hcju/isabelleps huggingface dataset:
mem_zero: fixes p :: "'a" shows "0 \<in> sub.tangent_space p"
RETURN_as_SPEC_refine_old: fixes M :: "'b nres"   and R :: "('b \<times> 'c) set"   and v :: "'c" assumes "M \<le> \<Down> R (SPEC (\<lambda>x. x = v))" shows "M \<le> \<Down> R (RETURN v)"
acc_wfD: fixes r :: "('a \<times> 'a) set"   and x :: "'a" assumes "wf r" shows "x \<in> Wellfounded.acc r"
elims_2: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal set" assumes "validate_clause x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and C :: "'b literal set" where "x = I" and "xa = C" and "\<forall>s. ground_clause (subst_cl C s) \<longrightarrow> validate_ground_clause I (subst_cl C s)"
preserves_section_retraction: fixes e :: "'a \<times> 'a"   and m :: "'a \<times> 'a" assumes "VV.ide (VV.comp e m)" shows "ide ((fst e \<star> snd e) \<cdot> (fst m \<star> snd m))"
iT_Plus_neg_mem_iff2: fixes k :: "nat"   and x :: "nat"   and I :: "nat set" assumes "k \<le> x" shows "(x - k \<in> I \<oplus>- k) = (x \<in> I)"
count_nonempty: fixes cs :: "('b \<Rightarrow> bool) list"   and a :: "'b"   and k :: "nat" assumes "cs \<noteq> []" shows "count cs a k < length cs"
Prop1A: fixes c :: "OBJ"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes " \<Turnstile> c : Sec \<Phi> " shows "secure c"
assms_2: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "P n [] n"
complete_bipartite_digraph_def: fixes m :: "nat"   and n :: "nat"   and G :: "('b, 'c) pre_digraph" shows "K\<^bsub>m,n\<^esub> G \<equiv> graph G \<and> (\<exists>U V. verts G = U \<union> V \<and> U \<inter> V = {} \<and> card U = m \<and> card V = n \<and> arcs_ends G = U \<times> V \<union> V \<times> U)"
even_word_imp_odd_next: fixes x :: "'b word" assumes "even (unat x)" shows "x + 1 = 0 \<or> odd (unat (x + 1))"
sha1compressstart: fixes r :: "nat"   and b :: "bit list"   and A :: "bit list"   and B :: "bit list"   and C :: "bit list"   and D :: "bit list"   and E :: "bit list" shows "sha1compressstart r b A B C D E = sha1compress r (sha1expand b 79) A B C D E"
aux: assumes "trimono F" shows "gfp F (f 0) = FAIL"
rel_refl_strong: fixes x :: "'b err"   and Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z. z \<in> set_err x \<Longrightarrow> Ra z z" shows "rel_err Ra x x"
rep_transfer: shows "rel_fun cr_uint16 (=) (\<lambda>x. x) Rep_uint16"
trans: fixes x :: "'a\<^sub>m"   and y :: "'a\<^sub>m"   and z :: "'a\<^sub>m" assumes "bo x y"   and "bo y z"   and "x \<in> A" shows "bo x z"
add_col_sub_row_carrier_3: fixes A :: "'a mat"   and n :: "nat"   and a :: "'a"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat n n" shows "add_col_sub_row a k l A \<in> carrier_mat n n"
iMODb_0_iMOD_subset_conv: fixes r' :: "nat"   and m' :: "nat"   and r :: "nat"   and m :: "nat" shows "([ r', mod m', 0 ] \<subseteq> [ r, mod m ]) = (r' mod m = r mod m \<and> r \<le> r')"
delayed_pool_write_def: fixes s :: "'b sparc_state" shows "delayed_pool_write s \<equiv> let dwp0 = get_delayed_pool s; dwp1 = delayed_pool_minus dwp0; wl = get_delayed_write dwp1; s1 = delayed_write_all wl s; s2 = delayed_pool_rm_list wl s1 in s2"
assms: shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
none2: fixes p :: "real"   and Da :: "'e pmf"   and Db :: "'f pmf" assumes "0 < p"   and "p < 1" shows "set_pmf (bernoulli_pmf p \<bind> (\<lambda>b. if b then map_pmf Inl Da else map_pmf Inr Db)) \<inter> {f. \<exists>e. Inr e = f} \<noteq> {}"
