hcju/isabelleps huggingface dataset:
red_plus_cs: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.red F p q" shows "relation.cs (punit.red F) (p + r) (q + r)"
safe_distance_1r_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and \<delta> :: "real" shows "safe_distance_1r a\<^sub>e v\<^sub>e \<delta> = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / a\<^sub>e / 2"
succss_closed_dfs': fixes ys :: "'b"   and xs :: "'a list" assumes "invariant ys"   and "list_all isNode xs"   and "nodeAbs ` succss (set_of ys) \<subseteq> nodeAbs ` (set xs \<union> set_of ys)" shows "nodeAbs ` succss (set_of (dfs ys xs)) \<subseteq> nodeAbs ` set_of (dfs ys xs)"
length_take_nths_00: fixes xs :: "'b list" shows "length (take_nths 0 (Suc 0) xs) = length (take_nths (Suc 0) (Suc 0) xs) \<or> length (take_nths 0 (Suc 0) xs) = length (take_nths (Suc 0) (Suc 0) xs) + 1"
code: fixes A :: "'c llist set" shows "lSup A = (if \<forall>x\<in>A. lnull x then LNil else LCons (THE x. x \<in> lhd ` (A \<inter> {xs. \<not> lnull xs})) (lSup (ltl ` (A \<inter> {xs. \<not> lnull xs}))))"
separated_by_this_wall_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and D :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "separated_by {f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X} C D"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X" shows "D \<in> g \<turnstile> ChamberComplex.chamber_system X"
cSup_least: fixes X :: "'k multiset set"   and z :: "'k multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X \<subseteq># z"
less_induct: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and P :: "'g \<Rightarrow> bool"   and a :: "'g" assumes "class.wellorder less_eq less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
cases: fixes x :: "auth_query" obtains "x = Forward"   | "x = Drop"   | "x = Look"
sifum_example_def: shows "sifum_example \<equiv> sifum_lang_no_dma ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars \<and> sifum_example_axioms"
diffeomorphism_def: fixes k :: "enat"   and S :: "'b set"   and T :: "'c set"   and p :: "'b \<Rightarrow> 'c"   and p' :: "'c \<Rightarrow> 'b" shows "diffeomorphism k S T p p' = (k-smooth_on S p \<and> k-smooth_on T p' \<and> homeomorphism S T p p')"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) atm" shows "GE.satL \<xi> (Neg a) = (\<not> GE.satA \<xi> a)"
g: fixes x :: "'a" assumes "x \<in> S" shows "f x = exp (\<i> * complex_of_real (g x))"
wf_ciD3_ci_app: fixes ci :: "('a, 'b) check_instr"   and ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list"   and pc' :: "nat" assumes "ci_app ci ins P h stk loc C M pc frs"   and "ins_jump_ok ins pc'" shows "ci_app ci ins P h stk loc C M pc' frs"
residual_r_pnat_def: fixes x :: "pnat"   and y :: "pnat" shows "x \<rightarrow> y \<equiv> y - x"
IH_2: assumes "v \<noteq> 0" shows "\<exists>x. x \<^bold>\<in> v \<and> (\<forall>y. y \<^bold>\<in> v \<longrightarrow> Rep_hf x \<le> Rep_hf y)"
nat_filtration_info: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "f \<circ> pseudo_proj_True n = f"
one: shows "0 + 1 \<cdot> 1 = 1"
simps: fixes a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'s"   and a6 :: "'a literal multiset" shows "ord_resolve_rename a1 a2 a3 a4 a5 a6 = (\<exists>CAs n AAs As DA \<rho> \<rho>s \<sigma> E. a1 = CAs \<and> a2 = DA \<and> a3 = AAs \<and> a4 = As \<and> a5 = \<sigma> \<and> a6 = E \<and> length CAs = n \<and> length AAs = n \<and> length As = n \<and> (\<forall>i<n. poss (AAs ! i) \<subseteq># CAs ! i) \<and> negs (mset As) \<subseteq># DA \<and> \<rho> = hd (renamings_apart (DA # CAs)) \<and> \<rho>s = tl (renamings_apart (DA # CAs)) \<and> ord_resolve (CAs \<cdot>\<cdot>cl \<rho>s) (DA \<cdot> \<rho>) (AAs \<cdot>\<cdot>aml \<rho>s) (As \<cdot>al \<rho>) \<sigma> E)"
