hcju/isabelleps huggingface dataset:
assms_4: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, uu_). castmsg (I \<sigma>) a)"
C: shows "\<forall>\<^sub>F x in F. h x / g x / h x = inverse (g x)"
assms_3: fixes t1 :: "('a, unit) RBT_Impl.rbt"   and t2 :: "('a, unit) RBT_Impl.rbt"   and color :: "color"   and a :: "'a"   and b :: "unit" assumes "P t1"   and "P t2"   and "t1 \<noteq> rbt.Empty" shows "P (Branch color t1 a b t2)"
add_rat_1: fixes y :: "int poly \<times> rat \<times> rat"   and r1 :: "rat" assumes "invariant_1_2 y" shows "invariant_1_2 (add_rat_1 r1 y) \<and> real_of_1 (add_rat_1 r1 y) = real_of_rat r1 + real_of_1 y"
prime_elem_dvd_prod_msetE: fixes p :: "'a"   and A :: "'a multiset" assumes "prime_elem p"   and "p dvd \<Prod>\<^sub># A" obtains a :: "'a" where "a \<in># A" and "p dvd a"
bind_eq_OkE: fixes x :: "('a, 'c) result"   and f :: "'c \<Rightarrow> ('a, 'b) result"   and z :: "'b" assumes "x \<bind> f = Ok z" obtains xa :: "'c" where "x = Ok xa \<and> f xa = Ok z"
wtimes_exact: fixes w1 :: "32 Word.word"   and r1 :: "real"   and w2 :: "32 Word.word"   and r2 :: "real" assumes "w1 \<equiv>\<^sub>E r1"   and "w2 \<equiv>\<^sub>E r2" shows "wtimes w1 w2 \<equiv>\<^sub>E r1 * r2"
non_zero_mult_rep_non_zero: fixes v :: "(real, 3) vec"   and C :: "cltn2" assumes "v \<noteq> 0" shows "v v* cltn2_rep C \<noteq> 0"
affine_scheme_def: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and X :: "'e set"   and is_open :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'e set \<Rightarrow> 'f set"   and \<rho> :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and add_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str :: "'e set \<Rightarrow> 'f"   and one_str :: "'e set \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'd set"   and \<phi>\<^sub>f :: "'d set set \<Rightarrow> ('d set \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> 'f" shows "affine_scheme R addition multiplication zero unit X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str f \<phi>\<^sub>f \<equiv> comm_ring R addition multiplication zero unit \<and> locally_ringed_space X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str \<and> iso_locally_ringed_spaces X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str (comm_ring.spectrum R addition multiplication zero unit) (comm_ring.is_zariski_open R addition multiplication zero unit) (comm_ring.sheaf_spec R addition multiplication zero unit) (comm_ring.sheaf_spec_morphisms R addition multiplication zero unit) comm_ring.\<O>b (comm_ring.add_sheaf_spec R addition multiplication zero) (comm_ring.mult_sheaf_spec R addition multiplication zero) (comm_ring.zero_sheaf_spec R addition multiplication zero unit) (comm_ring.one_sheaf_spec R addition multiplication zero unit) f \<phi>\<^sub>f"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>0 r) (tab\<^sub>0 r)\<^sup>* \<eta> \<epsilon>"
chain_tllist_terminal: fixes A :: "('c, 'b) tllist set" assumes "Complete_Partial_Order.chain local.tllist_ord A" shows "Complete_Partial_Order.chain (flat_ord b) {terminal xs |xs. xs \<in> A \<and> tfinite xs}"
A\<^sub>A_not_None: shows "A\<^sub>A_of A\<^sub>C x = Some (v, v')"
mod_eq_imp_eq: fixes b :: "nat"   and x :: "nat"   and y :: "nat" assumes "b \<le> x"   and "x < b + N"   and "b \<le> y"   and "y < b + N"   and "x mod N = y mod N" shows "x = y"
