hcju/isabelleps huggingface dataset:
interaction_bounded_by_start: fixes "consider" :: "'i \<Rightarrow> bool"   and gpv :: "('j, 'i, 'k) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
SUP_outer_measure_of_incseq: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "\<And>n. A n \<subseteq> space M"   and "incseq A" shows "(SUP n. outer_measure_of M (A n)) = outer_measure_of M (\<Union> (range A))"
rep_of_id: fixes l :: "nat list"   and i :: "nat" assumes "ufa_invar l"   and "i < length l"   and "l ! i = i" shows "rep_of l i = i"
simps_37: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> While x61 x62"
unit_SB_def: fixes e :: "'c" shows "unit\<^sub>S\<^sub>B e = (\<lambda>\<sigma>. {(e, \<sigma>)})"
monom_power: fixes a :: "'a"   and b :: "nat"   and n :: "nat" shows "monom a b ^ n = monom (a ^ n) (b * n)"
frontier_def: shows "frontier \<equiv> map_fun id antichain (\<lambda>M. minimal_antichain {t. 0 < zcount M t})"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "((x1)\<^sub>p = (y1)\<^sub>p) = (x1 = y1)"
elims: fixes x :: "'a word \<times> nat"   and xa :: "'a word \<times> nat"   and y :: "('a word \<times> nat) option" assumes "ipcidr_conjunct x xa = y" obtains base1 :: "'a word" and m1 :: "nat" and base2 :: "'a word" and m2 :: "nat" where "x = (base1, m1)" and "xa = (base2, m2)" and "y = (if ipset_from_cidr base1 m1 \<inter> ipset_from_cidr base2 m2 = {} then None else if ipset_from_cidr base1 m1 \<subseteq> ipset_from_cidr base2 m2 then Some (base1, m1) else Some (base2, m2))"
f2: shows "(cmod (jozsa_algo $$ (1, 0)))\<^sup>2 = (cmod (complex_of_real (\<Sum>k<2 ^ n. (- 1) ^ (f k + 1) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)))))\<^sup>2"
find_match_rewrite_first: fixes cs :: "(Term_Class.term \<times> 'a) list"   and t :: "'a"   and env :: "(name, 'a) fmap"   and pat :: "Term_Class.term"   and rhs :: "'a" assumes "find_match cs t = Some (env, pat, rhs)" shows "rewrite_first cs t (subst rhs env)"
assms: shows "\<forall>i. D (\<lambda>t. f t $ i) \<mapsto> (\<lambda>h. h *\<^sub>R f' x $ i) at x within T"
a\<alpha>: shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
hcomp_conc_comm: fixes P :: "conc_interface"   and Q :: "conc_interface" shows "P \<otimes>\<^sub>c Q \<simeq> Q \<otimes>\<^sub>c P"
xdc: shows "x /\<^sub>R c \<in> ball a r"
reservation: fixes c :: "cars" shows "\<forall>ts v. (\<forall>ts'. ts \<^bold>\<midarrow>r( c ) \<^bold>\<rightarrow> ts' \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts' c = ext v \<and> restrict v (res ts') c = lan v \<and> |lan v| = 1) = (0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (clm ts) c = lan v \<and> |lan v| = 1)"
WT_plus_oracleD1: fixes \<I>l :: "('a, 'b) \<I>"   and \<I>r :: "('c, 'd) \<I>"   and s :: "'s" assumes "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd>" shows "\<I>l \<turnstile>c left s \<surd>"
ns'_def: shows "ns' = (Restr {((a, x), b, y) |a x b y. (a, b) \<in> {(a\<^sub>1, a\<^sub>2), (a\<^sub>1, a\<^sub>3), (a\<^sub>2, a\<^sub>3)} \<and> (x, y) \<in> ns} ({(a\<^sub>1, x) |x. x \<in># X} \<union> {(a\<^sub>2, y) |y. y \<in># Y} \<union> {(a\<^sub>3, z) |z. z \<in># Z}))\<^sup>="
powr_def: fixes x :: "'c"   and a :: "'c" shows "x powr a \<equiv> if x = (0::'c) then 0::'c else exp (a * ln x)"
srs: shows "secure_refinement_simple \<R>\<^sub>A \<R> abs_steps"
Y_big_def: shows "Y\<^sub>\<succeq> = {#ys \<in># mset (chop 5 xs). \<not> median ys \<prec> M#}"
