hcju/isabelleps huggingface dataset:
Fubini: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (M \<Otimes>\<^sub>M N)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>M \<partial>N = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>N \<partial>M"
N_2: shows "fb.N 2 = 1 / (2\<^sup>2 * (2 - 1))"
mhd_is_static_simp: fixes m :: "ref_ty \<times> mhead" shows "is_static (mhd m) = is_static m"
satisfies_bounded_NFAND: fixes \<Phi> :: "('a, 'k) aformula set"   and \<AA> :: "'i" assumes "finite \<Phi>" shows "(\<AA> \<Turnstile>\<^sub>b NFAND \<Phi>) = (\<forall>\<phi>\<in>\<Phi>. \<AA> \<Turnstile>\<^sub>b \<phi>)"
mC_ZOC_eff_cont0_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and I0 :: "nat set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_ZOC_eff_cont0 theta c d s t I0 F \<equiv> (\<forall>i\<in>I0. s \<approx> eff c s i \<and> (local.cont c s i, d) \<in> theta) \<and> (\<forall>j\<in>F I0. t \<approx> eff d t j \<and> (c, local.cont d t j) \<in> theta)"
assms_1: shows "J \<inter>\<^sub>\<circ> K = []\<^sub>\<circ>"
length_snth_enc: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and n :: "nat" assumes "formula \<Sigma>" shows "length (snd (formula.stream_enc \<Sigma> (w, I) !! n)) = length I"
AE_iffI: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure"   and Q :: "'a \<Rightarrow> bool" assumes "almost_everywhere M P"   and "AE x in M. P x = Q x" shows "almost_everywhere M Q"
has_derivative_subst: fixes f :: "'b \<Rightarrow> 'c"   and df :: "'b \<Rightarrow> 'c"   and x :: "'b"   and d :: "'b \<Rightarrow> 'c" assumes "(f has_derivative df) (at x)"   and "df = d" shows "(f has_derivative d) (at x)"
simps_8: fixes \<alpha> :: "('b, 'c + 'e, 'd) hp"   and \<sigma> :: "'e \<Rightarrow> ('b, 'c, 'd) formula" shows "PPsubst \<alpha>** \<sigma> = PPsubst \<alpha> \<sigma>**"
interior_halfspace_ge: fixes a :: "'b"   and b :: "real" assumes "a \<noteq> (0::'b)" shows "interior {x. b \<le> a \<bullet> x} = {x. b < a \<bullet> x}"
jw_Roll: fixes \<alpha> :: "Syntax.tvar"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and \<tau> :: "ty" assumes "atom \<alpha> \<sharp> \<Gamma>"   and "\<Gamma> \<turnstile>\<^sub>W e : subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>" shows "\<Gamma> \<turnstile>\<^sub>W Roll e : Mu \<alpha> \<tau>"
asm: shows "good_context s \<and> s'' = delayed_pool_write s \<and> fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = True \<and> fst v2 = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s'')) + 1) mod NWINDOWS)) (cpu_reg_val WIM s'') = 0 \<and> get_addr (snd v2) s'' AND 3 \<noteq> 0"
sappend_snil_id_right: fixes xs :: "[:'a:]" shows "xs :@ [::] = xs"
map_scale_spmf: fixes f :: "'d \<Rightarrow> 'c"   and r :: "real"   and p :: "'d spmf" shows "map_spmf f (scale_spmf r p) = scale_spmf r (map_spmf f p)"
abs_conv_abscissa_mult_leI: fixes f :: "'a fds"   and d :: "ereal"   and g :: "'a fds" assumes "abs_conv_abscissa f \<le> d"   and "abs_conv_abscissa g \<le> d" shows "abs_conv_abscissa (f * g) \<le> d"
ang_vec_a_sym: fixes z1 :: "complex"   and z2 :: "complex" shows "\<angle>a z1 z2 = \<angle>a z2 z1"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (F \<circ> G) C.map \<epsilon>'.map"
walk_decomp_2: fixes xs :: "'a list"   and ys :: "'a list" assumes "H.walk (xs @ ys)" shows "H.walk ys"
transpose_mat_eq: fixes A :: "'b mat"   and B :: "'b mat" shows "(A\<^sup>T = B\<^sup>T) = (A = B)"
hyps_1: fixes y :: "ty"   and \<Gamma> :: "binding list"   and S :: "ty"   and T :: "ty" assumes "size_ty y < size_ty Q"   and "\<Gamma>\<turnstile>S<:y"   and "\<Gamma>\<turnstile>y<:T" shows "\<Gamma>\<turnstile>S<:T"
pp_of_term_of_pair: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and k :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.pp_of_term pair_of_term (term_of_pair (t, k)) = t"
IH: assumes "k < K" shows "multiplicity p (j + k choose k) = 0"
