hcju/isabelleps huggingface dataset:
safe_distance_4r_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" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_normal.safe_distance_4r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = (v\<^sub>o + a\<^sub>o * \<delta> - v\<^sub>e)\<^sup>2 / 2 / (a\<^sub>o - a\<^sub>e) - v\<^sub>o * \<delta> - 1 / 2 * a\<^sub>o * \<delta>\<^sup>2 + v\<^sub>e * \<delta>"
map_iterator_no_cond_rule_P: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option"   and I :: "'b set \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and f :: "'b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "map_iterator it m"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
has_lower_bound: fixes U :: "'a set"   and V :: "'a set" assumes "U \<in> I"   and "V \<in> I" shows "\<exists>W\<in>I. W \<subseteq> U \<inter> V"
prems_3: shows "P,E \<turnstile> \<langle>\<lparr>C'\<rparr>blocks (p # ps', Ts, vs, e),(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"
fps_const_def: fixes c :: "'b" shows "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else (0::'b))"
stake_cycle_le: fixes u :: "'b list"   and n :: "nat" assumes "u \<noteq> []"   and "n < length u" shows "stake n (cycle u) = take n u"
unlabel_Cons_2: fixes b :: "'b \<times> 'a"   and A :: "('b \<times> 'a) list" shows "unlabel (b # A) = snd b # unlabel A"
supd_assn_def: fixes f :: "state \<Rightarrow> state"   and P :: "'b assn" shows "(f .; P) = (\<lambda>Y s. P Y (f s))"
simps: fixes x :: "'c regular"   and y :: "'c dense Filters.filter" shows "stone_phi.pairs_uminus (x, y) = (- x, stone_phi x)"
elims: fixes x :: "('c \<times> 'd) list"   and y :: "'c \<Rightarrow> 'd option" assumes "map_of x = y" obtains "x = []" and "y = Map.empty"   | a :: "'c" and b :: "'d" and ps :: "('c \<times> 'd) list" where "x = (a, b) # ps" and "y = (\<lambda>x. if x = a then Some b else map_of ps x)"
assms_2: shows "\<forall>n\<ge>m. e n \<le> e m"
eventually_nonneg_def: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'c" shows "eventually_nonneg F f = (\<forall>\<^sub>F x in F. (0::'c) \<le> f x)"
multi_self_add_other_not_self_zmset: fixes M :: "'b zmultiset"   and x :: "'b" shows "(M = add_zmset x M) = False"
sample_max_sample: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "stutter_sampler f \<sigma>" shows "\<natural>(\<sigma> \<circ> f) = \<natural>\<sigma>"
add_Limit: fixes \<alpha> :: "V"   and x :: "V" assumes "Limit \<alpha>" shows "x + \<alpha> = \<Squnion> ((+) x ` elts \<alpha>)"
xy_roots: shows "x < y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"
simps_45: fixes x8 :: "expr"   and x2 :: "expr" shows "Throw x8 \<noteq> Expr x2"
sum_less_eqE: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'b"   and x :: "'c" assumes "comm_monoid_add_class.sum f A \<le> y"   and "x \<in> A"   and "finite A" shows "f x \<le> y"
mult_to_poly: fixes c :: "'a"   and d :: "'a" shows "[:c:] * [:d:] = [:c * d:]"
sel_3: fixes x :: "quat" shows "Im2 (inverse x) = - Im2 x / ((quat.Re x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2)"
less: shows "r'\<^sub>2 \<le> r'\<^sub>1"
finite_gpv_parametric': fixes C :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and A :: "'j \<Rightarrow> 'k \<Rightarrow> bool" shows "(rel_\<I> C R ===> rel_gpv'' A C R ===> (=)) finite_gpv finite_gpv"
minus_finite_3_def: fixes x :: "finite_3"   and y :: "finite_3" shows "x - y = x + - y"
local_typedef2: shows "\<exists>Rep Abs. type_definition Rep Abs {0..<int n}"
prems_3: shows "a \<in> read_only (share (x # sb) S)"
lt_setE: fixes t :: "'t"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "t \<in> lt_set F" obtains f :: "'t \<Rightarrow>\<^sub>0 'c" where "f \<in> F" and "f \<noteq> 0" and "lt f = t"
fix_eq: shows "(\<mu> X. SKIP |-| STOP |-| Mprefix A (\<lambda>x. X)) = (\<Lambda> X. SKIP |-| STOP |-| Mprefix A (\<lambda>x. X))\<cdot>(\<mu> X. SKIP |-| STOP |-| Mprefix A (\<lambda>x. X))"