WTThrow: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and C :: "char list" assumes "P,E \<turnstile> e :: Class C" shows "P,E \<turnstile> throw e :: Void"
sum_drop_zero: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<Sum>k = 0..Suc n. if 0 < k then f (k - 1) else (0::'a)) = sum f {0..n}"
id_eq_3: fixes v :: "i"   and x :: "'b"   and y :: "'b"   and z :: "'b" shows "[x \<^bold>= y \<^bold>& y \<^bold>= z \<^bold>\<rightarrow> x \<^bold>= z in v]"
assms_2: fixes \<Phi> :: "'a ltln set" assumes "\<Phi> \<in> \<A>" shows "finite \<Phi>"
l_unit: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e" assumes "Group G"   and "a \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<cdot>\<^bsub>G\<^esub> a = a"
not_distinct_split_distinct: fixes xs :: "'a list" assumes "\<not> distinct xs" obtains y :: "'a" and ys :: "'a list" and zs :: "'a list" where "distinct ys" and "y \<in> set ys" and "xs = ys @ [y] @ zs"
distinct_3: fixes nat :: "nat"   and list :: "NominalSequents.var list"   and list' :: "char list"   and var' :: "NominalSequents.var"   and form' :: "form" shows "At nat list \<noteq> list' \<nabla> [var'].form'"
snth_sset: fixes s :: "'d stream"   and n :: "nat" shows "s !! n \<in> sset s"
Seq: fixes P :: "'c com"   and pre :: "'c set"   and rely :: "('c \<times> 'c) set"   and guar :: "('c \<times> 'c) set"   and mid :: "'c set"   and Q :: "'c com"   and post :: "'c set" assumes "\<turnstile> P sat [pre, rely, guar, mid]"   and "\<turnstile> Q sat [mid, rely, guar, post]" shows "\<turnstile> P;; Q sat [pre, rely, guar, post]"
less_sets_Un2: fixes A :: "'a set"   and B :: "'a set"   and B' :: "'a set" shows "(A \<lless> B \<union> B') = (A \<lless> B \<and> A \<lless> B')"
ide_char: fixes a :: "'c" shows "J'.ide a = (J'.arr a \<and> J'.dom a = a \<and> J'.cod a = a)"
\<R>\<^sub>G_inv: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "bij \<alpha>"   and "bij \<beta>" shows "(\<R>\<^sub>G (inv \<alpha>) (inv \<beta>) s' = s) = (\<R>\<^sub>G \<alpha> \<beta> s = s')"
f0: fixes F :: "nat \<Rightarrow> 'a poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>"   and "i \<noteq> 0" shows "(f i = (0::'a)) = (k < i)"
bounded_linear_zero: shows "bounded_linear (\<lambda>x. 0::'e)"
dp_consistency_def: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool" shows "dp_consistency lookup update P \<equiv> mem_correct lookup update P"
tendsto_ine\<^sub>N_iff_limitin: fixes N :: "'a quasinorm"   and u :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "defect N = 1" shows "tendsto_ine\<^sub>N N u x = limitin (topology\<^sub>N N) u x sequentially"
ex_assn_def: fixes P :: "'a \<Rightarrow> assn" shows "\<exists>\<^sub>Ax. P x \<equiv> Abs_assn (\<lambda>h. \<exists>x. h \<Turnstile> P x)"
Tn_measure_preserving: fixes n :: "nat" shows "T ^^ n \<in> measure_preserving M M"
Cons_tuv_1: fixes tu :: "'a event list"   and t :: "'a event list"   and u :: "'a event list"   and v :: "'a event list" assumes "tu setinterleaves ((t, u), A)"   and "tuv setinterleaves ((tu, v), A)" shows "\<exists>uv. uv setinterleaves ((u, v), A) \<and> tuv setinterleaves ((t, uv), A)"
rel_arg_cong: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "(x, y) \<in> Id" shows "(f x, f y) \<in> Id"
abel_semigroup_axioms: shows "abel_semigroup (\<iinter>)"
prems_1: shows "set (x # as) \<subseteq> carrier G"
distinct_26: fixes x81 :: "nat"   and x82 :: "msg"   and x2 :: "nat" shows "Crypt x81 x82 \<noteq> Number x2"
index_transpose_mat_2: fixes A :: "'a Matrix.mat" shows "dim_row A\<^sup>t = dim_col A"
