hcju/isabelleps huggingface dataset:
simps_2: fixes final :: "'c \<Rightarrow> bool"   and s :: "('d, 'e) locks \<times> (('e \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option) \<times> 'f) \<times> ('e \<Rightarrow> 'g wait_set_status option) \<times> 'e set"   and t :: "'e"   and ct :: "'e conditional_action"   and cts :: "'e conditional_action list" shows "final_thread.cond_action_oks final s t (ct # cts) = (final_thread.cond_action_ok final s t ct \<and> final_thread.cond_action_oks final s t cts)"
not_S_Var: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "'b"   and t :: "('a, 'b) Term.term" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "\<not> fst (kbo.kbo w w0 scf least pr_strict pr_weak (Var x) t)"
rtranclp_sup_rtranclp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "(sup R\<^sup>*\<^sup>* S\<^sup>*\<^sup>*)\<^sup>*\<^sup>* = (sup R S)\<^sup>*\<^sup>*"
span_vec_eq: fixes X :: "(real, 'a) vec set" shows "vec.span X = span X"
adds_term_adds_pp: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and u :: "'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_term pair_of_term u v = (term_powerprod.component_of_term pair_of_term u = term_powerprod.component_of_term pair_of_term v \<and> term_powerprod.adds_pp pair_of_term (term_powerprod.pp_of_term pair_of_term u) v)"
V01_def: shows "V\<^sub>0\<^sub>1 = {f \<in> \<R>. \<phi> (the (f 0)) = f \<or> \<phi> (the (f 1)) = f}"
a_2: shows "f \<in> O(\<lambda>x. real x powr p * (1 + integral {a..real x} (\<lambda>u. u powr p' / u powr (p + 1))))"
is_generator_set: shows "cspan (set canonical_basis) = UNIV"
Semi: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c1 :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c2 :: "com"   and R :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "C \<turnstile> {P} c1 {Q}"   and "C \<turnstile> {Q} c2 {R}" shows "C \<turnstile> {P} c1; c2 {R}"
transpose_transpose: fixes A :: "(('b, 'c) vec, 'd) vec" shows "Finite_Cartesian_Product.transpose (Finite_Cartesian_Product.transpose A) = A"
exhaust: fixes y :: "'a basic_type" obtains "y = OclAny"   | "y = OclVoid"   | "y = Boolean"   | "y = Real"   | "y = Integer"   | "y = UnlimitedNatural"   | "y = String"   | x8 :: "'a" where "y = \<langle>x8\<rangle>\<^sub>\<T>"   | x9 :: "('a, String.literal) phantom" where "y = Enum x9"
min_int_poly_irreducible: fixes x :: "'b" shows "irreducible (min_int_poly x)"
n_less_a: shows "n < mod_type_class.to_nat a"
that: fixes I :: "'a \<Rightarrow> 'e option" assumes "ts_inv (\<lambda>_. P) I ts m" shows "thesis"
\<tau>inv: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and tl1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s1' :: "'x1 \<times> 'm1"   and tl2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s2' :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2"   and "t \<turnstile> s1 -1-tl1\<rightarrow> s1'"   and "t \<turnstile> s2 -2-tl2\<rightarrow> s2'"   and "t \<turnstile> s1' \<approx> s2'"   and "tl1 \<sim>m tl2" shows "\<tau>move1 s1 tl1 s1' = \<tau>move2 s2 tl2 s2'"
R_trans: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a \<le> b"   and "b \<le> c" shows "a \<le> c"
func_times: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f * g = (\<lambda>x. f x * g x)"
carrier_is_subcring: fixes R :: "('c, 'd) ring_scheme" assumes "cring R" shows "subcring (carrier R) R"
star_absorb: fixes star :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.left_zero_kleene_algebra star one times sup less_eq less bot"   and "times x y = bot" shows "times x (star y) = x"
fin_1: shows "finite {x \<in> C. 0 < u x}"
length_ys: shows "length (sorted_list_of_set J) = k"
Id_SPEC_refine: fixes S :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "S \<le> SPEC \<Phi>" shows "S \<le> \<Down> Id (SPEC \<Phi>)"