f_Exec_N_append: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "nat"   and b :: "nat"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_N trans_fun (a + b) xs c = f_Exec_Comp_N trans_fun b (xs \<up> a) (f_Exec_Comp_N trans_fun a xs c)"
standard_def: fixes C :: "('c \<times> 'd) set"   and b :: "'c"   and t :: "'c"   and idt :: "'c"   and G :: "('c, 'd) labeled_graph" shows "standard C b t idt G \<equiv> graph G \<and> vertices G \<noteq> {} \<and> ident_rel idt G \<and> getRel b G = {} \<and> getRel t G = {(x, y). x \<in> vertices G \<and> y \<in> vertices G} \<and> (\<forall>(l, v)\<in>C. getRel l G = {(v, v)})"
succ_ss_V: fixes u :: "nat" shows "cf.E `` {u} \<subseteq> cf.V"
zero_not_in_totatives: fixes n :: "nat" shows "0 \<notin> totatives n"
MSB_append: fixes I1 :: "nat fset list"   and I2 :: "nat fset list" shows "MSB (I1 @ I2) = max (MSB I1) (MSB I2)"
adds_minus: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and s :: "'a"   and t :: "'a" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus s t" shows "plus (minus t s) s = t"
\<b>_one_gr: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "finite X"   and "valid_decomp X ps"   and "standard_decomp k ps"   and "(h, U) \<in> set (ps\<^sub>+)" shows "poly_deg h < \<b> ps (Suc 0)"
CompDiag_TensorDiag: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term"   and w :: "'c local.term" assumes "Diag t"   and "Diag u"   and "Diag v"   and "Diag w"   and "Seq t v"   and "Seq u w" shows "(t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> (v \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> w) = t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> v \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> w"
star_inductl_var: fixes x :: "'a"   and y :: "'a" assumes "x ; y \<le> y" shows "x\<^sup>\<star> ; y \<le> y"
lb1: shows "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> lb1 < x"
cod_simp: fixes f :: "'a1 \<times> 'a2" assumes "A1xA2.arr f" shows "A1xA2.cod f = (A1.cod (fst f), A2.cod (snd f))"
succ_tran_def: fixes G :: "'a graph"   and v :: "'a" shows "succ_tran G v = {e2. (v, e2) \<in> (edges G)\<^sup>+}"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_mult_hom hom"   and "one_hom_1 hom" shows "monoid_mult_hom_1 hom"
valid_Act: fixes P :: "'state"   and \<alpha> :: "'act"   and x :: "('b, 'pred, 'act) formula" shows "P \<Turnstile> Act \<alpha> x = (\<exists>\<alpha>' x' P'. Act \<alpha> x = Act \<alpha>' x' \<and> P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> P' \<Turnstile> x')"
degree0_coeffs: fixes p :: "'a poly" assumes "Polynomial.degree p = 0" shows "\<exists>a. p = [:a:]"
flow_attainability_axioms: shows "flow_attainability \<Delta>''"
isl_def: fixes sum :: "'a + 'b" shows "isl sum = (\<exists>x1. sum = Inl x1)"
Kleene_iter_lpfp: fixes f :: "'a \<Rightarrow> 'a"   and p :: "'a"   and k :: "nat" assumes "mono f"   and "f p \<le> p" shows "(f ^^ k) bot \<le> p"
simps_1: fixes ts :: "'b set list" shows "partitioning1 [] ts = ts"
simps_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a" shows "[] \<Squnion>\<^bsub>f\<^esub> y = y"
that: fixes s :: "'a \<Rightarrow>\<^sub>0 nat" assumes "s \<in> keys p"   and "t = except s {x}" shows "thesis"
BlockRedsFinal: fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>0 :: "char list \<Rightarrow> val option"   and V :: "char list"   and T :: "ty" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "final e\<^sub>2"   and "s\<^sub>0 = (h\<^sub>0, l\<^sub>0(V := None))" shows "P \<turnstile> \<langle>{V:T; e\<^sub>0},(h\<^sub>0, l\<^sub>0)\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2(V := l\<^sub>0 V))\<rangle>"
rep_eq: fixes x :: "Vars.var set"   and xa :: "CoCalls" shows "Rep_ttree (ccTTree x xa) = valid_lists x xa"