ChamberComplexRetraction_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" shows "ChamberComplexRetraction X f \<equiv> ChamberComplexEndomorphism X f \<and> ChamberComplexRetraction_axioms X f"
one_closed: shows "\<one>\<^bsub>Characters G\<^esub> \<in> carrier (Characters G)"
in_set_conv_nth: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)"
a0i_running_def: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "a0i_running A B M \<equiv> {(s, s'). s' = s\<lparr>signals := addSignal (signals s) (Running A B M)\<rparr>}"
cases: fixes a1 :: "'sort"   and a2 :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wls a1 a2" obtains xs :: "'varSort" and x :: "'var" where "a1 = asSort xs" and "a2 = Var xs x"   | delta :: "'opSym" and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" where "a1 = stOf delta" and "a2 = Op delta inp binp" and "wlsInp delta inp" and "wlsBinp delta binp"
\<tau>_cont: shows "\<tau> \<midarrow>p\<rightarrow> \<tau> p"
cr_option_gpv_def: fixes x :: "'e option"   and gpv :: "('e, 'f, 'g) gpv" shows "cr_option_gpv x gpv = (gpv = (lift_spmf \<circ> return_pmf) x)"
word_ao_absorbs_3: fixes x :: "'b word"   and y :: "'b word" shows "x AND (x OR y) = x"
ivl_less: fixes x :: "'b ivl"   and y :: "'b ivl" shows "(x < y) = (low x < low y \<or> low x = low y \<and> high x < high y)"
search_enum_nonempty_eq0: fixes b :: "nat"   and i :: "nat"   and y :: "nat" assumes "s b \<down>= i"   and "\<phi> i y \<down>"   and "e_length b \<le> y" shows "\<exists>x. search_enum x b \<down>= 0"
CD_on_def: fixes ds :: "D2 set"   and X :: "X4 set" shows "StableNoDecomp.CD_on ds X = (\<Union>d\<in>ds. StableNoDecomp.Cd d X)"
Commutativity: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<Psi> \<otimes> \<Psi>' \<simeq> \<Psi>' \<otimes> \<Psi>"
mismatchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto>[Rel] Q"   and "Rel \<subseteq> Rel'" shows "[a\<noteq>b]P \<leadsto>[Rel'] [a\<noteq>b]Q"
union_diff2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "finite A"   and "finite B" shows "abel_semigroup_set.F f z g (A \<union> B) = f (f (abel_semigroup_set.F f z g (A - B)) (abel_semigroup_set.F f z g (B - A))) (abel_semigroup_set.F f z g (A \<inter> B))"
U_consts_3: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n false\<^sub>n U\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<phi>"
wf_jvm_prog_def: fixes P :: "jvm_prog" shows "wf_jvm_prog P \<equiv> \<exists>\<Phi>. wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
simps_2: fixes y :: "name"   and e :: "lam"   and \<theta> :: "(name \<times> lam) list"   and x :: "name" shows "(y, e) # \<theta> maps x to (if x = y then e else lookup \<theta> x)"
lifelock_free_is_non_terminating: fixes P :: "'b process" shows "lifelock_free P = non_terminating P"
SMP_ikI: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "t \<in> SMP (trms\<^sub>s\<^sub>t S)"
intro: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder less_eq less"   and "class.order_axioms less_eq" shows "class.order less_eq less"
dvd_times_right_cancel_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero" shows "dvd.dvd times (times b a) (times c a) = dvd.dvd times b c"
Cast1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and C :: "ty" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>Cast C e,s\<rangle> -ta\<rightarrow> \<langle>Cast C e',s'\<rangle>"
pre_buf: shows "wf_mbuf2' \<sigma> j n R \<phi> \<psi> buf"
vertical_chop_assoc2: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and v3 :: "view"   and v4 :: "view" shows "v=v1--v2 \<and> v1=v3--v4 \<longrightarrow> (\<exists>v'. v=v3--v' \<and> v'=v4--v2)"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> \<NN>' = vcard \<NN>'"