indexed_eval_inj_on_carrier: fixes L :: "('d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a, 'e) ring_scheme"   and i :: "'d" assumes "\<And>P. P \<in> carrier L \<Longrightarrow> L.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> L.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = L.indexed_const \<zero>\<^bsub>L\<^esub>" shows "inj_on (\<lambda>Ps. L.indexed_eval Ps i) (carrier (poly_ring L))"
refl: fixes x :: "'a" assumes "x \<in> A" shows "x \<sqsubseteq> x"
Op_def: fixes delta :: "'f"   and inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option" shows "Op delta inp binp \<equiv> asTerm (qOp delta (pickInp inp) (pickBinp binp))"
that: fixes i :: "nat" assumes "length ys = length xs"   and "i < length xs"   and "take i xs = take i ys"   and "(xs ! i, ys ! i) \<in> r" shows "thesis"
Liminf_mono: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and F :: "'c filter" assumes "\<forall>\<^sub>F x in F. f x \<le> g x" shows "Liminf F f \<le> Liminf F g"
assms: shows "t \<in> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A)"
reachable_imp_dfs: fixes y :: "'a \<times> 'b \<times> 'a"   and xs :: "('a \<times> 'b \<times> 'a) list" assumes "y \<in> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0"   and "list_all (\<lambda>t. t \<in> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0) xs"   and "y \<in> reachable (set xs)" shows "\<exists>y'. id y' = id y \<and> y' \<in> set (dfs [] xs)"
iff_qf: fixes p :: "fm"   and q :: "fm" assumes "qfree p"   and "qfree q" shows "qfree (Cooper.iff p q)"
SBWriteVolatile: fixes a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'c memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "'d"   and \<O> :: "'e"   and \<R> :: "'f"   and \<S> :: "'g" shows "(Write True a (D, f) A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>, sb @ [Write\<^sub>s\<^sub>b True a (D, f) (f \<theta>) A L R W], m, \<D>, \<O>, \<R>, \<S>)"
dimXAB: shows "\<forall>n. X n * Y n \<in> carrier_mat m m \<and> A * B \<in> carrier_mat m m"
inverse_fds_def: fixes f :: "'a fds" shows "inverse f = fds (dirichlet_inverse (fds_nth f) (inverse (fds_nth f 1)))"
dfz: fixes i :: "nat" assumes "0 < i"   and "i < n" shows "(deriv ^^ i) f \<xi> = 0"
complement_unique: fixes a :: "'a"   and x :: "'a"   and y :: "'a" assumes "a \<^bold>\<sqinter> x = \<zero>"   and "a \<^bold>\<squnion> x = \<one>"   and "a \<^bold>\<sqinter> y = \<zero>"   and "a \<^bold>\<squnion> y = \<one>" shows "x = y"
zmult_pos_bignumTr: fixes a :: "int"   and z :: "int"   and x :: "int" assumes "0 < a" shows "\<exists>l. \<forall>m>l. z < x + int m * a"
gen_length_def: fixes n :: "nat"   and xs :: "'a list" shows "List.gen_length n xs = n + length xs"
hyps: shows "<\<nu>x>a<y>.P \<longmapsto> \<alpha> \<prec> P'"
F2_get_f: shows "Fubx_get F2 f = Some fd2"
sup_apply: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "sup f g x = sup (f x) (g x)"
enum_0_bot: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s" shows "(a = kuhn_simplex.enum base upd 0) = (\<forall>a'\<in>s. a \<le> a')"
product_nn_integral_insert_rev: fixes I :: "nat set"   and i :: "nat"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "finite I"   and "i \<notin> I"   and "f \<in> borel_measurable (Pi\<^sub>M (insert i I) (\<lambda>v. stock_measure (\<Gamma> v)))" shows "integral\<^sup>N (Pi\<^sub>M (insert i I) (\<lambda>v. stock_measure (\<Gamma> v))) f = \<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x(i := y)) \<partial>Pi\<^sub>M I (\<lambda>v. stock_measure (\<Gamma> v)) \<partial>stock_measure (\<Gamma> i)"
C_eq_RD_aux: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" shows "Cp p x = Cp (removeDuplicates p) x"