bidirected_digraphI_subdivision: fixes G :: "'b pair_pre_digraph"   and H :: "'b pair_pre_digraph" assumes "subdivision_pair G H" shows "pair_bidirected_digraph H"
MM0: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> G) \<longrightarrow> \<box>F \<longrightarrow> \<box>G"
ldropn_def: fixes n :: "nat"   and xs :: "'c llist" shows "ldropn n xs = (ltl ^^ n) xs"
M7: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>[F]_v \<and> \<box>[G]_v \<longrightarrow> \<box>[F \<and> G]_v"
simps_2: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" shows "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
sum_weights_append: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and ys :: "'b list"   and xs :: "'b list" assumes "graph weights" shows "foldr (+) (ys @ xs) (0::'b) = foldr (+) ys (0::'b) + foldr (+) xs (0::'b)"
get_proc_return: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "get_proc (sourcenode a) = p"
tarjan_params_def: shows "tarjan_params = \<lparr>on_init = RETURN \<lparr>sccs = {}, lowlink = Map.empty, tj_stack = []\<rparr>, on_new_root = tarjan_disc, on_discover = \<lambda>u. tarjan_disc, on_finish = tarjan_fin, on_back_edge = tarjan_back, on_cross_edge = tarjan_back, is_break = \<lambda>s. False\<rparr>"
gtr_rev_cases: fixes c :: "'c multiset"   and e :: "'d"   and s' :: "'e"   and ce' :: "'e multiset"   and gtrs :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'e \<times> 'e multiset) set" assumes "(c, e, add_mset s' ce') \<in> gtr gtrs" obtains s :: "'c" and ce :: "'c multiset" where "c = add_mset s ce__" and "((s, ce__), e, s', ce') \<in> gtrs"   | cc :: "'e multiset" and ss :: "'c" and ss' :: "'e" and ce :: "'c multiset" where "c = add_mset ss ce" and "ce' = add_mset ss' cc" and "((ss, ce), e, ss', add_mset s' cc) \<in> gtrs"
colossless_bind_gpvI: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and f :: "'f \<Rightarrow> ('g, 'd, 'e) gpv" assumes "colossless_gpv \<I> gpv"   and "\<And>x. x \<in> results_gpv \<I> gpv \<Longrightarrow> colossless_gpv \<I> (f x)" shows "colossless_gpv \<I> (gpv \<bind> f)"
imp_simps_6: fixes P :: "fm"   and Q :: "fm" assumes "P \<noteq> T"   and "P \<noteq> F"   and "P \<noteq> Q"   and "Q \<noteq> T"   and "Q \<noteq> F" shows "imp P Q = Imp P Q"
tagged_division_finer: fixes p :: "('a \<times> 'a set) set"   and a :: "'a"   and b :: "'a"   and d :: "'a \<Rightarrow> 'a set" assumes "p tagged_division_of cbox a b"   and "gauge d" obtains q :: "('a \<times> 'a set) set" where "q tagged_division_of cbox a b" and "d fine q" and "\<forall>(x, k)\<in>p. k \<subseteq> d x \<longrightarrow> (x, k) \<in> q"
uminus_real_alg_def: shows "uminus \<equiv> (rep_real_alg ---> Real_Alg_Quotient) uminus_3"
continuous_on_inverse_open_map: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "continuous_on S f"   and "f ` S = T"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"   and "\<And>U. openin (top_of_set S) U \<Longrightarrow> openin (top_of_set T) (f ` U)" shows "continuous_on T g"
ordinal_minus_0: fixes x :: "ordinal" shows "x - 0 = x"
span_superset: fixes S :: "'c set" shows "S \<subseteq> l.vs2.span S"
are_all_operator_effects_consistent_def: fixes ops :: "('c, 'd) strips_operator_scheme list" shows "STRIPS_Semantics.are_all_operator_effects_consistent ops \<equiv> list_all (\<lambda>op. list_all (STRIPS_Semantics.are_operator_effects_consistent op) ops) ops"
fimage_iff: fixes b :: "V"   and r :: "V"   and A :: "V" shows "(b \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A) = (\<exists>a\<in>elts A. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r)"
