree = r tree' \<and> val tree = val tree' \<and> lchild tree = lchild tree' \<and> rchild tree = rchild tree'" shows "tree = tree'"
closedin_Un: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "closedin U S"   and "closedin U T" shows "closedin U (S \<union> T)"
generated_ring_disjoint_Union: fixes A :: "'a set set" assumes "finite A"   and "A \<subseteq> generated_ring"   and "disjoint A" shows "\<Union> A \<in> generated_ring"
ext_inject: fixes hostsL :: "'b list"   and flows_fixL :: "('b \<times> 'b) list"   and flows_stateL :: "('b \<times> 'b) list"   and more :: "'c"   and hostsL' :: "'b list"   and flows_fixL' :: "('b \<times> 'b) list"   and flows_stateL' :: "('b \<times> 'b) list"   and more' :: "'c" shows "(\<lparr>hostsL = hostsL, flows_fixL = flows_fixL, flows_stateL = flows_stateL, \<dots> = more\<rparr> = \<lparr>hostsL = hostsL', flows_fixL = flows_fixL', flows_stateL = flows_stateL', \<dots> = more'\<rparr>) = (hostsL = hostsL' \<and> flows_fixL = flows_fixL' \<and> flows_stateL = flows_stateL' \<and> more = more')"
non_empty_mat_mult: fixes nr :: "nat"   and n :: "nat"   and A :: "'a list list"   and nc :: "nat"   and B :: "'a list list" assumes "mat nr n A"   and "mat n nc B"   and "A \<noteq> []"   and "B \<noteq> []" shows "A \<circ> B \<noteq> []"
iface_rec: fixes G :: "prog"   and I :: "qtname"   and i :: "iface"   and f :: "qtname \<Rightarrow> iface \<Rightarrow> 'a set \<Rightarrow> 'a" assumes "iface G I = Some i"   and "ws_prog G" shows "iface_rec G I f = f I i ((\<lambda>J. iface_rec G J f) ` set (isuperIfs i))"
big_mono: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. norm (f x) \<le> norm (g x)" shows "f \<in> O[F](g)"
continuous_map_fst: fixes X :: "'a topology"   and Y :: "'b topology" shows "continuous_map (prod_topology X Y) X fst"
has_fps_expansion_def: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps" shows "f has_fps_expansion F = (0 < fps_conv_radius F \<and> (\<forall>\<^sub>F z in nhds (0::'a). eval_fps F z = f z))"
h1: shows "(\<Sum>i\<le>0. isolate_variable_sparse p var i * Var var ^ i) = isolate_variable_sparse p var 0 * Var var ^ 0"
LIMSEQ_decseq_INF: fixes X :: "nat \<Rightarrow> 'a" assumes "bdd_below (range X)"   and "decseq X" shows "X \<longlonglongrightarrow> Inf (range X)"
R: fixes C :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "C \<in> components S"   and "continuous_on C f"   and "f ` C \<subseteq> T"   and "continuous_on C g"   and "g ` C \<subseteq> T" shows "homotopic_with_canon (\<lambda>x. True) C T f g"
sigma: fixes i :: "nat" shows "\<sigma> i = \<phi> (the (eval r_limr [i]))"
Cauchy_Bseq: fixes X :: "nat \<Rightarrow> 'a" assumes "Cauchy X" shows "Bseq X"
assms: shows "e \<in> UNIV \<rightarrow> I"
wf: fixes r :: "('a \<times> 'a) set" assumes "trans r"   and "wf r" shows "wf (pex r)"
equal_pair_count_Suc: fixes a :: "'b"   and ps :: "('b \<times> 'b) list" shows "equal_pair_count ((a, a) # ps) = Suc (equal_pair_count ps)"
fin_outer_D_is_reachable: fixes D :: "'v set" assumes "outer_invar {} D" shows "D = reachable"
fz: shows "f z = (1 + cos (complex_of_real pi * h z)) / 2"
embed_in_Field: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "embed r r' f"   and "a \<in> Field r" shows "f a \<in> Field r'"
A: shows "\<forall>s\<in>set (toList (snd ec)). envObs a (snd s) = envObs a (tLast t)"
maxim_def: fixes A :: "'a set" shows "maxim A \<equiv> THE b. isMaxim A b"
simps_1: fixes s :: "name"   and env :: "(name, Term_Class.term) fmap" shows "subst (Free s) env = (case fmlookup env s of None \<Rightarrow> Free s | Some t \<Rightarrow> t)"
arcs_H: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d" assumes "subdiv_step G rev_G H rev_H u v w uv uw vw" shows "arcs H = {uw, rev_H uw, vw, rev_H vw} \<union> arcs G - {uv, rev_G uv}"
strict_mono_sequence_partition: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "strict_mono f"   and "f 0 \<le> x"   and "filterlim f at_top sequentially" shows "\<exists>k. x \<in> {f k..<f (Suc k)}"
pdivides_zero: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)" shows "p pdivides\<^bsub>poly_ring R\<^esub> []"
space_subprob_algebra: fixes A :: "'c measure" shows "space (subprob_algebra A) = {M. subprob_space M \<and> sets M = sets A}"