Q_subgroup_FreeS: shows "Group \<langle>\<Union>h\<in>FreeGroup S. lconjby h ` P'\<rangle> \<and> order.greater_eq (FreeGroup S) \<langle>\<Union>h\<in>FreeGroup S. lconjby h ` P'\<rangle>"
mapl_FGl_def: fixes l1 :: "'q \<Rightarrow> 'ad"   and l2 :: "'r \<Rightarrow> 'ae"   and l3 :: "'x \<Rightarrow> 'af" shows "mapl_FGl l1 l2 l3 = map_FGl l1 l2 l3 id id id id id id id id"
scf_lowering_strategyproof_def: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and agents' :: "'a set"   and alts' :: "'b set"   and set_ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool" shows "scf_lowering_strategyproof agents alts scf agents' alts' set_ext \<equiv> strategyproof_scf agents alts scf set_ext \<and> liftable_set_extension alts' alts set_ext \<and> scf_lowering agents alts scf agents' alts'"
l_distr: fixes x :: "int"   and y :: "int"   and z :: "int" assumes "x \<in> UNIV"   and "y \<in> UNIV"   and "z \<in> UNIV" shows "(x + y) * z = x * z + y * z"
simps_5: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
Term_imp_is_tm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "Term x" obtains t :: "tm" where "x = \<lbrakk>\<guillemotleft>t\<guillemotright>\<rbrakk>e"
m_o_anti_mono: fixes X :: "char list set"   and o1 :: "'av st option"   and o2 :: "'av st option" assumes "finite X"   and "top_on\<^sub>o o1 (- X)"   and "top_on\<^sub>o o2 (- X)"   and "o1 \<le> o2" shows "m\<^sub>o o2 X \<le> m\<^sub>o o1 X"
h2_def: fixes g :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and fs :: "('e \<times> 'f) \<times> 'g" shows "h2 g fs = (let (a, b) = fst fs in g a b)"
cr_transaction_def: shows "cr_transaction \<equiv> \<lambda>x y. x = the_Transaction y"
check_mult_l_dom_err_impl_def: fixes pd :: "bool"   and p :: "uint64"   and ia :: "bool"   and i :: "uint64" shows "check_mult_l_dom_err_impl pd p ia i = (if pd then ''The polynomial with id '' @ show (nat_of_uint64 p) @ '' was not found'' else []) @ (if ia then ''The id of the resulting id '' @ show (nat_of_uint64 i) @ '' was already given'' else [])"
singular_chain_add: fixes p :: "nat"   and X :: "'b topology"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X a"   and "singular_chain p X b" shows "singular_chain p X (a + b)"
qmodule_module: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "submodule R M H" shows "R module M /\<^sub>m H"
nsum_eq: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j \<in> carrier M"   and "\<forall>j\<le>n. g j \<in> carrier M"   and "\<forall>j\<le>n. f j = g j" shows "\<Sigma>\<^sub>e M f n = \<Sigma>\<^sub>e M g n"
nabla_closure: fixes x :: "'a" shows "d \<nabla>x = \<nabla>x"
M2: shows "mat_of_row y2 \<in> carrier_mat 1 nr2"
real_restrict_merge: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and I :: "nat set"   and J :: "nat set"   and n :: "nat"   and tI :: "'b option list"   and tJ :: "'b option list" assumes "getIJ getIJ"   and "I \<inter> J = {}"   and "wf_tuple n I tI"   and "wf_tuple n J tJ" shows "restrict I (New_max.merge tI tJ) = restrict I tI \<and> restrict J (New_max.merge tI tJ) = restrict J tJ"
spmf_conv_measure_spmf: fixes p :: "'a spmf"   and x :: "'a" shows "spmf p x = Sigma_Algebra.measure (measure_spmf p) {x}"
wf_inorm: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp" assumes "M2L_Equivalence_Checking.wf \<Sigma> n r" shows "M2L_Equivalence_Checking.wf \<Sigma> n (inorm r)"
HTset_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and \<Theta> :: "(((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool) \<times> com \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool)) set" shows "HTset \<pi> \<Theta> \<equiv> \<forall>(P, c, Q)\<in>\<Theta>. HT \<pi> P c Q"
SUP_upper: fixes i :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "i \<in> A" shows "f i \<le> Sup (f ` A)"
prefixI: fixes ys :: "'a list"   and xs :: "'a list"   and zs :: "'a list" assumes "ys = xs @ zs" shows "prefix xs ys"