measure_UN_bound: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure"   and B :: "real" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"   and "\<And>I'. \<lbrakk>I' \<subseteq> I; finite I'\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure M (\<Union> (A ` I')) \<le> B" shows "Sigma_Algebra.measure M (\<Union> (A ` I)) \<le> B"
induct: fixes P :: "('g \<Rightarrow> 'g \<Rightarrow> bool) \<Rightarrow> 'g list \<Rightarrow> bool"   and a0 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "'g list" assumes "\<And>Pa. P Pa []"   and "\<And>Pa x. P Pa [x]"   and "\<And>Pa x y xs. P Pa (y # xs) \<Longrightarrow> P Pa (x # y # xs)" shows "P a0 a1"
rbt_map_entry_inv1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and t :: "('a, 'b) rbt" shows "inv1 (ord.rbt_map_entry less k f t) = inv1 t"
assms_6: shows "\<^bold>g [^] z2 \<otimes> g' [^] z2' = a \<otimes> h [^] e'"
gauss_jordan_single_2: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan_single A = C" shows "C \<in> carrier_mat nr nc"
vector_smult_distrib: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and a :: "'a"   and x :: "('a, 'c) Finite_Cartesian_Product.vec" shows "A *v (a *s x) = a *s (A *v x)"
IH1: assumes "Vagree \<nu> \<nu>' (FVDiff t1)"   and "Iagree I J {Inl x |x. x \<in> SIGT t1}" shows "frechet I t1 (fst \<nu>) (snd \<nu>) = frechet J t1 (fst \<nu>') (snd \<nu>')"
G1: shows "snd (mk_eqcl (Some l # xs) zs i T) ! j < length (fst (mk_eqcl (Some l # xs) zs i T)) + i"
assms_1: shows "(a, b) \<in> set (arcs i j xs)"
coupling_unfinished: fixes Q :: "'v \<Rightarrow> nat option"   and V :: "'v \<Rightarrow> nat option"   and D :: "'v \<Rightarrow> enat"   and S :: "'v set" assumes "coupling Q V D S" shows "unfinished_dnodes' V = unfinished_dnodes S"
t_comm: fixes x :: "'a"   and y :: "'a" shows "tx \<cdot> ty = ty \<cdot> tx"
return_def: fixes a :: "'c" shows "return a \<equiv> \<lambda>s. ((a, s), False)"
simps: fixes h :: "nat"   and m :: "'a \<Rightarrow> nat"   and S :: "char list \<Rightarrow> 'a"   and X :: "char list set" assumes "Measure_fun h m" shows "Measure_fun.top_on_st S X = (S = \<lambda>x. \<top> on X)"
fmrel_fmdom_eq: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('e, 'c) fmap"   and y :: "('e, 'd) fmap" assumes "fmrel P x y" shows "fmdom x = fmdom y"
domintros_12: fixes v :: "exp0"   and va :: "(pat \<times> exp0) list" shows "is_value_dom (Mat v va)"
iTILL_add_Min: fixes n :: "nat"   and k :: "nat" shows "iMin ([\<dots>n] \<oplus> k) = k"
rep_eq: fixes x :: "'b \<Longrightarrow> 'a"   and xa :: "'a upred" shows "\<lbrakk>\<forall> x \<bullet> xa\<rbrakk>\<^sub>e = (\<lambda>b. \<forall>v. \<lbrakk>xa\<rbrakk>\<^sub>e (put\<^bsub>x\<^esub> b v))"
MtransT_invert: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "'state" assumes "a1 \<rightarrow>*t a2" obtains cf :: "('test, 'atom) com \<times> 'state" and cf' :: "('test, 'atom) com \<times> 'state" and s'' :: "'state" where "a1 = cf" and "a2 = s''" and "cf \<rightarrow>*c cf'" and "cf' \<rightarrow>t s''"
dc_on_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "dc_on P A = (\<forall>C. chain_on P C A \<and> C \<noteq> {} \<longrightarrow> (\<exists>x\<in>A. glb P C x))"
natural_extension_2: shows "qmpt.invertible_qmpt J.natural_extension_measure J.natural_extension_map"
assms_2: fixes x :: "'a" assumes "P x" shows "\<not> Q x"