init_pnet_p_NodeS: fixes i :: "nat"   and s :: "'s"   and R :: "nat set"   and p :: "net_tree" assumes "NodeS i s R \<in> init (pnet np p)" shows "p = \<langle>i; R\<rangle>"
Diff_Int_distrib: fixes C :: "'d set"   and A :: "'d set"   and B :: "'d set" shows "C \<inter> (A - B) = C \<inter> A - C \<inter> B"
measure_eq_AE: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure" assumes "AE x in M. (x \<in> A) = (x \<in> B)"   and "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M A = Sigma_Algebra.measure M B"
fps_of_ratfps_shift: fixes n :: "nat"   and x :: "'b ratfps" shows "fps_of_ratfps (ratfps_shift n x) = fps_shift n (fps_of_ratfps x)"
prob_space_def: fixes M :: "'d measure" shows "prob_space M \<equiv> finite_measure M \<and> prob_space_axioms M"
div_free_DF: fixes A :: "'b set" shows "D (DF A) = {}"
wellformed_items_\<J>_k_u_if_0: fixes k :: "nat"   and u :: "nat" assumes "wellformed_items (\<J> k 0)" shows "wellformed_items (\<J> k u)"
union_properdiv_set: fixes n :: "nat" assumes "1 \<le> n" shows "divisor_set n = properdiv_set n \<union> {n}"
funpow_dist_0_eq: fixes y :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "y \<in> orbit f x" shows "(funpow_dist f x y = 0) = (x = y)"
assms: shows "\<forall>P. peak ars P \<longrightarrow> (\<exists>\<sigma>' \<tau>'. DD ars r (fst P, snd P, \<sigma>', \<tau>'))"
pbernpoly_bound: shows "\<forall>x. \<bar>pbernpoly n x\<bar> \<le> D"
subspace_diff: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "csubspace S"   and "x \<in> S"   and "y \<in> S" shows "x - y \<in> S"
elims_2: fixes x :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" assumes "invpst x" obtains "x = \<langle>\<rangle>"   | l :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" and xa :: "'c \<times> 'd" and uu_ :: "'e" and mkp :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (xa, uu_, mkp), r\<rangle>" and "invpst l \<and> invpst r \<and> is_min2 mkp (set (Tree2.inorder l @ xa # Tree2.inorder r))"
l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs type_wf \<equiv> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs"
inf_admS_def: fixes f :: "'a admS"   and g :: "'a admS" shows "inf f g \<equiv> mklr (unlr f \<inter> unlr g)"
simps_20: fixes f1 :: "i32 \<Rightarrow> 'a"   and f2 :: "i64 \<Rightarrow> 'a"   and f3 :: "f32 \<Rightarrow> 'a"   and f4 :: "f64 \<Rightarrow> 'a"   and x4 :: "f64" shows "(case ConstFloat64 x4 of ConstInt32 x \<Rightarrow> f1 x | ConstInt64 x \<Rightarrow> f2 x | ConstFloat32 x \<Rightarrow> f3 x | ConstFloat64 x \<Rightarrow> f4 x) = f4 x4"
leD: fixes y :: "'b word"   and x :: "'b word" assumes "y \<le>s x" shows "\<not> x <s y"
interaction_any_bounded_converter_of_callee: fixes conv :: "'h \<Rightarrow> 'i \<Rightarrow> ('k \<times> 'h, 'j, 'l) gpv"   and n :: "enat"   and s :: "'h" assumes "\<And>s x. interaction_any_bounded_by (conv s x) n" shows "interaction_any_bounded_converter (CNV conv s) n"
squarefree_factorial_semiring': fixes n :: "'a" assumes "n \<noteq> (0::'a)" shows "squarefree n = (\<forall>p\<in>#prime_factorization n. multiplicity p n = 1)"
rOpen: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "x \<in> supp N"   and "\<And>C. Prop C \<Psi> P (M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle>) P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> yvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "distinct xvec"   and "distinct yvec"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* M"   and "yvec \<sharp>* C"   and "x \<sharp> C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) (M\<lparr>\<nu>*(xvec @ x # yvec)\<rparr>\<langle>N\<rangle>) P'"