ca_open: fixes e :: "db"   and \<rho> :: "nat \<rightarrow> ValD"   and \<Gamma> :: "nat \<Rightarrow> db" assumes "\<forall>v. freedb e v \<longrightarrow> \<rho>\<cdot>v \<triangleleft> \<Gamma> v \<and> closed (\<Gamma> v)" shows "evalDdb e\<cdot>\<rho> \<triangleleft> closing_subst e \<Gamma> 0"
slp_callstack_length_equal: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and cfs' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "n -as\<rightarrow>\<^bsub>sl\<^esub>* n'" obtains cf' :: "('var \<Rightarrow> 'val option) \<times> 'ret" where "transfers (kinds as) (cf # cfs) = cf' # cfs" and "transfers (kinds as) (cf # cfs') = cf' # cfs'"
rel_witness_option2: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" assumes "rel_option A x y" shows "rel_option (\<lambda>(a, b') b. b = b' \<and> A a b') (rel_witness_option (x, y)) y"
Program: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'a"   and is' :: "instr list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')" shows "sbh_computation.concurrent_step (ts, m, \<S>) (ts[i := (p', is @ is', \<theta>, sb @ [Prog\<^sub>s\<^sub>b p p' is'], \<D>, \<O>, \<R>)], m, \<S>)"
det_rows_sum: fixes M :: "('c, 'd) sq_matrix"   and T :: "'d set"   and a :: "'d \<Rightarrow> 'e \<Rightarrow> ('c, 'd) vec"   and S :: "'e set" shows "Square_Matrix.det (upd_rows M T (\<lambda>i. sum (a i) S)) = (\<Sum>f\<in>T \<rightarrow>\<^sub>E S. Square_Matrix.det (upd_rows M T (\<lambda>i. a i (f i))))"
col_carrier_vec: fixes j :: "nat"   and nc :: "nat"   and A :: "'b mat"   and nr :: "nat" assumes "j < nc"   and "A \<in> carrier_mat nr nc" shows "col A j \<in> carrier_vec nr"
app_vfield_vpairE: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r" obtains b :: "V" where "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<or> \<langle>b, a\<rangle> \<in>\<^sub>\<circ> r"
lambda_system_algebra: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f" shows "algebra \<Omega> (lambda_system \<Omega> M f)"
pconj_comm: fixes a :: "real"   and b :: "real" shows "a .& b = b .& a"
connected_connected_component: fixes S :: "'c set"   and x :: "'c" shows "connected (connected_component_set S x)"
relation_subset_crossp: shows "relation \<subseteq> carrier \<times> carrier"
case_2: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "(case ENV x2 of LOC x \<Rightarrow> f1 x | ENV x \<Rightarrow> f2 x) = f2 x2"
prpl_of_lasso_def: fixes L :: "('c, 'd) lasso_scheme" shows "prpl_of_lasso L \<equiv> (lasso_reach L, lasso_va L # lasso_cysfx L)"
gcd_red_nat: fixes x :: "nat"   and y :: "nat" shows "gcd x y = gcd y (x mod y)"
On_CTRL_def: shows "On_CTRL = Abs_seqauto (On_CTRL_States, On_CTRL_Init, On_CTRL_Labels, On_CTRL_Delta)"
assms_3: shows "(flow0 x \<circ> s) \<longlonglongrightarrow> u"
empty_problem_bound: fixes PROB :: "(('d, bool) fmap \<times> ('d, bool) fmap) set" assumes "prob_dom PROB = {}" shows "problem_plan_bound PROB = 0"
clausal_logic_compact: fixes N :: "'b literal multiset set" shows "(\<not> satisfiable N) = (\<exists>DD\<subseteq>N. finite DD \<and> \<not> satisfiable DD)"
left_plus_below_circ: fixes x :: "'a" shows "x\<^sup>+ \<le> x\<^sup>\<star>"
writes_bind: fixes S :: "('l, 'm, 'n) prog set"   and f :: "('l, 'm, 'o) prog"   and h :: "'l"   and g :: "'o \<Rightarrow> ('l, 'm, 'p) prog"   and h' :: "'l" assumes "\<And>h2. writes S f h h2"   and "\<And>x h2. \<lbrakk>h \<turnstile> f \<rightarrow>\<^sub>r x; h \<turnstile> f \<rightarrow>\<^sub>h h2\<rbrakk> \<Longrightarrow> writes S (g x) h2 h'" shows "writes S (Heap_Error_Monad.bind f g) h h'"
to_int_mod_ring_rebase: fixes x :: "'a mod_ring" shows "to_int_mod_ring (@x) = to_int_mod_ring x"
wt_args_ge_length_times_\<epsilon>\<^sub>h: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and ss :: "('s, 'v) tm list"   and i :: "nat"   and \<zeta> :: "('s, 'v) hd" assumes "legal_zpassign A" shows "of_nat (length ss) * zhmset_of \<epsilon>\<^sub>h \<le> wt_args i A \<zeta> ss"