simps: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and S :: "'b SDG_node set"   and n :: "'b SDG_node"   and a :: "'c" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.relevant_varsp sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S n a = (\<exists>as n' V. a = V \<and> CFG.intra_path sourcenode targetnode kind valid_edge (parent_node n) as (parent_node n') \<and> n' \<in> SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S \<and> SDG.SDG_Use sourcenode targetnode kind valid_edge get_proc procs Use ParamUses V n' \<and> (\<forall>n''. SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n'' \<and> parent_node n'' \<in> set (CFG.sourcenodes sourcenode as) \<longrightarrow> \<not> SDG.SDG_Def sourcenode targetnode kind valid_edge get_proc procs Def ParamDefs V n''))"
assms_2: shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"
pred_subset_eq2: fixes R :: "('d \<times> 'e) set"   and S :: "('d \<times> 'e) set" shows "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) = (R \<subseteq> S)"
assms_4: shows "f' \<equiv> \<lambda>y. \<integral>\<^sup>+ x. f <|x, y|> \<partial>stock_measure t1"
lrel_def: shows "lrel \<equiv> inv_image (measure id <*lex*> measure id <*lex*> measure id) (\<lambda>i. (k - cc (s i), tweight (tts (s i) (tkt_thread (tts (s i)) (cc (s i)))), A.dist_step (tkt_thread (tts (s i)) (cc (s i))) i))"
convex_hull_scaling: fixes c :: "real"   and S :: "'b set" shows "convex hull (*\<^sub>R) c ` S = (*\<^sub>R) c ` (convex hull S)"
prj_char_1: shows "Maps.PRJ\<^sub>0 \<lbrakk>\<lbrakk>t\<^sub>0\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>u\<^sub>1\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>t\<^sub>0u\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>"
a_star_redu_AndL1_elim: fixes x :: "name"   and M :: "trm"   and y :: "name"   and R :: "trm" assumes "AndL1 x.M y \<longrightarrow>\<^sub>a* R" shows "\<exists>M'. R = AndL1 x.M' y \<and> M \<longrightarrow>\<^sub>a* M'"
discI_1: fixes status :: "status" assumes "status = FAILED" shows "is_failed status"
mt_mndet: fixes P :: "'a \<Rightarrow> 'a process" shows "mndet {} P = STOP"
coreflexive_symmetric: fixes x :: "'a" assumes "x \<le> top" shows "local.comp_inf.symmetric x"
cltn2_compose_abs: fixes M :: "((real, 3) vec, 3) vec"   and N :: "((real, 3) vec, 3) vec" assumes "invertible M"   and "invertible N" shows "cltn2_compose (cltn2_abs M) (cltn2_abs N) = cltn2_abs (M ** N)"
cases: fixes x :: "('c \<times> 'd) list" obtains "x = []"   | p :: "'c \<times> 'd" and ps :: "('c \<times> 'd) list" where "x = p # ps"
Implies_def: fixes P :: "fml"   and Q :: "fml" shows "(P \<rightarrow> Q) = (Q || ! P)"
not_weak_iff: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "total_pseudo_ordered_set A less_eq"   and "x \<in> A"   and "y \<in> A" shows "(\<not> less_eq y x) = (less_eq x y \<and> x \<noteq> y)"
simps_1: fixes x :: "'a" shows "rbt_del x Empty = Empty"
preserves_seq: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.CC.seq g f" shows "C.CC.seq (C.CC.map g) (C.CC.map f)"
listmap_gs_in_listmap: fixes h :: "'d \<Rightarrow> 'c \<Rightarrow> real"   and p :: "'d \<Rightarrow> real"   and fs :: "'d list" shows "(\<lambda>x. \<Prod>g\<leftarrow>fs. h g x powr p g) \<in> prod_list (map powr_closure (map h fs))"
lastMarked_def: fixes t :: "('b \<times> bool) list" shows "lastMarked t = last (markedElements t)"
take_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'a list"   and y :: "'a" assumes "n \<le> m" shows "take n (xs[m := y]) = take n xs"
cOfFax_def: fixes f :: "'fsym" shows "cOfFax f = Pr (Guard (resOf f)) [Fn (Oldf f) (GE.getTvars (arOf f))]"
orthogonal_matrix_id: shows "orthogonal_matrix (mat (1::'a))"