padr_padl_lindep: fixes A :: "'a vec set"   and n :: "nat"   and B :: "'a vec set"   and m :: "nat" assumes "A \<subseteq> carrier_vec n"   and "\<not> module.lin_dep class_ring (M n) A"   and "B \<subseteq> carrier_vec m"   and "\<not> module.lin_dep class_ring (M m) B" shows "\<not> module.lin_dep class_ring (M (n + m)) ((\<lambda>v. v @\<^sub>v 0\<^sub>v m) ` A \<union> (@\<^sub>v) (0\<^sub>v n) ` B)"
Zero_Mem_Iff: shows "{} \<turnstile> Zero IN Var i IFF SyntaxN.Ex j (Var j EQ Zero AND Var j IN Var i)"
bind_map_pmf: fixes f :: "'g \<Rightarrow> 'f"   and M :: "'g pmf"   and g :: "'f \<Rightarrow> 'e pmf" shows "map_pmf f M \<bind> g = M \<bind> (\<lambda>x. g (f x))"
"1_2": shows "(T, S) \<in> \<langle>A\<rangle>Rs"
frac_add_leIFF: fixes d :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> d"   and "d < 1 - frac a"   and "d < 1 - frac b" shows "(frac a \<le> frac b) = (frac (a + d) \<le> frac (b + d))"
projection_cadjoint: fixes M :: "'a set" assumes "closed_csubspace M" shows "projection M\<^sup>\<dagger> = projection M"
Collect_conj_eq: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "rel_ext (\<lambda>x. P x \<and> Q x) = rel_ext P \<inter> rel_ext Q"
card_of_mono2: fixes r :: "('e \<times> 'e) set"   and r' :: "('f \<times> 'f) set" assumes "r \<le>o r'" shows "|Field r| \<le>o |Field r'|"
euler_err_fas_nth_def: fixes X0 :: "'a"   and h :: "floatarith"   and CX :: "floatarith list"   and i :: "nat" shows "euler_err_fas_nth X0 h CX i = h ^\<^sub>e 2 / floatarith.Num 2 * ode_d_fa_nth 0 CX (ode_fa CX) i"
nondet_assign_def: fixes x :: "'d" shows "x ::= ?  = (\<lambda>s. {HS_VC_Spartan.vec_upd s x k |k. True})"
comp2_def: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'c"   and y :: "'d" shows "(f o2 g) x y \<equiv> f (g x y)"
overlap_symmetry: fixes x :: "i"   and y :: "i" assumes "O x y" shows "O y x"
a_subid_aux1': fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> y \<le> y"
is_ChurchRosser_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "relation.is_ChurchRosser r \<equiv> \<forall>a b. relation.srtc r a b \<longrightarrow> relation.cs r a b"
finite_edges: fixes e :: "'a set" assumes "e \<in> E" shows "finite e"
mod_mod_trivial: fixes divide :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and one :: "'g"   and modulo :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and euclidean_size :: "'g \<Rightarrow> nat"   and a :: "'g"   and b :: "'g" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (modulo a b) b = modulo a b"
supp: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "VectorSpaceHom smult V smult' T" shows "supp T \<subseteq> V"
pdevs_val_zero_pdevs: fixes e :: "nat \<Rightarrow> real" shows "pdevs_val e zero_pdevs = (0::'a)"
hyps: fixes c' :: "('p, 's, 'm) configuration" assumes "trace c t c'"   and "snd (cs c cid) \<noteq> NotStarted"   and "channel cid = Some (p, q)" shows "snd (cs c' cid) \<noteq> NotStarted"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(I x1 = I y1) = (x1 = y1)"
vconst_on_atI: fixes a :: "V"   and A :: "V"   and c :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "vconst_on A c\<lparr>a\<rparr> = c"
mult2_ns_conv: fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "mult2_ns ns s = mult2_s ns s \<union> multpw ns"
Dfb_apply: fixes x :: "'a" shows "blinfun_apply (Dfb x) = Df x"
map_sel_1: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c tree" shows "root (map_tree f a) = f (root a)"
simps: fixes Ps :: "('d multiset \<Rightarrow> 'a) list"   and i :: "'d" shows "indexed_eval_aux Ps i = foldr (\<lambda>P Q. Q \<Otimes> i \<Oplus> P) Ps (indexed_const \<zero>)"