append_rhs_preserves_cls: fixes rhs :: "'b trm set"   and r :: "'b rexp" shows "rhss (Append_rexp_rhs rhs r) = rhss rhs"
SMP_D_aux2: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "t \<in> SMP (set M)"   and "\<exists>x. t = Var x"   and "finite_SMP_representation arity Ana \<Gamma> M" shows "\<exists>m\<in>set M. \<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = m \<cdot> \<delta>"
find_lambda_measures_decreases2: fixes ls :: "nat"   and c :: "nat" assumes "ls < c" shows "((Suc ls, c), ls, c) \<in> find_lambda_measures"
trace_callee_resource_of_oracle: fixes callee :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf" shows "trace_callee run_resource (map_spmf (RES callee) (return_spmf s)) = trace_callee callee (return_spmf s)"
simps_6: fixes n :: "nat"   and \<phi>\<^sub>1 :: "'a formula"   and \<phi>\<^sub>2 :: "'a formula" shows "rexp_of n (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2) = INTERSECT [rexp_of n \<phi>\<^sub>1, rexp_of n \<phi>\<^sub>2, ENC n (FOV (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2))]"
trans: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a \<le>s b"   and "b \<le>s c" shows "a \<le>s c"
prems_2: shows "(l # ls) ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"
n_def: shows "n = (\<lambda>s\<in>S. P_inf s (\<lambda>\<omega>. (HLD S1 suntil HLD S2) (s ## \<omega>)))"
sprefE: fixes u :: "'b list"   and v :: "'b list" assumes "u <p v" obtains z :: "'b list" where "u \<cdot> z = v" and "z \<noteq> \<epsilon>"
run_asset_i_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_asset_i n s))) \<notin> used s"
stable_on_Xd: fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "stable_on ds X"   and "x \<in> X" shows "Xd x \<in> ds"
ll_on_open_it_axioms: shows "ll_on_open_it S g Y"
common_boundary_subdiv_exists_refl: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" assumes "\<forall>(k, \<gamma>)\<in>boundary twoC. valid_path \<gamma>" shows "common_boundary_sudivision_exists (boundary twoC) (boundary twoC)"
rep_eq: fixes x :: "'b permutation"   and xa :: "'b permutation" shows "(\<rightarrow>) (x + xa) = (\<rightarrow>) x \<circ> (\<rightarrow>) xa"
finite: fixes M :: "'b dfa" assumes "dfa M" shows "finite (dfa.states M)"
unfold_llist_id: fixes xs :: "'b llist" shows "unfold_llist lnull lhd ltl xs = xs"
fact_less_mono_nat: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "m < n" shows "fact m < fact n"
OUT: fixes b :: "'v" assumes "b \<in> B \<Gamma>" shows "d_OUT (current_of_bipartite f) b = 0"
self_minus_gauss_cnj: fixes z :: "gauss_int" shows "z - gauss_cnj z = of_int (2 * ImZ z) * \<i>\<^sub>\<int>"
front_tickFree_append: fixes s :: "'a event list"   and t :: "'a event list" assumes "tickFree s"   and "front_tickFree t" shows "front_tickFree (s @ t)"
simps: fixes vs :: "nat list"   and f :: "facetype" shows "vertices (Face vs f) = vs"
uncurry_simp: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and a :: "'c"   and b :: "'d" shows "uncurry f (a, b) = f a b"
orderI: fixes a :: "'g"   and b :: "'g" assumes "a = a \<sqinter> b" shows "a \<le> b"
maps_over_def: fixes A :: "'a set"   and B :: "'b set" shows "maps_over A B = {m. dom m \<subseteq> A \<and> ran m \<subseteq> B}"
that_6: fixes CTa :: "nat \<Rightarrow> classDef option"   and e :: "exp"   and m :: "nat"   and e' :: "exp list"   and e'' :: "exp list"   and eia :: "exp"   and ei' :: "exp"   and el :: "exp list"   and er :: "exp list" assumes "x = CTa"   and "xa = MethodInvk e m e'"   and "xb = MethodInvk e m e''"   and "CTa \<turnstile> eia \<rightarrow> ei'"   and "Execute.append el (eia # er) e'"   and "Execute.append el (ei' # er) e''" shows "thesis"
pairs_sup_closed: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter"   and z :: "'c"   and w :: "'d Filters.filter" assumes "triple phi"   and "(x, y) \<in> triple.pairs phi"   and "(z, w) \<in> triple.pairs phi" shows "triple.pairs_sup (x, y) (z, w) \<in> triple.pairs phi"
