hcju/isabelleps huggingface dataset:
transpose_minus_1: fixes Q :: "'b mat" assumes "dim_row Q = dim_col Q" shows "(Q - 1\<^sub>m (dim_row Q))\<^sup>T = Q\<^sup>T - 1\<^sub>m (dim_row Q)"
InvariantConflictClauseCharacterization_def: fixes conflictFlag :: "bool"   and conflictClause :: "nat"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list" shows "InvariantConflictClauseCharacterization conflictFlag conflictClause F M \<equiv> conflictFlag \<longrightarrow> conflictClause < length F \<and> clauseFalse (F ! conflictClause) (elements M)"
known_ptr_def: shows "known_ptr = (\<lambda>ptr. CharacterDataClass.known_ptr ptr \<or> is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)"
below_lubmapI: fixes m :: "'f \<Rightarrow> 'e option"   and k :: "'f"   and v :: "'e"   and e :: "'f \<Rightarrow> 'e \<Rightarrow> 'g" assumes "m k = Some v" shows "e k v \<sqsubseteq> lub (mapCollect e m)"
pseudo_complement: fixes uminus :: "'b \<Rightarrow> 'b"   and 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 bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "(inf x y = bot) = less_eq x (uminus y)"
intro: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" assumes "monoid f z" shows "monoid_list f z"
degree_prod_monic: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b poly" assumes "\<And>i. i < n \<Longrightarrow> degree (f i) = 1"   and "\<And>i. i < n \<Longrightarrow> coeff (f i) 1 = (1::'b)" shows "degree (prod f {0..<n}) = n \<and> coeff (prod f {0..<n}) n = (1::'b)"
max2_def: fixes a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" shows "max2 Bonk_Schramm_extension_unfolded_wo a b \<equiv> if (a, b) \<in> Bonk_Schramm_extension_unfolded_wo then b else a"
assms: shows "\<exists>x\<in>set xs. l \<le> f x"
component_single: fixes e :: "'a"   and v :: "'a" shows "choose_component e v = e * choose_component e v"
fbox_kcomp: fixes G :: "'b \<Rightarrow> 'd set"   and F :: "'d \<Rightarrow> 'c set"   and P :: "'c \<Rightarrow> bool" shows "|G ; F] P = |G] |F] P"
term_of_pair_pair: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and v :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "term_of_pair (term_powerprod.pp_of_term pair_of_term v, term_powerprod.component_of_term pair_of_term v) = v"
splus_mono_strict_left: fixes s :: "'a"   and t :: "'a"   and v :: "'a" assumes "s \<prec> t" shows "local.punit.splus s v \<prec> local.punit.splus t v"
eq_swap_rev: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 =\<^sub>S w\<^sub>2" shows "rev w\<^sub>1 =\<^sub>S rev w\<^sub>2"
simps: fixes x :: "'a"   and tn :: "'a tree \<times> nat" shows "nxt x tn = insert3 x tn"
product_integrable_prod: fixes I :: "'i set"   and f :: "'i \<Rightarrow> 'a \<Rightarrow> 'b" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> integrable (M i) (f i)" shows "integrable (Pi\<^sub>M I M) (\<lambda>x. \<Prod>i\<in>I. f i (x i))"
sees_methods_Object: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and D :: "char list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list"   and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "Mm = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms" shows "P \<turnstile> Object sees_methods Mm"
upt_minus_eq_append: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j"   and "i \<le> j - k" shows "[i..<j] = [i..<j - k] @ [j - k..<j]"
assms_2: shows "argmin f [i..j] \<in> {i'..j'}"
