hcju/isabelleps huggingface dataset:
Sup_llist_imp_exists_index: fixes x :: "'a"   and Xs :: "'a set llist" assumes "x \<in> Sup_llist Xs" shows "\<exists>i. enat i < llength Xs \<and> x \<in> lnth Xs i"
PiE_subgroup_product_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme"   and H :: "'f \<Rightarrow> 'g set" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "subgroup (Pi\<^sub>E I H) (product_group I G) = (\<forall>i\<in>I. subgroup (H i) (G i))"
converse_rtranclpE: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and z :: "'a" assumes "r\<^sup>*\<^sup>* x z" obtains "x = z"   | y :: "'a" where "r x y" and "r\<^sup>*\<^sup>* y z"
cod_simps_3: shows "src.cod.leg1 = src.leg1"
lnth_llist_of_tllist: fixes xs :: "('a, 'b) tllist" shows "lnth (llist_of_tllist xs) = tnth xs"
il_less_def: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
supported_def: fixes a :: "'c zmultiset"   and t :: "'c" shows "supported a t = (\<exists>s<t. zcount a s < 0)"
changes_poly_neg_inf_pders: fixes p :: "real poly" shows "changes_poly_neg_inf (pders p) = int (degree p)"
iso_def: fixes G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" shows "Group.iso G H = {h \<in> hom G H. bij_betw h (carrier G) (carrier H)}"
prems_4: shows "ps (S t i) p \<noteq> None"
sequent_def: fixes P :: "'b upred"   and Q :: "'b upred" shows "(P \<tturnstile> Q) = (Q \<sqsubseteq> P)"
Adjunct_segment_eq: fixes X :: "'a set" assumes "S_inductive_set D"   and "Chain D X"   and "Algebra2.Sup D X \<notin> X" shows "segment (Iod D (insert (Algebra2.Sup D X) X)) (Algebra2.Sup D X) = X"
intro: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set" assumes "c_manifold' charts k"   and "submanifold_axioms S" shows "submanifold charts k S"
safe_distance_normal_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" shows "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> \<equiv> safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<and> safe_distance_normal_axioms \<delta>"
hyps_1: shows "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> ROut M C"
\<R>_typed_step: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c\<^sub>1 :: "('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"   and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1} \<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"   and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>" shows "\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>1, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>"
low_rank_det_zero: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "det A \<noteq> (0::'a)" shows "vec_space.rank n A = n"
suminf_indicator: fixes A :: "nat \<Rightarrow> 'b set"   and x :: "'b" assumes "disjoint_family A" shows "(\<Sum>n. indicator (A n) x) = indicator (\<Union> (range A)) x"
euler_consistent_solution: fixes t' :: "real" assumes "t' \<in> {t0..t0 + e}" shows "consistent solution t' (t0 + e) (euler_C TYPE('a)) 1 (euler_increment f)"
plus_\<I>_sel_1: fixes \<I>l :: "('i, 'k) \<I>"   and \<I>r :: "('j, 'l) \<I>" shows "outs_\<I> (\<I>l \<oplus>\<^sub>\<I> \<I>r) = outs_\<I> \<I>l <+> outs_\<I> \<I>r"
agreement_def: fixes sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and c :: "('a, 'c, 'b) configuration" assumes "flpSystem sends" shows "flpSystem.agreement c \<equiv> \<forall>v1. <\<bottom>, outM v1> \<in># msgs c \<longrightarrow> (\<forall>v2. <\<bottom>, outM v2> \<in># msgs c = (v2 = v1))"
of_int_of_nat_eq: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (int n) = semiring_1.of_nat one plus zero n"
validTrans: fixes trn :: "('state, 'act, 'out) trans" shows "validTrans trn = (step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn))"
simps_2: fixes p :: "nat" shows "showsp_bool p False = shows_string ''False''"
metric_Cauchy_iff2: fixes X :: "nat \<Rightarrow> 'c" shows "Cauchy X = (\<forall>j. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < inverse (real (Suc j)))"