initPD_def: fixes sh :: "sheap"   and C :: "char list" shows "initPD sh C \<equiv> \<exists>sfs i. sh C = \<lfloor>(sfs, i)\<rfloor> \<and> (i = Done \<or> i = Processing)"
induct: fixes x1 :: "'c list"   and x2 :: "'c list"   and P :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "x1 <~~> x2"   and "P [] []"   and "\<And>y x l. P (y # x # l) (x # y # l)"   and "\<And>xs ys z. \<lbrakk>xs <~~> ys; P xs ys\<rbrakk> \<Longrightarrow> P (z # xs) (z # ys)"   and "\<And>xs ys zs. \<lbrakk>xs <~~> ys; P xs ys; ys <~~> zs; P ys zs\<rbrakk> \<Longrightarrow> P xs zs" shows "P x1 x2"
imeq: shows "(hull) affine ` (\<lambda>a. c - {a}) ` (c - b) = (\<lambda>a. affine hull (c - {a})) ` (c - b)"
poly_weak_mono_all_pos: fixes p :: "('d monom \<times> 'c) list" assumes "(0::'c) \<le> eval_poly (\<lambda>w. 0::'c) p"   and "poly_weak_mono_all p" shows "p \<ge>p zero_poly"
att_or_snd_hd: fixes a :: "'a attree"   and list :: "'a attree list"   and aa :: "'a set"   and b :: "'a set" assumes "\<turnstile>a # list \<oplus>\<^sub>\<or>\<^bsup>(aa, b)\<^esup>" shows "snd (attack a) \<subseteq> b"
aff_eq: shows "affine hull (S - y ` K) = affine hull S"
intro: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"   and "tyenv_wellformed mds \<Gamma> \<S> P"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"   and "pred P mem\<^sub>1"   and "pred P mem\<^sub>2"   and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x" shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c, mds, mem\<^sub>2\<rangle>"
distinct_5: fixes x1 :: "'a"   and x4 :: "'c set" shows "Fu x1 \<noteq> Abs x4"
notin_range_Some: fixes x :: "'a option" shows "(x \<notin> range Some) = (x = None)"
hm_is_it'_def: fixes l :: "('c \<times> 'd) list list"   and ht :: "('c, 'd) hashtable"   and l' :: "('c \<times> 'd) list"   and it :: "nat \<times> ('c \<times> 'd) list \<times> ('c, 'd) hashtable" shows "hm_is_it' l ht l' it \<equiv> is_hashtable l ht * \<up> (let (i, r, ht') = it in ht = ht' \<and> l' = concat (take i l) @ rev r \<and> distinct (map fst l') \<and> i \<le> length l \<and> (r = [] \<longrightarrow> i = 0))"
assms_4: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set" assumes "(h, U) \<in> set qs" shows "cone (h, U) \<subseteq> P[X]"
fundantivertex_inj_on: shows "inj_on fundantivertex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
chain': shows "Complete_Partial_Order.chain (ord_spmf (=)) ((\<lambda>p. p \<upharpoonleft> A) ` Y)"
INF_mono': fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Inf (f ` A)) (Inf (g ` A))"
deg_in_aug_minf: fixes p :: "'a" assumes "p \<in> carrier R" shows "deg R S X p \<in> Z\<^sub>-\<^sub>\<infinity>"
frameImpResPres: fixes F :: "'b frame"   and G :: "'b frame"   and x :: "name" assumes "F \<hookrightarrow>\<^sub>F G" shows "\<lparr>\<nu>x\<rparr>F \<hookrightarrow>\<^sub>F \<lparr>\<nu>x\<rparr>G"
seq_mono_right: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "(c\<^sub>0::'a::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'a::refinement_lattice)" shows "(d::'a::refinement_lattice) ; (c\<^sub>0::'a::refinement_lattice) \<sqsubseteq> d ; (c\<^sub>1::'a::refinement_lattice)"
from_nat_to_nat: fixes j :: "nat"   and k :: "'a" assumes "from_nat j = k"   and "j < CARD('a)" shows "to_nat k = j"
adm_wf_def: fixes R :: "('a \<times> 'a) set"   and F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" shows "adm_wf R F = (\<forall>f g x. (\<forall>z. (z, x) \<in> R \<longrightarrow> f z = g z) \<longrightarrow> F f x = F g x)"