that: fixes k :: "'u"   and v :: "'v" assumes "min s P = Some (k, v)"   and "(k, v) \<in> rel_of (\<alpha> s) P"   and "\<forall>(k', v')\<in>rel_of (\<alpha> s) P. k \<le> k'" shows "thesis"
S_geq_S_succ_plus_ell: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and vs :: "'a set"   and lss :: "('a, 'b) fmap list"   and s' :: "('a, 'b) fmap" assumes "s \<in> valid_states PROB"   and "top_sorted_abs (\<lambda>x y. y \<in> state_successors (prob_proj PROB vs) x) lss"   and "s' \<in> state_successors (prob_proj PROB vs) s"   and "set lss = valid_states (prob_proj PROB vs)" shows "problem_plan_bound (snapshot PROB (fmrestrict_set vs s)) + S vs lss PROB (fmrestrict_set vs s') + 1 \<le> S vs lss PROB (fmrestrict_set vs s)"
Pd'_Above: fixes Y :: "'a set"   and d :: "'b" assumes "Y \<subseteq> above (Pd' (Xd x)) x"   and "Y \<noteq> {}" shows "Above (Pd' d) Y \<subseteq> Above (Pd'_tax d) Y"
unit_in_hom_2: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>F.unit a : F.map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright>"
le_approx1: fixes P :: "'c process"   and Q :: "'c process" assumes "P \<sqsubseteq> Q" shows "D Q \<subseteq> D P"
cross7_simps_17: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a - (b - c) = a + c - b"
GoodRead_def: fixes mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and l :: "Locs" shows "GoodRead mm rs p l \<equiv> #l \<in> #MemLoc \<and> (rs!p)$ = $(mm!l)"
eexp_ineq_iff_1: fixes x :: "ereal" shows "(eexp x \<le> 1) = (x \<le> 0)"
n_subid: fixes x :: "'a" shows "!x \<sqsubseteq> (1::'a)"
antisym: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "b \<preceq> a"   and "a \<preceq> b" shows "a = b"
Lp_cases_1_PInf: fixes p :: "ennreal" assumes "1 \<le> p" obtains p2 :: "real" where "p = ennreal p2" and "1 < p2" and "1 < p"   | "p = 1"   | "p = \<infinity>"
homeomorphism_grouping_points_exists_gen: fixes S :: "'a set"   and U :: "'a set"   and K :: "'a set"   and T :: "'a set" assumes "openin (top_of_set S) U"   and "openin (top_of_set (affine hull S)) S"   and "U \<noteq> {}"   and "finite K"   and "K \<subseteq> S"   and "S \<subseteq> T"   and "T \<subseteq> affine hull S"   and "connected S" obtains f :: "'a \<Rightarrow> 'a" and g :: "'a \<Rightarrow> 'a" where "homeomorphism T T f g" and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S" and "bounded {x. \<not> (f x = x \<and> g x = x)}" and "\<And>x. x \<in> K \<Longrightarrow> f x \<in> U"
fe_inv_def: shows "fe_inv \<equiv> \<lambda>S V v it (V', r). (r \<longrightarrow> tgt \<in> local.reachable) \<and> (\<not> r \<longrightarrow> tgt \<notin> V' \<and> insert v V \<subseteq> V' \<and> E `` {v} - it \<subseteq> V' \<and> V' \<subseteq> local.reachable \<and> S \<subseteq> insert v V \<and> E `` (V' - S) \<subseteq> V' \<union> it \<and> E `` (V' - insert v S) \<subseteq> V')"
eq: fixes x :: "nat" assumes "x \<noteq> i" shows "map of_int_hom.vec_hom fs' ! x = map of_int_hom.vec_hom fs ! x"
exposed_face_of: fixes T :: "'b set"   and S :: "'b set" shows "(T exposed_face_of S) = (T face_of S \<and> (T = {} \<or> T = S \<or> (\<exists>a b. a \<noteq> (0::'b) \<and> S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b})))"
zero_less_Suc: fixes n :: "nat" shows "n \<in># Suc"
assms: fixes l :: "cell list"   and r :: "cell list"   and n :: "nat" assumes "P (l, r)" shows "\<not> is_final (steps0 (1, l, r) p n)"