poly_subst_map_indets: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and g :: "'e \<Rightarrow> 'd"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f (map_indets g p) = poly_subst (f \<circ> g) p"
IH: fixes n :: "nat"   and p :: "'a poly" assumes "n \<le> k" shows "\<exists>q. poly ((pderiv ^^ k) ([:- c, 1::'a:] ^ n * p)) c = fact n * poly q c"
restr_eqI: fixes A :: "'b set"   and A' :: "'b set"   and F :: "'b \<Rightarrow> 'c"   and F' :: "'b \<Rightarrow> 'c" assumes "A = A'"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "restrict F A = restrict F' A'"
inner_Basis: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and u :: "'a"   and v :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis" shows "inner u v = (if u = v then 1 else 0)"
hb_consistent_append: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and suffix :: "'c list"   and prefix :: "'c list" assumes "happens_before hb_weak hb"   and "happens_before.hb_consistent hb suffix"   and "happens_before.hb_consistent hb prefix"   and "\<And>s p. \<lbrakk>s \<in> set suffix; p \<in> set prefix\<rbrakk> \<Longrightarrow> \<not> hb s p" shows "happens_before.hb_consistent hb (prefix @ suffix)"
type_less_eq_x_Tuple: fixes \<tau> :: "'a OCL_Types.type"   and \<xi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" assumes "\<tau> \<le> Tuple \<xi>" obtains \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" where "\<tau> = Tuple \<pi>" and "subtuple (\<le>) \<pi> \<xi>"
pseudoinverse: fixes \<omega> :: "real"   and x :: "real" assumes "a < \<omega>"   and "\<omega> < b" shows "(\<omega> \<le> f x) = (I \<omega> \<le> x)"
le_cases: fixes x :: "'a"   and y :: "'a" obtains "x \<le> y"   | "y \<le> x"
hoare_diagram2: fixes D :: "'b \<times> 'b \<Rightarrow> 'e \<Rightarrow> 'e"   and X :: "'a \<Rightarrow> 'b \<Rightarrow> 'e" assumes "dmono D"   and "\<forall>u i j. \<Turnstile> X u i{| D (i, j) |}SUP_L_P X (pair u i) j" shows "\<Turnstile> \<Squnion>range X{| pt D |}(\<Squnion>range X \<sqinter> - grd (step D))"
antisym: fixes b :: "'a::order"   and a :: "'a::order" assumes "(b::'a::order) \<le> (a::'a::order)"   and "(a::'a::order) \<le> (b::'a::order)" shows "(a::'a::order) = (b::'a::order)"
m'_less: fixes i :: "nat" assumes "i < \<phi> 0" shows "m' i = m i"
exists_path_component_of_superset: fixes X :: "'a topology"   and S :: "'a set" assumes "path_connectedin X S"   and "topspace X \<noteq> {}" obtains C :: "'a set" where "C \<in> path_components_of X" and "S \<subseteq> C"
synthetic_div_def: fixes p :: "'b poly"   and c :: "'b" shows "synthetic_div p c = fst (synthetic_divmod p c)"
insert_ops_rem_cons: fixes x :: "'b \<times> 'b option"   and xs :: "('b \<times> 'b option) list" assumes "insert_ops (x # xs)" shows "insert_ops xs"
prime_field_def: fixes p :: "int" shows "prime_field TYPE('b) p \<equiv> mod_ring_locale TYPE('b) p"
nonintSfmla_nonintSI: shows "nonintSfmla [] = nonintSI"
zip_Nil: fixes ys :: "'d list" shows "[] || ys = []"
h1: shows "\<forall>i. insertion (nth_default 0 (L[var := C])) (isolate_variable_sparse p var i) = insertion (nth_default 0 (L[var := (A + B * C) / D])) (isolate_variable_sparse p var i)"
D_seq: fixes P :: "'c process"   and Q :: "'c process" shows "D (P `;` Q) = div_seq P Q"
1: shows "card (atoms (tseytin_tran F)) \<le> card (atoms (tseytin_toatom F F) \<union> \<Union> (atoms ` set (tseytin_tran1 (tseytin_toatom F) F)))"
simps_85: fixes x91 :: "'f"   and x92 :: "'d set"   and x93 :: "('d, 'e, 'f) com"   and x51 :: "'d set"   and x52 :: "('d, 'e, 'f) com"   and x53 :: "('d, 'e, 'f) com" shows "Guard x91 x92 x93 \<noteq> Cond x51 x52 x53"