strict_coboundedI2: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b < c" shows "a \<sqinter> b < c"
that: fixes a :: "int"   and b :: "int" assumes "q = Fract a b"   and "0 < b"   and "coprime a b" shows "C"
SD_efficient_support_subset: fixes p :: "'alt pmf"   and p' :: "'alt pmf" assumes "SD_efficient R p"   and "set_pmf p' \<subseteq> set_pmf p"   and "p \<in> lotteries" shows "SD_efficient R p'"
interval_lebesgue_integral_def: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" shows "interval_lebesgue_integral M a b f = (if a \<le> b then set_lebesgue_integral M (einterval a b) f else - set_lebesgue_integral M (einterval b a) f)"
hom_induced_hom: fixes S :: "'a set"   and P :: "('a \<times> bool) list set"   and f :: "'a \<Rightarrow> 'b" assumes "GroupByPresentationInducedFun S P f" shows "GroupHom (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (GroupByPresentationInducedFun.induced_hom S P f)"
inj_right_lang_imp_eq_app_right_index: assumes "inj_on MN.right_lang (dfa.states Canon.DFA)" shows "bij_betw (\<lambda>q. eq_app_right MN.language `` {MN.path_to q}) MN.accessible (UNIV // eq_app_right MN.language)"
Infinitesimal_HFinite_mult2: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Infinitesimal"   and "y \<in> HFinite" shows "y * x \<in> Infinitesimal"
unequal: fixes a :: "'a array"   and a' :: "'a array" shows "(a \<noteq> a') = a =!!= a'"
good_item_simps_77: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and xa :: "'var"   and x1 :: "'var" assumes "good Y"   and "fresh xsa xa Y" shows "Y #[x1 // xa]_xsa = Y"
intros_2: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi>' :: "nat \<Rightarrow> 'a"   and k' :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<sigma> :: "'b \<Rightarrow> 'c"   and \<sigma>' :: "'b \<Rightarrow> 'c" assumes "IFC_def.cs prob \<pi>' k' = IFC_def.cs prob \<pi> k"   and "\<pi> = IFC_def.path prob \<sigma>"   and "\<pi>' = IFC_def.path prob \<sigma>'"   and "IFC_def.kth_state prob \<sigma> k \<restriction> IFC_def.reads prob (\<pi> k) \<noteq> IFC_def.kth_state prob \<sigma>' k' \<restriction> IFC_def.reads prob (\<pi> k)" shows "IFC_def.contradicts prob (\<sigma>', k') (\<sigma>, k)"
swap: fixes g :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and B :: "'m multiset"   and A :: "'l multiset" shows "(\<Sum>i\<in>#A. \<Sum>\<^sub># (image_mset (g i) B)) = (\<Sum>j\<in>#B. \<Sum>i\<in>#A. g i j)"
difg_Suc: shows "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> (difg m has_real_derivative difg (Suc m) t) (at t)"
norm: fixes k :: "nat" shows "cmod (principal_dchar n k) = (if coprime k n then 1 else 0)"
cases: fixes n :: "'node SDG_node"   and a :: "'node SDG_node" assumes "sum_SDG_slice1p n a" obtains "a = n" and "valid_SDG_node n"   | n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a = n''__" and "n''__ s\<longrightarrow>\<^bsub>cd\<^esub> n'__" and "sum_SDG_slice1p n n'__"   | n'' :: "'node SDG_node" and V :: "'var" and n' :: "'node SDG_node" where "a = n''a__" and "n''a__ s-V__\<rightarrow>\<^sub>d\<^sub>d n'a__" and "sum_SDG_slice1p n n'a__"   | n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a = n''b__" and "n''b__ s-p__\<rightarrow>\<^bsub>call\<^esub> n'b__" and "sum_SDG_slice1p n n'b__"   | n'' :: "'node SDG_node" and p :: "'pname" and V :: "'var" and n' :: "'node SDG_node" where "a = n''c__" and "n''c__ s-pa__:V\<rightarrow>\<^bsub>in\<^esub> n'c__" and "sum_SDG_slice1p n n'c__"   | n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a = n''" and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'" and "sum_SDG_slice1p n n'"
EqOnUC_eqOnUC: fixes env :: "pvar \<Rightarrow> nat"   and p :: "pvar"   and i :: "nat"   and p' :: "pvar"   and i' :: "nat"   and u :: "'U"   and c :: "'C" assumes "env p = i"   and "env p' = i'" shows "sem (EqOnUC p p' u c) env = eqOnUC i i' u c"