function_times_one_l: fixes a :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier F" shows "\<one>\<^bsub>F\<^esub> \<otimes>\<^bsub>F\<^esub> a = a"
ivl_diff: fixes i :: "'c"   and n :: "'c"   and m :: "'c" assumes "i \<le> n" shows "{i..<m} - {i..<n} = {n..<m}"
r_coset_subset_G: fixes H :: "nat set"   and x :: "nat" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x \<subseteq> carrier G"
supp_sum_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> 'b"   and S :: "'c set" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.supp_sum plus zero f S = comm_monoid_add.sum plus zero f (monoid_add.support_on zero S f)"
routing_ipassmt_wi_disjoint: fixes tbl :: "'b routing_rule list"   and a1 :: "char list"   and a2 :: "char list"   and b1 :: "'b wordinterval"   and b2 :: "'b wordinterval" assumes "valid_prefixes tbl"   and "a1 \<noteq> a2"   and "(a1, b1) \<in> set (routing_ipassmt_wi tbl)"   and "(a2, b2) \<in> set (routing_ipassmt_wi tbl)" shows "wordinterval_to_set b1 \<inter> wordinterval_to_set b2 = {}"
general_product_character: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>z. \<forall>y. F y \<longrightarrow> P z y" shows "\<forall>y. O y (\<pi>x. F x) = (\<exists>z. (\<forall>y. F y \<longrightarrow> P z y) \<and> O y z)"
row_mat_of_cols: fixes i :: "nat"   and nr :: "nat"   and ls :: "'b vec list" assumes "i < nr" shows "row (mat_of_cols nr ls) i = vec (length ls) (\<lambda>j. ls ! j $ i)"
lookup_raw_map_raw: fixes xs :: "('b \<times> 'c) list \<times> 'b nat_term_order"   and f :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'd"   and k :: "'b"   and v :: "'c" assumes "ko_ntm.oalist_inv xs"   and "snd (f (k, 0::'c)) = (0::'d)"   and "\<And>a b. key_compare (key_order_of_nat_term_order_inv (snd xs)) (fst (f a)) (fst (f b)) = key_compare (key_order_of_nat_term_order_inv (snd xs)) (fst a) (fst b)" shows "lookup_ko_ntm (map_raw f xs) (fst (f (k, v))) = snd (f (k, lookup_ko_ntm xs k))"
cone_empty: fixes n :: "nat" shows "gram_schmidt.cone n {} = {0\<^sub>v n}"
uniqButlastNotUniqListImpliesLastMemButlast: fixes l :: "'b list" assumes "uniq (butlast l)"   and "\<not> uniq l" shows "last l \<in> set (butlast l)"
assms: assumes "I s" shows "M \<le> \<Down> R (WHILE\<^bsup>I\<^esup> b f s)"
iMOD_iFROM_subset_same: fixes r :: "nat"   and m :: "nat" shows "[ r, mod m ] \<subseteq> [r\<dots>]"
submoduleE_5: fixes H :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Module.submodule H R M"   and "a \<in> H"   and "b \<in> H" shows "a \<oplus>\<^bsub>M\<^esub> b \<in> H"
prems: shows "\<Gamma> \<turnstile> App p s \<leftrightarrow> u : T\<^sub>2'"
igSubstIGVar1_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igSubstIGVar1 MOD \<equiv> \<forall>ys y Y xs x. igWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igSubst MOD ys Y y (igVar MOD xs x) = igVar MOD xs x"
lspasl_e: fixes Gamma :: "bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and Delta :: "bool" assumes "Gamma \<and> (h1,h2\<triangleright>h0) \<and> (h2,h1\<triangleright>h0) \<longrightarrow> Delta" shows "Gamma \<and> (h1,h2\<triangleright>h0) \<longrightarrow> Delta"
invar_cons_down: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" assumes "invar (t # bq)" shows "invar bq"
encode12_inv2: fixes i :: "nat"   and dims :: "nat list"   and vars :: "nat set"   and j :: "nat" assumes "i < partial_state.d1 dims vars"   and "j < partial_state.d2 dims vars" shows "partial_state.encode2 dims vars (partial_state.encode12 dims vars (i, j)) = j"