sym_rtrancl: fixes r :: "('b \<times> 'b) set" assumes "sym r" shows "sym (r\<^sup>*)"
1: shows "eSubstAbs MOD ys eY y eA \<noteq> ERR"
make_affine_def: fixes p :: "'a \<times> 'a \<times> 'a" shows "make_affine p = (let (x, y, z) = p in if z = \<zero> then Infinity else Point (x \<oslash> z) (y \<oslash> z))"
span_alt: fixes B :: "'b set" assumes "B \<subseteq> U\<^sub>M" shows "local.span B = {x \<in> U\<^sub>M. \<exists>f. x = (\<Sum>x | x \<in> U\<^sub>M \<and> f x \<noteq> (0::'a). f x *s x) \<and> finite {x \<in> U\<^sub>M. f x \<noteq> (0::'a)} \<and> {x \<in> U\<^sub>M. f x \<noteq> (0::'a)} \<subseteq> B}"
simps_1: fixes l :: "'a set list" shows "qsort [] l = []"
Solution_axioms: shows "Solution inPorts outPorts nodeOf hyps vertices labelsIn labelsOut freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx inst edges"
psiFreshVec_3: fixes xvec :: "name list"   and C :: "('e, 'f, 'g) psiCase" shows "xvec \<sharp>* Case C = xvec \<sharp>* C"
ZusammenziehungS: fixes \<Gamma> :: "'b formula list"   and \<DD> :: "'b formula"   and \<Theta> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<DD> # \<DD> # \<Theta>" shows "\<Gamma> \<Rightarrow> \<DD> # \<Theta>"
k_def: shows "k \<equiv> \<lambda>c. {x. 0 \<le> c \<bullet> x}"
IH: fixes ys :: "'b list"   and zs :: "'b list"   and f :: "'a \<Rightarrow> 'b option" assumes "length xs \<le> length ys"   and "length xs \<le> length zs"   and "\<forall>i<length xs. ys ! i = zs ! i" shows "f(xs [\<mapsto>] ys) = f(xs [\<mapsto>] zs)"
is_SNF_Smith_nx2: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n 2" shows "is_SNF A (Smith_nx2 A)"
bij_comp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "bij f"   and "bij g" shows "bij (g \<circ> f)"
cINF_greatest: fixes A :: "'c set"   and m :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> m \<le> f x" shows "m \<le> Inf (f ` A)"
N_geometric: shows "\<forall>M\<ge>N. \<forall>n. \<bar>real_of_int (B * b (n + M + 1)) / real_of_int (prod a {M..n + M + 1})\<bar> < 1 / 2 ^ n"
valid_strategy_supergame: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and V' :: "'a set"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "G'.strategy p \<sigma>"   and "ParityGame.strategy (G'.subgame V') p \<sigma>'"   and "\<And>v. v \<in> V' \<Longrightarrow> \<not> \<not> (\<exists>w\<in>V\<^bsub>G'.subgame V'\<^esub>. v \<rightarrow>\<^bsub>G'.subgame V'\<^esub> w)" shows "G'.strategy p (override_on \<sigma> \<sigma>' V')"
"11_1": shows "R.path (b @ [a]) (target w q)"
minus_zero_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v - 0\<^sub>v n = v"
order_trans: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "prefix x y"   and "prefix y z" shows "prefix x z"
maxt12: shows "wmin (wtimes l\<^sub>1 l\<^sub>2) (wtimes u\<^sub>1 l\<^sub>2) \<equiv>\<^sub>E min (rl1 * rl2) (ru1 * rl2)"
assms_2: shows "strategy_attracts_via p \<sigma> v0 A W"
assms_3: shows "xs \<prec> cs\<^bsup>\<pi>\<^esup> k"
assms_4: shows "v \<in> set (precondition_of op)"
cases: fixes a1 :: "cmd"   and a2 :: "char list" assumes "newname a1 a2" obtains V :: "char list" where "a1 = Skip" and "a2 = V__"   | V :: "char list" and V' :: "char list" and e :: "expr" where "a1 = V':=e" and "a2 = Va__" and "Va__ \<notin> {V'} \<union> rhs_aux e"   | c1 :: "cmd" and V :: "char list" and c2 :: "cmd" where "a1 = c1__;; c2__" and "a2 = Vb__" and "newname c1__ Vb__" and "newname c2__ Vb__"   | c1 :: "cmd" and V :: "char list" and c2 :: "cmd" and b :: "expr" where "a1 = if (b__) c1 else c2" and "a2 = Vc__" and "newname c1 Vc__" and "newname c2 Vc__" and "Vc__ \<notin> rhs_aux b__"   | c :: "cmd" and V :: "char list" and b :: "expr" where "a1 = while (b) c" and "a2 = V" and "newname c V" and "V \<notin> rhs_aux b"