cmp_rfn_def: fixes G :: "('e, 'f) Group_scheme"   and r :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and s :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "cmp_rfn G r f s g = (\<lambda>i. if i < s * r then f (Suc (i div s)) \<diamondop>\<^bsub>G\<^esub> f (i div s) \<inter> g (i mod s) else {\<one>\<^bsub>G\<^esub>})"
has_locks_unlock_lock_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "has_lock l t" shows "has_locks (unlock_lock l) t = has_locks l t - 1"
simps_2: fixes a :: "('a, 'b) Node" shows "FingerTreeStruc.viewRn (Single a) = Some (a, Empty)"
lattice_def: fixes L :: "('e, 'f) gorder_scheme" shows "lattice L \<equiv> upper_semilattice L \<and> lower_semilattice L"
\<ll>_ide_simp: fixes a :: "'d" assumes "ide a" shows "\<ll> a = \<l>[a]"
eq_divide_eq_1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(one = divide b a) = (a \<noteq> zero \<and> a = b)"
cont_paths: fixes f :: "'b \<Rightarrow> 'c ttree" assumes "cont f" shows "cont (\<lambda>x. paths (f x))"
chopped: fixes a :: "(real mpoly \<times> nat) list"   and b :: "(real mpoly \<times> nat) list"   and v :: "nat"   and p :: "real mpoly"   and f :: "nat \<Rightarrow> real" assumes "(a, b) = chophelper (to_list v p) []" shows "insertion f p = (\<Sum>(term, i)\<leftarrow>b. insertion f term * f v ^ i)"
epath_shift: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and "structure" :: "'f set"   and sat :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and srs :: "('e \<times> 'd) list"   and steps :: "('e \<times> 'd) stream" assumes "Soundness eff rules structure sat"   and "RuleSystem_Defs.epath eff rules (srs @- steps)" shows "RuleSystem_Defs.epath eff rules steps"
abs_of_rat: fixes r :: "rat" shows "\<bar>of_rat r\<bar> = of_rat \<bar>r\<bar>"
simulation_silent1_aux: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and s1' :: "'g"   and \<mu>1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "bisimulation_into_delay trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "bisim s1 s2"   and "\<tau>trsys.silent_move trsys1 \<tau>move1 s1 s1'" shows "bisim s1' s2 \<and> \<mu>1\<^sup>+\<^sup>+ s1' s1 \<or> (\<exists>s2'. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>+\<^sup>+ s2 s2' \<and> bisim s1' s2')"
insert_order: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "root_order k t" shows "root_order k (abs_split.insert k x t)"
elims_3: fixes x :: "'a extT"   and xa :: "'a extT" assumes "\<not> x < xa" obtains x :: "'a extT" and y :: "'a extT" where "xa__ = x" and "xa = y" and "\<not> (x \<le> y \<and> \<not> y \<le> x)"
par: shows "S'.par (\<Phi> (\<Psi> f')) f'"
f: shows "eval f (z # xs) \<notin> {None, Some 0}"
IH: shows "trms\<^sub>l\<^sub>s\<^sub>t A = (\<Union>l. trms\<^sub>l\<^sub>s\<^sub>t (proj l A))"
assms_2: shows "s\<^sub>2' = s\<^sub>1(r' \<mapsto> y)"
k_PBD_def: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int" shows "k_PBD point_set block_collection u_block_size \<equiv> k_\<Lambda>_PBD point_set block_collection 1 u_block_size"
div_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "div_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; rs1 = get_operand_w5 (op_list ! 1); rd = get_operand_w5 (op_list ! 3) in do operand2 \<leftarrow> gets (get_operand2 op_list); if uint operand2 = 0 then do raise_trap division_by_zero; return () od else div_comp instr rs1 rd operand2 od"
hoare_triple_D2: shows "\<turnstile>\<^sub>p {Q1} hadamard_n n;; Utrans_P vars1 mat_Ph;; hadamard_n n {Q2}"
open_A: shows "open {s. s0 < ereal (Re s)}"
finite_gpvD: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and out :: "'d"   and c :: "('f, 'd, 'e) rpv"   and input :: "'e" assumes "finite_gpv \<I> gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out" shows "finite_gpv \<I> (c input)"