poly_reflect_poly_0: fixes p :: "'b poly" shows "poly (reflect_poly p) (0::'b) = lead_coeff p"
lattice_of_swap: fixes fs :: "'a vec list"   and i :: "nat"   and j :: "nat"   and gs :: "'a vec list" assumes "set fs \<subseteq> carrier_vec n"   and "i < length fs"   and "j < length fs"   and "i \<noteq> j"   and "gs = fs[i := fs ! j, j := fs ! i]" shows "lattice_of gs = lattice_of fs"
igVarIPresIGWls_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igVarIPresIGWls MOD \<equiv> \<forall>xs x. igWls MOD (asSort xs) (igVar MOD xs x)"
cUNION_cinsert: fixes f :: "'e \<Rightarrow> 'd cset"   and x :: "'e"   and A :: "'e cset" shows "cUnion (cimage f (cinsert x A)) = cUn (f x) (cUnion (cimage f A))"
disc_1: fixes x1 :: "'m" shows "isl (Inl x1)"
bind_fv2_changed': fixes v :: "variable \<times> typ"   and lev :: "nat"   and tm :: "Core.term" assumes "bind_fv2 v lev tm \<noteq> tm" shows "occs (case v of (x, xa) \<Rightarrow> Fv x xa) tm"
assms_1: shows "ocircline_set H = ocircline_set H'"
f2r_def: fixes f :: "'c \<Rightarrow> 'd set" shows "\<R> f = {(x, y). s2p (f x) y}"
exec_step_ind_Putfield_NoField: fixes r :: "val"   and stk :: "val list"   and a :: "nat"   and D :: "char list"   and fs :: "Objects.fields"   and h :: "heap"   and P :: "jvm_prog"   and F :: "char list"   and C :: "char list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "r = hd (tl stk)"   and "a = the_Addr r"   and "(D, fs) = the (h a)"   and "r \<noteq> Null"   and "\<nexists>t b. P \<turnstile> D has F,b:t in C" shows "exec_step_ind (StepI (Putfield F C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NoSuchFieldError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
d_def: shows "d \<equiv> y ^ (2 * n + 1) / (1 - y\<^sup>2) / real (2 * n + 1)"
E: fixes a :: "'a set"   and b :: "'b set" assumes "a \<in> Ea"   and "b \<in> Eb" shows "a \<times> b \<in> sets (sigma (space A \<times> space B) {a \<times> b |a b. a \<in> Ea \<and> b \<in> Eb})"
epi_implies_arr: fixes f :: "'a" assumes "C.epi f" shows "C.arr f"
mono_funpower: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "mono f" shows "mono (funpower f n)"
change_int_def: fixes f :: "nat \<Rightarrow> nat"   and i :: "char list \<Rightarrow> tv" shows "change_int f i \<equiv> \<lambda>s. change_tv f (i s)"
is_prefix'_def: fixes xs :: "'b list"   and ys :: "'b list" shows "is_prefix' xs ys \<equiv> CATCH (ebind (check_prefix xs ys) (\<lambda>_. ERETURN True)) (\<lambda>_. ERETURN False)"
times_mreal_def: fixes x :: "mreal"   and y :: "mreal" shows "x * y \<equiv> mreal_plus x y"
a_adopt_node_locs_def: fixes remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_adopt_node\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs remove_child_locs set_disconnected_nodes_locs = l_adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_adopt_node_locs remove_child_locs set_disconnected_nodes_locs"
enumerator_increasing: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "ls \<in> set (enumerator m nmax)" shows "increasing ls"
eval_pochhammer_poly: fixes n :: "nat"   and k :: "'a" shows "poly (pochhammer_poly n) k = pochhammer k n"
finfun_comp_apply: fixes g :: "'c \<Rightarrow> 'b"   and f :: "('a, 'c) finfun" shows "finfun_apply (finfun_comp g f) = g \<circ> finfun_apply f"
n2l_l2n_map_id: fixes zs :: "'b list list" assumes "\<And>x. x \<in> set zs \<Longrightarrow> x \<noteq> []" shows "map (n2l \<circ> l2n) zs = zs"
map_raw_cong: fixes xs :: "('b \<times> 'c) list \<times> 'd"   and f :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'e"   and g :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'e" assumes "\<And>kv. kv \<in> set (fst xs) \<Longrightarrow> f kv = g kv" shows "map_raw f xs = map_raw g xs"