closed_Collect_eq: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "closed {x. f x = g x}"
content_cbox_cart: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" assumes "cbox a b \<noteq> {}" shows "content (cbox a b) = (\<Prod>i\<in>UNIV. b $ i - a $ i)"
igSwapIGVar_def: fixes MOD :: "('d, 'e, 'f, 'g, 'h, 'i, 'j, 'k, 'l) model_scheme" shows "igSwapIGVar MOD \<equiv> \<forall>zs z1 z2 xs x. igSwap MOD zs z1 z2 (igVar MOD xs x) = igVar MOD xs (x @xs[z1 \<and> z2]_zs)"
inv_closed: fixes x :: "(bool \<times> 'a) list" assumes "x \<in> carrier \<F>\<^bsub>gens\<^esub>" shows "inv\<^bsub>\<F>\<^bsub>gens\<^esub>\<^esub> x \<in> carrier \<F>\<^bsub>gens\<^esub>"
inf_sup_distrib2: fixes 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 y :: "'b"   and z :: "'b"   and x :: "'b" assumes "class.distrib_lattice inf less_eq less sup" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
ldrop_eq_LNil: fixes n :: "enat"   and xs :: "'c llist" shows "(ldrop n xs = LNil) = (llength xs \<le> n)"
IH: fixes Di :: "('a \<times> 'b) list" assumes "Di \<in> set (subseqs D)"   and "list_all (\<lambda>d. fst d = i) Di" shows "Di \<in> set (subseqs (dbproj i D))"
simps_4: fixes p :: "perm"   and \<alpha> :: "'g"   and t :: "('e, 'f, 'g) Tree" shows "p \<bullet> tAct \<alpha> t = tAct (p \<bullet> \<alpha>) (p \<bullet> t)"
intro: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real" assumes "\<And>x y. 0 \<le> dist x y"   and "\<And>x y. (dist x y = 0) = (x = y)"   and "\<And>x y. dist x y = dist y x" shows "semimetric dist"
induct: fixes P :: "'c mat \<Rightarrow> 'c mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'c mat"   and a1 :: "'c mat"   and a2 :: "nat"   and a3 :: "nat" assumes "\<And>A B i j. \<lbrakk>\<And>x xa xaa. \<lbrakk>x = dim_row A; xa = dim_col A; i < x \<and> j < xa; xaa = A $$ (i, j); xaa = (0::'c); concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> (0::'c) then [i'] else []) [Suc i..<x]) = []\<rbrakk> \<Longrightarrow> P A B i (Suc j); \<And>x xa xaa x21 x22. \<lbrakk>x = dim_row A; xa = dim_col A; i < x \<and> j < xa; xaa = A $$ (i, j); xaa = (0::'c); concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> (0::'c) then [i'] else []) [Suc i..<x]) = x21 # x22\<rbrakk> \<Longrightarrow> P (swaprows i x21 A) (swaprows i x21 B) i j; \<And>x xa xaa xb. \<lbrakk>x = dim_row A; xa = dim_col A; i < x \<and> j < xa; xaa = A $$ (i, j); xaa \<noteq> (0::'c); xaa = (1::'c); xb = (\<lambda>i. A $$ (i, j))\<rbrakk> \<Longrightarrow> P (eliminate_entries xb A i j) (eliminate_entries xb B i j) (Suc i) (Suc j); \<And>x xa xaa xb. \<lbrakk>x = dim_row A; xa = dim_col A; i < x \<and> j < xa; xaa = A $$ (i, j); xaa \<noteq> (0::'c); xaa \<noteq> (1::'c); xb = inverse xaa\<rbrakk> \<Longrightarrow> P (multrow i xb A) (multrow i xb B) i j\<rbrakk> \<Longrightarrow> P A B i j" shows "P a0 a1 a2 a3"
alw_inv: fixes f :: "'b stream \<Rightarrow> 'c stream"   and P :: "'c stream \<Rightarrow> bool"   and s :: "'b stream" assumes "\<And>s. f (stl s) = stl (f s)" shows "alw P (f s) = alw (\<lambda>x. P (f x)) s"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_7: fixes l :: "'d strand_label"   and X :: "'f list"   and F :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and F' :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p ((l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle>) \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) = (\<lambda>(t, s). (l, t \<cdot> rm_vars (set X) \<delta>, s \<cdot> rm_vars (set X) \<delta>)) ` set F'"