simps_14: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Since \<phi> I \<psi>) j = min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)"
Hiy: shows "H $ i $ y \<noteq> (0::'a)"
subset: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "submonoid H G" shows "H \<subseteq> carrier G"
inter_gamma_subset_gamma_meet: fixes a1 :: "'av"   and a2 :: "'av" shows "\<gamma> a1 \<inter> \<gamma> a2 \<subseteq> \<gamma> (a1 \<sqinter> a2)"
symmetric_reachable': fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w" shows "w \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"
continuous_inj_imp_mono: fixes a :: "'a"   and x :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "a < x"   and "x < b"   and "continuous_on {a..b} f"   and "inj_on f {a..b}" shows "f a < f x \<and> f x < f b \<or> f b < f x \<and> f x < f a"
sel_3: fixes x1 :: "'c"   and x2 :: "'d"   and x3 :: "nat"   and x4 :: "('c, 'd) SkewBinomialTree list" shows "rank (Node x1 x2 x3 x4) = x3"
num_of_odd_nodes_def: fixes g :: "('a, 'b) graph" shows "num_of_odd_nodes g \<equiv> card (odd_nodes_set g)"
lock_thread_ok_has_lock: fixes ls :: "('a, 'b) locks"   and ts :: "'b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option"   and l :: "'a"   and t :: "'b" assumes "lock_thread_ok ls ts"   and "has_lock (ls $ l) t" obtains xw :: "'c \<times> 'a \<Rightarrow>f nat" where "ts t = \<lfloor>xw\<rfloor>"
pre_2: fixes \<sigma> :: "'\<sigma>"   and x :: "'xc"   and S :: "'xa set" assumes "c \<sigma>"   and "invar x"   and "\<alpha> x \<in> S"   and "I S \<sigma>"   and "S \<subseteq> S0"   and "\<forall>y\<in>S - {\<alpha> x}. R (\<alpha> x) y"   and "\<forall>y\<in>S0 - S. R y (\<alpha> x)" shows "I (S - {\<alpha> x}) (f x \<sigma>)"
obj_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "obj (src \<mu>)"
sSum_two_Submodules: fixes H :: "'a set"   and K :: "'a set" assumes "submodule R M H"   and "submodule R M K" shows "submodule R M (H \<minusplus> K)"
function_mult_eval_car: fixes a :: "nat \<Rightarrow> int"   and f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and g :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "g \<in> carrier (function_ring (carrier Zp) Zp)" shows "(f \<otimes>\<^bsub>function_ring (carrier Zp) Zp\<^esub> g) a = f a \<otimes> g a"
order_less_le_subst1: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and b :: "'b"   and c :: "'b" assumes "a < f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a < f c"
pmf_map_inj': fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d pmf"   and x :: "'d" assumes "inj f" shows "pmf (map_pmf f M) (f x) = pmf M x"
A: shows "\<forall>A. case A of (A, A') \<Rightarrow> P (A, A'\<lparr>ok := False\<rparr>) \<longrightarrow> P (A, A'\<lparr>ok := True\<rparr>)"
reach_snoc_iff: fixes tr :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and p :: "'a"   and bs :: "'b list"   and b :: "'b"   and k :: "'a" shows "reach tr p (bs @ [b]) k = (\<exists>q. reach tr p bs q \<and> k = tr q b)"
trss_xchange_context: fixes s :: "'f list"   and c :: "'f list multiset"   and w :: "('g, 'h) label list"   and s' :: "'f list"   and csp :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and cn :: "'f list multiset" assumes "((s, c), w, s', csp + c) \<in> trcl (trss fg)"   and "mon_c fg cn \<subseteq> mon_c fg c" shows "((s, cn), w, s', csp + cn) \<in> trcl (trss fg)"
mset_eq: shows "mset (map snd xs) = mset (map snd ys)"
Args_min_simps_1: fixes f :: "int \<Rightarrow> 'a"   and I :: "int set" shows "Args_min_on f I = {k. is_arg_min_on f I k}"