except_Diff_singleton: fixes p :: "'d \<Rightarrow>\<^sub>0 'e"   and t :: "'d" shows "except p (keys p - {t}) = monomial (lookup p t) t"
eigen_value_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and k :: "'a" shows "eigen_value A k = (\<exists>v. eigen_vector A v k)"
assms: shows "a \<in> set al \<or> F1 a = F2 a"
elims: fixes x :: "('c, 'd) bintree option"   and xa :: "('c, 'd) bintree option list"   and y :: "('c, 'd) bintree option list" assumes "add x xa = y" obtains xs :: "('c, 'd) bintree option list" where "x = None" and "xa = xsa__" and "y = xsa__"   | t :: "('c, 'd) bintree" where "x = Some ta__" and "xa = []" and "y = [Some ta__]"   | t :: "('c, 'd) bintree" and xs :: "('c, 'd) bintree option list" where "x = Some tb__" and "xa = None # xsb__" and "y = Some tb__ # xsb__"   | t :: "('c, 'd) bintree" and r :: "('c, 'd) bintree" and xs :: "('c, 'd) bintree option list" where "x = Some t" and "xa = Some r # xs" and "y = None # add (Some (merge t r)) xs"
le_factor: fixes x :: "'a"   and aa :: "'a"   and da :: "'a"   and ba :: "'a"   and ca :: "'a" assumes "x \<noteq> (0::'a)" shows "(aa * da * (ba * da) \<le> ca * ba * (ba * da)) = (aa * x * da * (ba * x * da) \<le> ca * (ba * x) * (ba * x * da))"
bisim1FAss2: fixes P :: "'a prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and v :: "'addr val" assumes "P,e2,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e\<bullet>F{D} := e2,h \<turnstile> (Val v\<bullet>F{D} := e', xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 e) + pc, xcp)"
col3: fixes X :: "'p"   and Y :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "X \<noteq> Y"   and "Col X Y A"   and "Col X Y B"   and "Col X Y C" shows "Col A B C"
set_sorted_list_of_multiset: fixes M :: "'b multiset" shows "set (sorted_list_of_multiset M) = set_mset M"
gt_one_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "less_eq one x" shows "less_eq y (times x y)"
complex_of_real_i: fixes r :: "real" shows "complex_of_real r * \<i> = Complex 0 r"
le: fixes x :: "real"   and y :: "real" assumes "real m - 1 \<le> x"   and "x \<le> y"   and "y \<le> real n" shows "f y \<le> f x"
IH: assumes "wf_theory \<Theta>"   and "distinct l"   and "\<forall>a\<in>set l. case a of (v, S) \<Rightarrow> typ_ok \<Theta> (Tv v S)"   and "wf_inst \<Theta> \<rho>" shows "inst_ok \<Theta> (map (\<lambda>a. case a of (v, S) \<Rightarrow> ((v, S), \<rho> v S)) l)"
map_id: fixes t :: "'b error_result" shows "map_error_result id t = t"
multiplicity_unit_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times x one" shows "factorial_semiring.multiplicity times one p x = 0"
uminus_eq_Minf: fixes x :: "'a extended" shows "(- x = -\<infinity>) = (x = \<infinity>)"
perm_def: shows "perm \<equiv> {f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"
in_arcs_add_arc_iff: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c"   and u :: "'d" shows "in_arcs (pre_digraph.add_arc G a) u = (if head G a = u then insert a (in_arcs G u) else in_arcs G u)"
simps_18: fixes v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "'a" shows "({v} WHILE va DO vb {vc} \<le> SKIP {vd}) = False"