simps_2: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "inf\<^sub>- (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = or (inf\<^sub>- \<phi>\<^sub>1) (inf\<^sub>- \<phi>\<^sub>2)"
UN_UN_flatten: fixes C :: "'d \<Rightarrow> 'c set"   and B :: "'f \<Rightarrow> 'd set"   and A :: "'f set" shows "\<Union> (C ` \<Union> (B ` A)) = (\<Union>y\<in>A. \<Union> (C ` B y))"
monoI2: fixes X :: "nat \<Rightarrow> 'a" assumes "\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m" shows "monoseq X"
servK_authentic_ter: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "evs \<in> kerbIV_gets" shows "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"
virt_to_phys_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word32" assumes "low_equal s1 s2" shows "virt_to_phys addr (mmu s1) (mem s1) = virt_to_phys addr (mmu s2) (mem s2)"
pfun_app_not_in_dom: fixes x :: "'a"   and f :: "('a, 'b) pfun" assumes "x \<notin> pdom f" shows "f(x)\<^sub>p = undefined"
IH: fixes s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option" assumes "\<Gamma>,High \<turnstile> c'"   and "\<langle>c',s1\<rangle> \<rightarrow>* \<langle>Skip,s2\<rangle>" shows "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"
dom_dom: fixes f :: "'a" assumes "S.arr f" shows "S.dom (S.dom f) = S.dom f"
strict_trans: fixes b :: "'k multiset"   and a :: "'k multiset"   and c :: "'k multiset" assumes "b \<subset># a"   and "c \<subset># b" shows "c \<subset># a"
from_nat_not_eq: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and a :: "nat"   and b :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "a \<noteq> mod_type.to_nat Rep b"   and "a < CARD('d)" shows "mod_type.from_nat Abs a \<noteq> b"
subdg_dghm_inc_is_ft_dghm: shows "dghm_inc \<BB> \<CC> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"
hyps: fixes st :: "nat"   and l :: "cell list"   and r :: "cell list" assumes "steps0 (Suc 0, tp) A stp = (st, l, r)"   and "tm_wf0 A" shows "st \<le> length A div 2"
hyps_3: assumes "i < n"   and "F \<subseteq> carrier_vec n"   and "f \<in> F \<rightarrow> UNIV" shows "module.lincomb (module_cpx_vec n) f F $ i = (\<Sum>v\<in>F. f v * v $ i)"
simps_10: fixes f1 :: "'a"   and f2 :: "('b, 'c) Node \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> ('b, 'c) Digit \<Rightarrow> ('b, 'c) FingerTreeStruc \<Rightarrow> ('b, 'c) Digit \<Rightarrow> 'a"   and x2 :: "('b, 'c) Node" shows "(case Single x2 of Empty \<Rightarrow> f1 | Single x \<Rightarrow> f2 x | Deep x xa xb xc \<Rightarrow> f3 x xa xb xc) = f2 x2"
induct: fixes P :: "'b err \<Rightarrow> bool"   and err :: "'b err" assumes "P Err"   and "\<And>x. P (OK x)" shows "P err"
assms: shows "\<lless>\<pi>\<^sub>1, eV\<ggreater> V\<rightarrow>i \<lless>\<pi>\<^sub>2, eV'\<ggreater>"
image_replicate_mset: fixes f :: "'c \<Rightarrow> 'b"   and n :: "nat"   and a :: "'c" shows "image_mset f (replicate_mset n a) = replicate_mset n (f a)"
assms: shows "moebius_pt M1 \<circ> conjugate = moebius_pt M2 \<circ> conjugate"
pred_fmap_def: shows "pred_fmap \<equiv> \<lambda>P. pred_fun (\<lambda>_. True) (pred_option P) \<circ> fmlookup"
norm_not_less_zero: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<not> norm x < 0"
da_elim_cases_14: fixes Env :: "env"   and B :: "lname set"   and jump :: "jump"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (Jmp jump)\<guillemotright> A" obtains "jump = Ret \<longrightarrow> Result \<in> B" and "nrm A = UNIV" and "brk A = (case jump of Break l \<Rightarrow> \<lambda>k. if k = l then B else UNIV | _ \<Rightarrow> \<lambda>k. UNIV)"
insert_before_wf2_is_l_insert_before_wf2: shows "l_insert_before_wf2 ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.insert_before Shadow_DOM.heap_is_wellformed"