Sup_eqI: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and x :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq y x"   and "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> less_eq z y) \<Longrightarrow> less_eq x y" shows "Sup A = x"
space_count_space: fixes \<Omega> :: "'c set" shows "space (count_space \<Omega>) = \<Omega>"
lemAddOverScale: fixes a :: "'a"   and u :: "'a Vector"   and b :: "'a" shows "a ** u \<oplus> b ** u = (a + b) ** u"
ex_\<rho>: shows "\<exists>\<rho>. tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> w) \<rho> (src w) (g \<star> w)"
indexle_def: fixes k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" shows "indexle k' u' k u = (indexlt k' u' k u \<or> k' = k \<and> u' = u)"
take_add: fixes i :: "nat"   and j :: "nat"   and xs :: "'a list" shows "take (i + j) xs = take i xs @ take j (drop i xs)"
int_tNN: fixes T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "Ik.wt T"   and "Ik.wtE \<xi>" shows "TE.int (eenv \<xi>) (tNN T) = Ik.int \<xi> T"
in_sets_Sup: fixes M :: "'b measure set"   and X :: "'b set"   and m :: "'b measure"   and A :: "'b set" assumes "\<And>m. m \<in> M \<Longrightarrow> space m = X"   and "m \<in> M"   and "A \<in> sets m" shows "A \<in> sets (Sup M)"
open_seq_invariant: fixes A :: "('a \<times> ('a, 'b, 'c, 'd) seqp, 'b seq_action) automaton"   and I :: "'b seq_action \<Rightarrow> bool"   and P :: "'a \<times> ('a, 'b, 'c, 'd) seqp \<Rightarrow> bool"   and i :: "nat"   and OA :: "((nat \<Rightarrow> 'a) \<times> ('a, 'b, 'c, 'd) seqp, 'b seq_action) automaton"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and ANY :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<TTurnstile> (I \<rightarrow>) P"   and "initiali i (init OA) (init A)"   and "automaton.trans OA = oseqp_sos \<Gamma> i"   and "automaton.trans A = seqp_sos \<Gamma>" shows "OA \<Turnstile> (act I, other ANY {i} \<rightarrow>) seql i P"
assms_6: shows "d' \<turnstile> ev \<mapsto> e'"
except_empty: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" shows "except p {} = p"
mojmir_accept_iff_token_set_accept: shows "\<UU>.accept = (\<exists>i<\<UU>.max_rank. finite \<UU>.fail \<and> finite (\<UU>.merge i) \<and> infinite (\<UU>.succeed i))"
inf_uminus: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "inf x (mbt_algebra.neg_assert one times inf dual bot x) = bot"
char_of_char: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "char" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "unique_euclidean_semiring_with_bit_shifts.char_of plus times one bit (comm_semiring_1.of_char times one plus zero c) = c"
compatible_iff_seq: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C" shows "(partial_magma.domains C g \<inter> partial_magma.codomains C f \<noteq> {}) = partial_magma.arr C (C g f)"
cyclic_group_commute: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "a \<otimes> b = b \<otimes> a"
mHom_0: fixes N :: "('i, 'b, 'j) Module_scheme"   and f :: "'a \<Rightarrow> 'i" assumes "R module N"   and "f \<in> mHom R M N" shows "f \<zero> = \<zero>\<^bsub>N\<^esub>"
intro: fixes x :: "'f \<Longrightarrow> 'e" assumes "\<And>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v" shows "weak_lens x"
eucl_inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "inf x y = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (inf (inner x i) (inner y i)) i) Basis"
ffold_rec: fixes x :: "'e \<times> nat"   and A :: "('e \<times> nat) fset"   and z :: "nat" assumes "x |\<in>| A" shows "ffold tcf z A = tcf x (ffold tcf z (A |-| {|x|}))"
bind_def: fixes P :: "'b pred"   and f :: "'b \<Rightarrow> 'a pred" shows "bind P f = \<Squnion> (f ` {x. eval P x})"