permute_unit_def: fixes p :: "perm"   and u :: "unit" shows "p \<bullet> u = u"
antisymmetric_bot_closed: shows "matrix_stone_relation_algebra.antisymmetric mbot"
iwlsFSwSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "iwlsFSwSb MOD \<equiv> iwlsFSw MOD \<and> igSubstAllIPresIGWlsAll MOD \<and> igSubstCls MOD"
tanh_def: fixes x :: "'b" shows "tanh x = sinh x / cosh x"
lubH_is_fixp: fixes H :: "'a set" assumes "H = {x. (x, f x) \<in> r \<and> x \<in> A}" shows "lub H cl \<in> fix f A"
coinduction: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr"   and "\<forall>R. R \<le> retr R \<longrightarrow> R \<le> (=)"   and "R \<le> retr (cong.gen_cong rel eval R)" shows "R \<le> (=)"
coerce_def: shows "COERCE('e, 'f) \<equiv> PRJ('f) oo EMB('e)"
Union_member: fixes F :: "'b set set"   and x :: "'b" shows "(\<exists>S\<in>F. x \<in> S) = (x \<in> \<Union> F)"
axioms_1: fixes L :: "('d, 'e) gorder_scheme" assumes "partial_order L" shows "weak_partial_order L"
class_add_states: fixes P :: "jvm_prog"   and mxs :: "nat"   and mxl :: "nat"   and C :: "char list"   and cdec :: "jvm_method class" shows "states P mxs mxl \<subseteq> states (class_add P (C, cdec)) mxs mxl"
wf_design_iff: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "design \<V> \<B> = (bl \<subseteq> \<V> \<and> finite \<V> \<and> bl \<noteq> {})"
finalConflictingState: fixes state :: "State" assumes "InvariantUniq (getM state)"   and "InvariantConsistent (getM state)"   and "InvariantImpliedLiterals (getF state) (getM state)"   and "\<not> applicableBackjump state"   and "formulaFalse (getF state) (elements (getM state))" shows "decisions (getM state) = []"
isGlb_dual_isLub: fixes S :: "'b set"   and cl :: "'b potype" shows "isGlb S cl = isLub S (dual cl)"
lunit_char_3: fixes a :: "'d" assumes "D.ide a" shows "\<exists>!f. \<guillemotleft>f : \<I>\<^sub>D \<otimes>\<^sub>D a \<rightarrow>\<^sub>D a\<guillemotright> \<and> \<I>\<^sub>D \<otimes>\<^sub>D f = (\<iota>\<^sub>D \<otimes>\<^sub>D a) \<cdot>\<^sub>D D.assoc' \<I>\<^sub>D \<I>\<^sub>D a"
mod_partition_Union: fixes m :: "nat"   and A :: "nat set" assumes "0 < m" shows "(\<Union>k. A \<inter> [k * m\<dots>,m - Suc 0]) = A"
group_isomorphisms_imp_iso: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'b" assumes "group_isomorphisms G H f g" shows "f \<in> Group.iso G H"
set: fixes x1 :: "'c"   and x2 :: "'c stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
lrev_work2_strict: shows "lrev_work2\<cdot>\<bottom> = \<bottom>"
simps_2: fixes x :: "('a, 'b) Node"   and xs :: "('a, 'b) Node list" shows "FingerTreeStruc.nlistToList (x # xs) = FingerTreeStruc.nodeToList x @ FingerTreeStruc.nlistToList xs"
hyps_14: shows "split t U S = (ps0 @ ps1, qs0 @ qs1)"
right_translations_of_subgroup_are_transformation_group: shows "transformation_group (translation ` K) G"
simps: fixes s :: "'s" shows "local.terminates_within s = (let (has_next, next) = g in if has_next s then map_option (\<lambda>n. n + 1) (local.terminates_within (snd (next s))) else Some 0)"
g2: shows "g \<in> lspace (lebesgue_on S) 2"
lcoeff_nonzero2: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "p \<noteq> \<zero>\<^bsub>UP R\<^esub>" shows "coeff (UP R) p (deg R p) \<noteq> \<zero>\<^bsub>R\<^esub>"
mono_bij_cInf: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "mono f"   and "bij f"   and "A \<noteq> {}"   and "bdd_below A" shows "f (\<Sqinter> A) = \<Sqinter> (f ` A)"
simps_1: fixes c :: "int \<times> int" shows "~\<^sub>p (C c) = C (~\<^sub>N c)"