induct: fixes x :: "('c, unit) Old_Datatype.node set"   and P :: "('c, unit) Old_Datatype.node set \<Rightarrow> bool" assumes "x \<in> sexp"   and "\<And>a. P (Leaf a)"   and "\<And>i. P (Numb i)"   and "\<And>M N. \<lbrakk>M \<in> sexp; P M; N \<in> sexp; P N\<rbrakk> \<Longrightarrow> P (Scons M N)" shows "P x"
Q1''_a_def: shows "Q1''_a \<equiv> [: p, t, lnk, lbl, mrk \<leadsto> p', t', lnk', lbl', mrk' . root = nil \<and> p' = nil \<and> t' = nil \<and> lnk' = lnk \<and> lbl' = lbl \<and> mrk' = mrk :]"
fresh_pgwt_empty_synth: fixes S :: "'fun set"   and t :: "('fun, 'var) Term.term" assumes "finite S"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "{} \<turnstile>\<^sub>c fresh_pgwt S (\<Gamma> t)"
base_case_sgas_R: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0" shows "set (characterize_consistent_signs_at_roots p [q]) \<subseteq> {[1], [0], [- 1]}"
zero_sheaf_spec_def: fixes U :: "nat set set" shows "cring0.zero_sheaf_spec U \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.zero_rel ({0}\<setminus>\<pp>) {0} ring0.subtraction ring0.subtraction 0 0"
elims: fixes x :: "'b heap"   and y :: "'b heap" assumes "del_min' x = y" obtains "x = heap.Empty" and "y = heap.Empty"   | xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "y = pass\<^sub>2' (pass\<^sub>1' hs)"
IH1: shows "readArgs g v n phis ms = (phis'_aux g v (set ms) phis, map (\<lambda>m. lookupDef g m v) ms)"
certC_def: fixes PAN :: "nat"   and Ka :: "nat"   and PS :: "nat"   and T :: "msg"   and signK :: "nat" shows "certC PAN Ka PS T signK = signCert signK \<lbrace>Hash \<lbrace>Nonce PS, Pan PAN\<rbrace>, Key Ka, T\<rbrace>"
swapVal_preserves_sWls: fixes SEM :: "('b, 'c, 'd, 'sort, 'e, 'f, 'g) semDom_scheme"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'f" shows "sWlsVal SEM (val ^[z1 \<and> z2]_zs) = sWlsVal SEM val"
top_impl_not_bot: fixes p :: "alt \<Rightarrow> real"   and b :: "alt" assumes "p <\<cdot> b" shows "\<not> b \<cdot>< p"
SFAccInitThrowReds: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. shp s D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,s,False\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},s,False\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
parallel_composition: fixes V :: "'a list"   and V' :: "'a list"   and SR :: "(('a \<times> ('b \<Rightarrow> 'c)) \<times> 'a list \<times> ('b \<Rightarrow> 'c)) set"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e" assumes "length V = length V'"   and "\<forall>i<length V. ([V ! i], [V' ! i]) \<in> Strong_Security.USdB SR DA d" shows "(V, V') \<in> Strong_Security.USdB SR DA d"
OUTfromChCorrect_data15: shows "OUTfromChCorrect data15"
seq_annir: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and seq :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seqstar :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.cka plus less_eq less one zero times star seq seqstar" shows "seq x zero = zero"
minarc_arc: fixes x :: "'a" shows "x \<noteq> bot \<longrightarrow> bijective (minarc x * top) \<and> bijective (minarc x[[top]])"
preferred_alts_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "preferred_alts R x = {y. R x y}"
set_of_nn: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and n :: "ref" assumes "Dag p low high t"   and "n \<in> set_of t" shows "n \<noteq> Null"