sub_minus_def: fixes x :: "'a"   and y :: "'a" shows "- x - - y = - (- - x \<squnion> - y)"
abs_eq: fixes xa :: "int"   and x :: "int" shows "integer_of_int xa div integer_of_int x = integer_of_int (xa div x)"
assms: fixes x :: "'b" assumes "x \<in> A" shows "conv_abscissa (f x) \<le> d"
simps_1: fixes k :: "'b" shows "form_of_lit (k\<^sup>+) = Atom k"
nfoldli_rule: fixes I :: "'c list \<Rightarrow> 'c list \<Rightarrow> 'd \<Rightarrow> bool"   and l0 :: "'c list"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and P :: "'d \<Rightarrow> bool" assumes "I [] l0 \<sigma>0"   and "\<And>x l1 l2 \<sigma>. \<lbrakk>l0 = l1 @ x # l2; I l1 (x # l2) \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (l1 @ [x]) l2)"   and "\<And>l1 l2 \<sigma>. \<lbrakk>l0 = l1 @ l2; I l1 l2 \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma>. \<lbrakk>I l0 [] \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "nfoldli l0 c f \<sigma>0 \<le> SPEC P"
lookup_zero: fixes k :: "'e" shows "lookup 0 k = (0::'d)"
idem_star: fixes s :: "'a"   and r :: "'b \<times> 'c"   and rs :: "('b \<times> 'c) list" assumes "contains s r" shows "s \<star> rs = s \<star> filter (\<lambda>x. x \<noteq> r) rs"
proj_eq_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.proj_eq times zero = (\<lambda>(x\<^sub>1, y\<^sub>1, z\<^sub>1) (x\<^sub>2, y\<^sub>2, z\<^sub>2). (z\<^sub>1 = zero) = (z\<^sub>2 = zero) \<and> times x\<^sub>1 z\<^sub>2 = times x\<^sub>2 z\<^sub>1 \<and> times y\<^sub>1 z\<^sub>2 = times y\<^sub>2 z\<^sub>1)"
set_eq_fold: fixes xs :: "'b list" shows "gcd_list xs = fold gcd xs (0::'b)"
is_volatile_Write\<^sub>s\<^sub>b_simps_1: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_volatile_Write\<^sub>s\<^sub>b (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = volatile"
reachable_imp_identifiers_finite_1: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "reachable s" shows "finite (RID\<^sub>G s)"
less_imp_neq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "less x y" shows "x \<noteq> y"
size_mset_add_mset: fixes x :: "'d \<times> 'e"   and M :: "('d \<times> 'e) multiset" shows "size_mset (add_mset x M) = pair_size x + size_mset M"
lock_thread_ok_has_lock: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option"   and l :: "'d"   and t :: "'e" assumes "lock_thread_ok ls ts"   and "has_lock (ls $ l) t" obtains xw :: "'f \<times> 'd \<Rightarrow>f nat" where "ts t = \<lfloor>xw\<rfloor>"
rel_distinct_1: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c"   and y21 :: "'b"   and y22 :: "('b, 'd) tllist" shows "\<not> tllist_all2 R1 R2 (TNil x1) (TCons y21 y22)"
opp_Point: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ell_field R" shows "ell_field.opp R (Point x y) = Point x (\<ominus>\<^bsub>R\<^esub> y)"
hequiv_names: fixes H :: "(('b, 'c) fm list \<times> 'c) set"   and i :: "'c"   and j :: "'c" assumes "hequiv H i j" shows "i \<in> names H"
prefix_fininf_le_inf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a stream"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "w\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "traces.le_inf ind w\<^sub>2 w\<^sub>3" shows "traces.le_fininf ind w\<^sub>1 w\<^sub>3"
self_src_iff_self_trg: fixes a :: "'a" assumes "ide a" shows "(a = some_src a) = (a = some_trg a)"
ran_map_upd: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and b :: "'c" assumes "m a = None" shows "ran (m(a \<mapsto> b)) = insert b (ran m)"
ground_fresh_1: fixes t :: "trm"   and i :: "'a" assumes "ground t" shows "atom i \<sharp> t"
fresherI1: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and r' :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "nsqn\<^sub>r r < nsqn\<^sub>r r'" shows "r \<sqsubseteq> r'"
Phfgz: shows "PROP P (h (f z, g z))"