smult_eq_iff: fixes b :: "'a"   and a :: "'a"   and p :: "'a poly"   and q :: "'a poly" assumes "b \<noteq> (0::'a)" shows "(smult a p = smult b q) = (smult (a / b) p = q)"
g: shows "g \<in> N \<rightarrow>\<^sub>M prob_algebra R"
p_l_r_def_2: shows "l \<equiv> rai_lb y"
range_dom_2: fixes f :: "('e \<times> 'f) set" shows "f\<inverse> `` Range f = Domain f"
R_Union: fixes M :: "('a \<times> 'b) set" assumes "\<forall>(A, B)\<in>M. R A B" shows "R (Sup (fst_proj M)) (Sup (snd_proj M))"
assms_2: shows "\<phi> (Trans s1 a1 ou1 s1')"
comp_inf_covector: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> (top \<sqinter> z)) = x \<sqinter> y \<sqinter> (top \<sqinter> z)"
subdivision_pair_divide: fixes e :: "'b \<times> 'b"   and H :: "'b pair_pre_digraph"   and w :: "'b"   and G :: "'b pair_pre_digraph" assumes "e \<in> parcs H"   and "w \<notin> pverts H"   and "subdivision_pair G H" shows "subdivision_pair G (subdivide H e w)"
start_heap_ok_def: shows "start_heap_ok = snd (snd start_heap_data)"
privateKey_notin_image_publicKey: fixes b :: "keymode"   and x :: "agent"   and c :: "keymode"   and AA :: "agent set" shows "privateKey b x \<notin> publicKey c ` AA"
poly_higher_pderiv_aux1': fixes k :: "nat"   and n :: "nat"   and c :: "'a"   and p :: "'a poly" assumes "k < n"   and "[:- c, 1::'a:] ^ n dvd p" shows "poly ((pderiv ^^ k) p) c = (0::'a)"
less_tail: fixes s :: "'c list"   and t :: "'c list" assumes "s \<noteq> []"   and "s < t" shows "tl s < tl t"
exec1_exec_n: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" assumes "P \<turnstile> c \<rightarrow> c'" shows "P \<turnstile> c \<rightarrow>^1 c'"
val_prod: fixes p :: "int"   and f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int" assumes "prime p"   and "f \<in> padic_set p"   and "g \<in> padic_set p"   and "f \<noteq> padic_zero p"   and "g \<noteq> padic_zero p" shows "padic_val p (padic_mult p f g) = padic_val p f + padic_val p g"
inner_Basis_add2: fixes b1 :: "'a"   and b2 :: "'a"   and v :: "'a" assumes "b1 \<in> S"   and "b2 \<in> S" shows "local.inner_Basis v (b1 + b2) = local.inner_Basis v b1 + local.inner_Basis v b2"
inj_comm_monoid_add_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_comm_monoid_add_hom hom \<equiv> comm_monoid_add_hom hom \<and> inj_monoid_add_hom hom"
diff_add_eq_diff_diff_swap: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus a (plus b c) = minus (minus a c) b"
not_final_thread_iff: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "not_final_thread s t = (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (\<not> final x \<or> ln \<noteq> no_wait_locks \<or> (\<exists>w. wset s t = \<lfloor>w\<rfloor>)))"
is_le_rel_def: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ordered_powerprod_lin.is_le_rel rel = (rel = (=) \<or> rel = (\<preceq>) \<or> rel = (\<prec>))"
dg_Set_is_arr_dg_Par_is_arr: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and T :: "V" assumes "T : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B" shows "T : A \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> B"
BrnNFT_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list" shows "BrnNFT cl dl \<equiv> \<Union>n\<in>theNFTBoth cl dl. {brnL cl n..<brnL cl n + brn (cl ! n)}"
mat_of_rows_carrier_2: fixes n :: "nat"   and vs :: "'a Matrix.vec list" shows "dim_row (mat_of_rows n vs) = length vs"
sum_def: shows "Sum_Type.sum = {f. (\<exists>a. f = Inl_Rep a) \<or> (\<exists>b. f = Inr_Rep b)}"
not_inf: shows "\<not> AlternativeSmallStep.inf \<Gamma> [c] [] s"