induct: fixes P :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b list"   and a2 :: "'b list" assumes "\<And>uu_ xs. P uu_ xs []"   and "\<And>rel v va. P rel [] (v # va)"   and "\<And>rel x xs y ys. \<lbrakk>x = y \<Longrightarrow> P rel xs ys; \<lbrakk>x \<noteq> y; rel x y\<rbrakk> \<Longrightarrow> P rel xs (y # ys); \<lbrakk>x \<noteq> y; \<not> rel x y\<rbrakk> \<Longrightarrow> P rel (x # xs) ys\<rbrakk> \<Longrightarrow> P rel (x # xs) (y # ys)" shows "P a0 a1 a2"
filter_pdevs_raw_def: fixes I :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a" shows "filter_pdevs_raw I X = (\<lambda>i. if I i (X i) then X i else (0::'a))"
disjoint_union: fixes class1 :: "'a set"   and class2 :: "'a set" assumes "class1 \<in> classes"   and "class2 \<in> classes"   and "class1 \<inter> class2 \<noteq> {}" shows "class1 = class2"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" shows "count_list (x # xs) y = (if x = y then count_list xs y + 1 else count_list xs y)"
assms: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho>' f' g'"
one_ge_zero: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.ordered_semiring_1 one times plus zero less_eq" shows "less_eq zero one"
unique_connecting_path: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "\<exists>!xs. v \<leadsto>xs\<leadsto> w"
epath_mono: fixes E :: "'b uprod set"   and E' :: "'b uprod set"   and u :: "'b"   and p :: "'b uprod list"   and v :: "'b" assumes "E \<subseteq> E'"   and "epath E u p v" shows "epath E' u p v"
zip_map: shows "zip [0..<degree X] (dense_list_of_pdevs X) = map (\<lambda>i. (i, pdevs_apply X i)) [0..<degree X]"
subset_Under_Above: fixes B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "B \<subseteq> Field r" shows "B \<subseteq> Under r (Above r B)"
primitive_part_smult: fixes a :: "'b"   and p :: "'b poly" shows "primitive_part (smult a p) = smult (unit_factor a) (primitive_part p)"
inj_onD: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and x :: "'g"   and y :: "'g" assumes "inj_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
H: shows "\<forall>U. U \<in> S \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)"
incidence_system_axioms: shows "incidence_system \<V> \<B>\<^sup>C"
inf_bot_right: fixes inf :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and bot :: "'x"   and x :: "'x" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf x bot = bot"
abs_pat_alpha: fixes pi :: "(name \<times> name) list"   and p :: "pat"   and t :: "btrm" assumes "(pi \<bullet> supp p) \<sharp>* t"   and "set pi \<subseteq> supp p \<times> pi \<bullet> supp p" shows "(\<lambda>[p]. t) = (\<lambda>[pi \<bullet> p]. pi \<bullet> t)"
case_distrib: fixes h :: "'d \<Rightarrow> 'c"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'e list \<Rightarrow> 'd"   and list :: "'e list" shows "h (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (case list of [] \<Rightarrow> h f1 | x1 # x2 \<Rightarrow> h (f2 x1 x2))"
preorder_top_closed: shows "top \<le> top \<and> top \<sqinter> top \<le> top"
fl: fixes A :: "'b set"   and B :: "'b set"   and s1 :: "'b \<Rightarrow> 'c"   and s2 :: "'b \<Rightarrow> 'c" assumes "A \<subseteq> B"   and "s1 = s2 on B" shows "s1 = s2 on A"
subset_snd_imageI: fixes A :: "'d set"   and B :: "'e set"   and S :: "('d \<times> 'e) set"   and x :: "'d" assumes "A \<times> B \<subseteq> S"   and "x \<in> A" shows "B \<subseteq> snd ` S"
assms_1: shows "x \<in> fv\<^sub>s\<^sub>s\<^sub>t T"