mk_all_fv_unchanged: fixes \<tau> :: "typ"   and x :: "variable"   and B :: "Core.term" shows "fv (mk_all x \<tau> B) = fv B - {(x, \<tau>)}"
simps: fixes H :: "'b pl set"   and a :: "'b pl" shows "(H |- a) = ((\<exists>p. a = p \<and> p \<in> H) \<or> (\<exists>p q. a = p -> q -> p) \<or> (\<exists>p q r. a = (p -> q -> r) -> (p -> q) -> p -> r) \<or> (\<exists>p. a = ((p -> false) -> false) -> p) \<or> (\<exists>p q. a = q \<and> H |- p -> q \<and> H |- p))"
vangle_eq_piD: fixes u :: "'b"   and v :: "'b" assumes "vangle u v = pi" shows "norm u *\<^sub>R v + norm v *\<^sub>R u = (0::'b)"
exactly_one_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and p :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final" shows "\<exists>!i. \<not> ps (distributed_system.s channel trans send recv init t i) p \<noteq> None \<and> ps (distributed_system.s channel trans send recv init t (i + 1)) p \<noteq> None"
p: shows "p = (\<Sum>a\<in>A. q a * a)"
st_vec_st_vec_of_pmf: fixes x :: "'a pmf"   and i :: "'a" shows "st_vec (st_vec_of_pmf x) $h i = pmf x i"
3: shows "[l] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> (m # ms) \<preceq> [l] \<langle>g\<rangle> (m # ms)"
assms_2: fixes N :: "nat" shows "0 \<le> f N"
ide_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VxV.ide f = (ide (fst f) \<and> ide (snd f))"
AddSA_Chi3: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d"   and T :: "'d" assumes "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A"   and "T \<notin> States SA"   and "T \<noteq> S" shows "Chi (A [++] (S, SA)) T = Chi A T"
surj: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ArrMap\<rparr> `\<^sub>\<circ> Hom \<AA> a b = Hom \<CC> ((\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>) ((\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>)"
not_distinct_conv_prefix: fixes as :: "'c list" shows "(\<not> distinct as) = (\<exists>xs y ys. y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys)"
exhaust: fixes y :: "BExpr" obtains x1 :: "Val \<Rightarrow> Val \<Rightarrow> bool" and x2 :: "Expr" and x3 :: "Expr" where "y = compB x1 x2 x3"
simps_2: fixes x :: "name"   and \<Psi>' :: "'b"   and F :: "'b frame" assumes "x \<sharp> \<Psi>'" shows "insertAssertion (\<lparr>\<nu>x\<rparr>F) \<Psi>' = \<lparr>\<nu>x\<rparr>insertAssertion F \<Psi>'"
ltlc_to_ltln'_correct_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc" shows "w \<Turnstile>\<^sub>n ltlc_to_ltln' False \<phi> = w \<Turnstile>\<^sub>c \<phi>"
cases: fixes x :: "'b list" obtains "x = []"   | xa :: "'b" and ys :: "'b list" where "x = xa # ys"
clD: fixes c :: "'b"   and L :: "'b set set"   and r :: "'b set"   and B :: "'b set" assumes "c \<in> cl L r"   and "B \<in> L"   and "r \<subseteq> B" shows "c \<in> B"
quotient_imp_continuous_map: fixes X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "quotient_map X X' f" shows "continuous_map X X' f"
lens_source_def: fixes X :: "'d \<Longrightarrow> 'c" shows "\<S>\<^bsub>X\<^esub> = {s. \<exists>v s'. s = put\<^bsub>X\<^esub> s' v}"
hyps_2: fixes T :: "secLevel" assumes "\<Gamma> \<turnstile> e2 : T"   and "x \<in> fv e2"   and "\<Gamma> x = Some High" shows "\<Gamma> \<turnstile> e2 : High"
transitionE_1: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P'" shows "\<exists>P'' P'''. P \<Longrightarrow>\<^sub>\<tau> P'' \<and> P'' \<longmapsto> \<alpha> \<prec> P''' \<and> P''' \<Longrightarrow>\<^sub>\<tau> P'"