subsumes_list_Cons_filter_iff: fixes leq :: "('a, 'b) Term.term literal \<Rightarrow> ('a, 'b) Term.term literal \<Rightarrow> bool"   and L :: "('a, 'b) Term.term literal"   and Ls :: "('a, 'b) Term.term literal list"   and Ks :: "('a, 'b) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term option" assumes "sorted_wrt leq (L # Ls)"   and "transp leq"   and "\<And>L K \<sigma> \<tau>. \<lbrakk>match_term_list [(atm_of L, atm_of K)] \<sigma> = Some \<tau>; is_pos L = is_pos K\<rbrakk> \<Longrightarrow> leq L K" shows "subsumes_list (L # Ls) (filter (leq L) Ks) \<sigma> = subsumes_list (L # Ls) Ks \<sigma>"
stable_no_blocking_on_def: fixes ds :: "D2 set"   and X :: "X5 set" shows "BSI.stable_no_blocking_on ds X = (\<forall>h X'. \<not> BSI.blocking_on ds X h X')"
pre_ranks_state_obtain: fixes r\<^sub>q :: "nat"   and r :: "'b \<Rightarrow> nat option"   and \<nu> :: "'a"   and q :: "'b" assumes "r\<^sub>q \<in> pre_ranks r \<nu> q - {max_rank}" obtains q' :: "'b" where "r q' = Some r\<^sub>q" and "q = \<delta> q' \<nu>"
ltl_to_gba_spec_def: fixes \<phi> :: "'i ltlc" shows "ltl_to_gba_spec \<phi> \<equiv> SPEC (\<lambda>gba. igba.lang gba = language_ltlc \<phi> \<and> igba gba \<and> finite ((g_E gba)\<^sup>* `` g_V0 gba))"
dfs_imp_reachable: fixes y :: "'rep"   and xs :: "'rep list" assumes "k_isNode y"   and "list_all k_isNode xs"   and "k_memb y (KBPAlg.dfs k_empt xs)" shows "\<exists>y'. simAbs y' = simAbs y \<and> y' \<in> KBPAlg.reachable (set xs)"
preserves_red: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "P t x m" shows "P t x' m'"
restrict_relation_empty: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "restrict_relation {} R = (\<lambda>_ _. False)"
prime_sum_upto_nonneg: fixes f :: "nat \<Rightarrow> real"   and x :: "real" assumes "\<And>n. 0 < n \<Longrightarrow> 0 \<le> f n" shows "0 \<le> prime_sum_upto f x"
pelims: fixes x :: "int poly"   and xa :: "int poly"   and xb :: "(rat \<times> rat \<times> rat) \<times> rat \<times> rat \<times> rat"   and y :: "(rat \<times> rat \<times> rat) \<times> rat \<times> rat \<times> rat" assumes "tighten_poly_bounds_binary x xa xb = y"   and "tighten_poly_bounds_binary_dom (x, xa, xb)" obtains cr1 :: "int poly" and cr2 :: "int poly" and l1 :: "rat" and r1 :: "rat" and sr1 :: "rat" and l2 :: "rat" and r2 :: "rat" and sr2 :: "rat" where "x = cr1" and "xa = cr2" and "xb = ((l1, r1, sr1), l2, r2, sr2)" and "y = (tighten_poly_bounds cr1 l1 r1 sr1, tighten_poly_bounds cr2 l2 r2 sr2)" and "tighten_poly_bounds_binary_dom (cr1, cr2, (l1, r1, sr1), l2, r2, sr2)"
sigma_lemma: fixes rg :: "recf"   and xs :: "nat list"   and x :: "nat" assumes "primerec rg (Suc (length xs))" shows "rec_exec (rec_sigma rg) (xs @ [x]) = UF.Sigma (rec_exec rg) (xs @ [x])"
induced_eq_verts_imp_eq: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph"   and G' :: "('c, 'd) pre_digraph" assumes "induced_subgraph G H"   and "induced_subgraph G' H"   and "verts G = verts G'" shows "G = G'"
emeasure_Collect_T: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "emeasure (K.T s) {x \<in> space (K.T s). P x} = \<integral>\<^sup>+ t. emeasure (K.T t) {x \<in> space (K.T t). P (t ## x)} \<partial>measure_pmf (K s)"
simps_1: fixes t :: "('c, 'd) SkewBinomialTree" shows "getMinTree [t] = t"
cm_unchanged_worklist: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('a, 'c) configuration"   and c1 :: "('a, 'c) configuration"   and loc :: "'a"   and t :: "'c"   and n :: "int"   and loc' :: "'a" assumes "dataflow_topology summary results_in"   and "dataflow_topology.next_change_multiplicity' c0 c1 loc t n"   and "loc' \<noteq> loc" shows "c_work c1 loc' = c_work c0 loc'"
abs_summable_on_cmult_left: assumes "inverse c \<noteq> (0::'b) \<Longrightarrow> (\<lambda>x. f x * c) abs_summable_on A" shows "(\<lambda>x. f x * c * inverse c) abs_summable_on A"
ex_iff_abs: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "(\<exists>x. P x) = (\<exists>n. P (abs_nat n))"
nullable_iff: fixes r :: "'a rexp"   and S :: "'a set" shows "nullable r = ([] \<in> lang S r)"