disc_5: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "\<not> isTrans (RecvMarker x51 x52 x53)"
ennreal_pmf_bind: fixes N :: "'b pmf"   and f :: "'b \<Rightarrow> 'a pmf"   and i :: "'a" shows "ennreal (pmf (N \<bind> f) i) = \<integral>\<^sup>+ x. ennreal (pmf (f x) i) \<partial>measure_pmf N"
LeftDerivationIntrosAt_implies_nonterminal: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" assumes "LeftDerivationIntrosAt a D L index" shows "is_nonterminal (ladder_\<alpha> a D L index ! ladder_i L index)"
intensional_simps_20: fixes y :: "'a \<Rightarrow> bool" shows "y \<and> y \<equiv> y"
notIn_Un_bij_betw3: fixes b :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and A' :: "'b set" assumes "b \<notin> A"   and "f b \<notin> A'" shows "bij_betw f A A' = bij_betw f (A \<union> {b}) (A' \<union> {f b})"
distinct_1: fixes x1 :: "'b \<times> 'b fs_set \<times> 'a"   and x2 :: "'b fs_set \<times> 'a" shows "AC x1 \<noteq> EF x2"
A_2: shows "(i, j) \<in> set (arcs a a xs)"
paidAnm_inbound: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat"   and m :: "nat" assumes "BIT_Off acts qs init"   and "n < length (BIT_Off.paid_A acts init)"   and "m < length (BIT_Off.paid_A acts init ! n)" shows "Suc (BIT_Off.paid_A acts init ! n ! (length (BIT_Off.paid_A acts init ! n) - Suc m)) < length init"
inv_convcomp: fixes c :: "'c \<Rightarrow>\<^sub>U 'd" shows "inv\<^sub>C c \<circ>\<^sub>C c = id\<^sub>C"
snd_sym_preproc_superset: fixes fs :: "('a \<Rightarrow>\<^sub>0 'd) list"   and gs :: "('a \<Rightarrow>\<^sub>0 'd) list" shows "set fs \<subseteq> set (snd (punit.sym_preproc gs fs))"
\<delta>ss_accs_precise: fixes t :: "'L tree"   and s :: "'Q set"   and q :: "'Q" assumes "accs \<delta>ss t s"   and "q \<in> s" shows "accs \<delta> t q"
lunit'_simps_5: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.cod (S.lunit' f) = S.trg f \<star>\<^sub>S f"
ENR_openin: fixes T :: "'a set"   and S :: "'a set" assumes "ENR T"   and "openin (top_of_set T) S" shows "ENR S"
rep_eq: fixes x :: "'c"   and xa :: "'c monom" shows "univariate_power x xa = univariate_power_list x (Rep_monom xa)"
propnot\<^sub>1_def: shows "propnot \<equiv> \<lambda>F. \<^bold>\<lambda>x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>"
less_csucc: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "\<kappa> \<subset>\<^sub>\<circ> csucc \<kappa>"
list_length_iff_distinct: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs = set ys"   and "distinct ys" shows "distinct xs = (length xs = length ys)"
eq_iff: fixes a :: "'e"   and b :: "'e" shows "(a = b) = (b \<le> a \<and> a \<le> b)"
convex_ball: fixes x :: "'c"   and e :: "real" shows "convex (ball x e)"
c_prod_def: fixes A :: "'a set"   and B :: "'a set" shows "A * B = {u * v |u v. u \<in> A \<and> v \<in> B}"
set_transfer_1: fixes R1 :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (tllist_all2 R1 R2) (rel_set R1) tset tset"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "x \<sqsubseteq> y"
vfsequence_vremove_vcons_vfsequence: fixes xs :: "V"   and xs' :: "V"   and x :: "V" assumes "vfsequence xs"   and "xs = xs' #\<^sub>\<circ> x" shows "vfsequence xs'"
ide_some_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "ide (some_trg \<mu>)"
ltl_const_equiv_const_1: fixes \<phi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>C true\<^sub>n = (eval \<phi> = Yes)"
wf\<^sub>t\<^sub>r\<^sub>m_code: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and t :: "('a, 'c) Term.term" assumes "intruder_model arity Ana" shows "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t = wf\<^sub>t\<^sub>r\<^sub>m' arity t"