hyps_1: shows "\<turnstile>\<^sub>! (r \<^bold>\<longleftrightarrow>\<^sub>! reduce r)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and edge_kind :: "('b, 'c, 'd, 'e) edge_kind" shows "P (case edge_kind of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = ((\<forall>x1. edge_kind = \<Up>x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. edge_kind = (x2)\<^sub>\<surd> \<longrightarrow> P (f2 x2)) \<and> (\<forall>x31 x32 x33 x34. edge_kind = x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34 \<longrightarrow> P (f3 x31 x32 x33 x34)) \<and> (\<forall>x41 x42 x43. edge_kind = x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 \<longrightarrow> P (f4 x41 x42 x43)))"
c1_euclidean_blinfunE: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on S (\<lambda>x. f' x i)" obtains bf' :: "'b \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'c" where "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative blinfun_apply (bf' x)) (at x within S)" and "continuous_on S bf'" and "\<And>x. x \<in> S \<Longrightarrow> blinfun_apply (bf' x) = f' x"
indep_system_axioms: shows "indep_system \<E>' (indep_in \<E>')"
simps_2: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a option" shows "Option.bind (Some x) f = f x"
OR3_imp_OR2: fixes B :: "agent"   and NA :: "msg"   and A :: "agent"   and K :: "nat"   and NB :: "msg"   and evs :: "event list" assumes "Says Server B \<lbrace>NA, Crypt (shrK A) \<lbrace>NA, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs"   and "B \<notin> bad"   and "evs \<in> otway" shows "\<exists>X. Says B Server \<lbrace>NA, Agent A, Agent B, X, Crypt (shrK B) \<lbrace>NA, NB, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"
zero_in_omega: shows "0 \<in>\<^sub>\<circ> \<omega>"
deleted_not_domA: fixes x :: "'d"   and \<Gamma> :: "('d \<times> 'e) list" shows "x \<notin> domA (delete x \<Gamma>)"
poly_simple_add_Cons: fixes a :: "'a"   and t :: "'a list" shows "[a] +++ ((0::'a) # t) = a # t"
assms: shows "g \<in> norm_pres_extensions E p F f"
map_bind_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'c spmf"   and g :: "'c \<Rightarrow> 'b spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
face_cycle_succ_def: fixes M :: "'e pre_map" shows "pre_digraph_map.face_cycle_succ M \<equiv> edge_succ M \<circ> edge_rev M"
quasi_isometry_betweenD_6: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry_between X Y f" shows "1 \<le> lambda"
FOREACH_cond_def: fixes c :: "'d \<Rightarrow> bool" shows "FOREACH_cond c \<equiv> \<lambda>(xs, \<sigma>). xs \<noteq> [] \<and> c \<sigma>"
mat_map_def: fixes f :: "'b \<Rightarrow> 'b" shows "mat_map f = map (vec_map f)"
starfun_minus: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b star" shows "- (*f* f) x = (*f* (\<lambda>x. - f x)) x"
cases: fixes x :: "'a literal" obtains t :: "'a trm" and s :: "'a trm" where "x = Pos (Eq t__ s__)"   | t :: "'a trm" and s :: "'a trm" where "x = Neg (Eq t s)"
read_only_unowned: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<O> \<inter> read_only \<S>\<^sub>s\<^sub>b = {}"
inv_all'_def: fixes p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.inv_all' p A \<equiv> \<forall>i j. i < dim_row A \<longrightarrow> j < dim_row A \<longrightarrow> p A i j"
0: fixes k :: "nat" shows "integral UNIV (indicat_real (\<Union> (S ` {..k}))) = 0"
pseudominimal_inD2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and x :: "'b"   and z :: "'b" assumes "ordering less_eq less"   and "ordering.pseudominimal_in less_eq less P x"   and "z \<in> P"   and "less z x" shows "z = ordering.bottom less_eq P"
rel_pmf_reflI: fixes p :: "'a pmf"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x" shows "rel_pmf P p p"
Abs_assn_inverse: fixes y :: "assn_raw" assumes "y \<in> Collect proper" shows "Rep_assn (Abs_assn y) = y"
mset_contains_eq: fixes m :: "'c"   and M :: "'c multiset" shows "(m \<in># M) = ({#m#} + (M - {#m#}) = M)"