wf_interp_for_formula_any_Inr: fixes w :: "'a list"   and P :: "nat set"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "wf_interp_for_formula (w, Inr P # I) \<phi>" shows "\<forall>P. finite P \<longrightarrow> wf_interp_for_formula (w, Inr P # I) \<phi>"
user_reg_state_mod_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = s1\<lparr>user_reg := new_user_reg\<rparr> \<and> t2 = s2\<lparr>user_reg := new_user_reg\<rparr>" shows "low_equal t1 t2"
tllist_all2_thdD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist" assumes "tllist_all2 P Q xs ys"   and "\<not> is_TNil xs \<or> \<not> is_TNil ys" shows "P (thd xs) (thd ys)"
c1_2: fixes n :: "nat" assumes "1 \<le> n" shows "norm (sum f {1..n} - g (real n)) \<le> c1 * norm (h (real n))"
simp_atom_eval: fixes a :: "atom"   and xs :: "real list" shows "aEval a xs = eval (simp_atom a) xs"
lift_Postdomination: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and nx :: "'g" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"   and "Postdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit nx" shows "Postdomination src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry NewExit"
document_ptr_kinds_M_eq: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "|h \<turnstile> object_ptr_kinds_M|\<^sub>r = |h' \<turnstile> object_ptr_kinds_M|\<^sub>r" shows "|h \<turnstile> document_ptr_kinds_M|\<^sub>r = |h' \<turnstile> document_ptr_kinds_M|\<^sub>r"
drop_perp_incident: fixes p :: "proj2"   and l :: "proj2_line" shows "proj2_incident p (drop_perp p l)"
simps: fixes a :: "nat \<Rightarrow> nat" shows "(a \<in> PrimRec1') = (a = (\<lambda>x. 0) \<or> a = Suc \<or> a = c_fst \<or> a = c_snd \<or> (\<exists>f g. a = (\<lambda>x. f (g x)) \<and> f \<in> PrimRec1' \<and> g \<in> PrimRec1') \<or> (\<exists>f g. a = c_f_pair f g \<and> f \<in> PrimRec1' \<and> g \<in> PrimRec1') \<or> (\<exists>f g. a = UnaryRecOp f g \<and> f \<in> PrimRec1' \<and> g \<in> PrimRec1'))"
sets_trivial_measure: fixes X :: "'a set" shows "sets (trivial_measure X) = {{}, X}"
inf_commute: fixes x :: "'f multiset"   and y :: "'f multiset" shows "x \<inter># y = y \<inter># x"
cmult_in_iff': fixes L :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) set"   and L' :: "'c filter \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'b) set"   and Lr :: "'a filter \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) set"   and c :: "'b"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "landau_symbol L L' Lr"   and "c \<noteq> (0::'b)" shows "((\<lambda>x. f x * c) \<in> L F g) = (f \<in> L F g)"
assms_1: shows "finite {a. f a \<noteq> (1::'a)}"
eval_simps'_3: fixes f :: "'c local.term" assumes "Arr f" shows "trg \<lbrace>f\<rbrace> = \<lbrace>Trg f\<rbrace>"
B_Issues_A: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and T3 :: "nat"   and evs :: "event list" assumes "Says B A (Crypt servK (Number T3)) \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV" shows "B Issues A with Crypt servK (Number T3) on evs"
small_steps_skip_term: fixes \<pi> :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and cs' :: "(com \<times> (char list \<Rightarrow> int \<Rightarrow> int)) option" shows "small_steps \<pi> (SKIP, s) cs' = (cs' = Some (SKIP, s))"
set_eq_subset: fixes A :: "'b set"   and B :: "'b set" shows "(A = B) = (order.greater_eq B A \<and> order.greater_eq A B)"
inf_Sup: fixes a :: "'a"   and B :: "'a set" shows "a \<sqinter> \<Squnion>B = \<Squnion>((\<sqinter>) a ` B)"
simps_2: fixes n :: "nat" shows "Partial_Recursive.list_decode (Suc n) = tuple_decode (pdec1 n) (pdec2 n)"
ENR_retract_of_ENR: fixes T :: "'a set"   and S :: "'a set" assumes "ENR T"   and "S retract_of T" shows "ENR S"
simps: fixes vs :: "nat list"   and f :: "facetype" shows "type (Face vs f) = f"
prism_4: fixes q :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_chain q S c" shows "chain_boundary (Suc q) (prism q c) = chain_map q g c - chain_map q f c - prism (q - 1) (chain_boundary q c)"