sat_fvi_cong: fixes \<phi> :: "'c MFOTL.formula"   and v :: "'c list"   and v' :: "'c list"   and \<sigma> :: "(char list \<times> 'c list) trace"   and i :: "nat" assumes "\<forall>x\<in>fv \<phi>. v ! x = v' ! x" shows "MFOTL.sat \<sigma> v i \<phi> = MFOTL.sat \<sigma> v' i \<phi>"
z_def: shows "z = (zeroes (length xs))[k := zk]"
slengthE: fixes xs :: "[:'a:]" assumes "xs \<noteq> \<bottom>" obtains n :: "int" where "slength\<cdot>xs = MkI\<cdot>n" and "0 \<le> n"
subst_simp_pair: fixes A :: "'a trm"   and B :: "'a trm"   and x :: "'a"   and M :: "'a trm" shows "SimplyTyped.Pair A B[x ::= M] = SimplyTyped.Pair A[x ::= M] B[x ::= M]"
get_parent_ok: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "type_wf h"   and "known_ptrs h"   and "node_ptr |\<in>| node_ptr_kinds h" shows "h \<turnstile> ok get_parent node_ptr"
region_compatible_suntil1: fixes \<phi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool"   and \<psi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool"   and x :: "('s \<times> ('c \<Rightarrow> real)) stream" assumes "(holds (\<lambda>x. \<phi> (reps x)) suntil holds (\<lambda>x. \<psi> (reps x))) (smap abss x)"   and "pred_stream (\<lambda>s. \<phi> (reps (abss s)) \<longrightarrow> \<phi> s) x"   and "pred_stream (\<lambda>s. \<psi> (reps (abss s)) \<longrightarrow> \<psi> s) x" shows "(holds \<phi> suntil holds \<psi>) x"
sams_chara: 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 A' :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "A' \<noteq> B"   and "Bet A B A'" shows "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F = Tarski_neutral_dimensionless.LeA Bet Cong D E F C B A'"
L: fixes f :: "complex \<Rightarrow> complex"   and B :: "real" assumes "f holomorphic_on interior T"   and "\<And>z. z \<in> interior T \<Longrightarrow> cmod (f z) \<le> B" shows "cmod (contour_integral \<gamma> f) \<le> L * B"
jnf_complexity_1_real: fixes A :: "real mat"   and n :: "nat"   and d :: "nat" assumes "A \<in> carrier_mat n n"   and "nonneg_mat A"   and "\<And>x. poly (char_poly A) x = 0 \<Longrightarrow> x \<le> 1"   and "\<lbrakk>poly (char_poly A) 1 = 0; d + 1 < order 1 (char_poly A)\<rbrakk> \<Longrightarrow> dim_gen_eigenspace A 1 (d + 1) = order 1 (char_poly A)" shows "\<exists>c1 c2. \<forall>k a. a \<in> elements_mat (A ^\<^sub>m k) \<longrightarrow> \<bar>a\<bar> \<le> c1 + c2 * real k ^ d"
Units_r_inv: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.monoid G"   and "x \<in> Units G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
functor_axioms: shows "functor hom.comp hom'.comp (\<lambda>\<mu>. e\<^sub>1 \<star> \<mu> \<star> d\<^sub>0)"
finite_maps_to_finite_type: fixes D :: "'a set" assumes "finite D"   and "finite UNIV" shows "finite (dom -` Pow D)"
cring_simprules_8: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "cring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
cylinder_prob: fixes A :: "nat set set"   and B :: "nat set set" assumes "A \<subseteq> S_edges"   and "B \<subseteq> S_edges"   and "A \<inter> B = {}" shows "prob (cylinder S_edges A B) = p ^ card A * (1 - p) ^ card B"
winning_strategy_def: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "G'.winning_strategy p \<sigma> v0 \<equiv> \<forall>P. vmc_path G' P v0 p \<sigma> \<longrightarrow> G'.winning_path p P"
not_cd_2: shows "\<forall>k. \<not> i' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> k"