weakly_preferred_by_women_def: fixes \<mu> :: "(M \<times> W) set"   and \<mu>' :: "(M \<times> W) set" shows "Non_Strict.weakly_preferred_by_women \<mu> \<mu>' = (\<forall>w. \<forall>m\<in>{m. (m, w) \<in> \<mu>}. \<exists>m'\<in>{m. (m, w) \<in> \<mu>'}. m' \<in> Non_Strict.w_weakly_prefers w \<mu>)"
R2: fixes xps :: "('a \<times> 'b) list"   and Y :: "('a \<times> 'b) set"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'c"   and X :: "('a \<times> 'b) set" assumes "(xps, Y) \<in> c_failures step out s\<^sub>0"   and "X \<subseteq> Y" shows "(xps, X) \<in> c_failures step out s\<^sub>0"
llist_all2_ltlI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist" assumes "llist_all2 P xs ys" shows "llist_all2 P (ltl xs) (ltl ys)"
nn_transfer_operator_cmult: fixes g :: "'a \<Rightarrow> ennreal"   and c :: "ennreal" assumes "g \<in> borel_measurable M" shows "AE x in M. nn_transfer_operator (\<lambda>x. c * g x) x = c * nn_transfer_operator g x"
encode_problem_parallel_correct_ii_2: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "k < length (\<Phi>\<inverse> \<Pi> \<A> t)" shows "are_all_operator_effects_consistent ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
birkhoff_sum_small_asymp_neg: fixes f :: "'a \<Rightarrow> real"   and e :: "real" assumes "integrable M f"   and "0 < e" shows "AE x in M. infinite {n. birkhoff_sum f n x \<in> {real n * expectation f - e..real n * expectation f}}"
equal_poly_mapping_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b" shows "equal_class.equal p q \<equiv> \<forall>t. lookup p t = lookup q t"
eq_OO: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(=) OO R = R"
Abs_formula_inverse: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "Rep_formula (Abs_formula t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
empty_iff: fixes c :: "'b::type" shows "((c::'b::type) \<in> {}) = False"
init_triv_filt_def: fixes M :: "'a measure"   and F :: "'b \<Rightarrow> 'a measure" shows "init_triv_filt M F = (Filtration.filtration M F \<and> sets (F \<bottom>) = {{}, space M})"
Conj_mono: fixes A :: "fm"   and H :: "fm set"   and B :: "fm"   and C :: "fm"   and D :: "fm" assumes "insert A H \<turnstile> B"   and "insert C H \<turnstile> D" shows "insert (A AND C) H \<turnstile> B AND D"
prems_2: shows "j \<noteq> gap i (Suc t)"
Q_freelist_freeword: fixes c :: "'w freeword" assumes "c \<in> FreeGroup S"   and "Abs_freelistfst (freeword c) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>" shows "c \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
fpairs_eqE: fixes r :: "V" assumes "fpairs r = r" obtains "\<And>x. x \<in>\<^sub>\<circ> r \<Longrightarrow> \<exists>a b. x = [a, b]\<^sub>\<circ>"
simps_1: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" assumes "valid_unMultigraph G" shows "valid_unMultigraph.is_trail G v [] v' = (v = v' \<and> v' \<in> nodes G)"
oexp_succ: fixes \<beta> :: "V"   and \<alpha> :: "V" assumes "Ord \<beta>" shows "\<alpha> \<up> ZFC_in_HOL.succ \<beta> = \<alpha> \<up> \<beta> * \<alpha>"
midpoint_eq_endpoint_1: fixes a :: "'a"   and b :: "'a" shows "(midpoint a b = a) = (a = b)"
is_ladder_L': shows "is_ladder (drop (ladder_stepdown_diff L) D) L'"
powsum2: fixes x :: "'a"   and n :: "nat"   and m :: "nat" shows "x\<^bsub>n\<^esub>\<^bsup>Suc m\<^esup> = x\<^bsub>n\<^esub>\<^bsup>m\<^esup> + x\<^bsup>n + Suc m\<^esup>"
le_inf_transp: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>3"
enum_n_top: fixes a :: "nat \<Rightarrow> nat" assumes "a \<in> s" shows "(a = enum n) = (\<forall>a'\<in>s. a' \<le> a)"
weak_reduction_bisimulation_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_bisimulation (symcl Rel) Cal"
closed_Int: fixes "open" :: "'d set \<Rightarrow> bool"   and S :: "'d set"   and T :: "'d set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "topological_space.closed open T" shows "topological_space.closed open (S \<inter> T)"
lgc_invar_lgc_step: fixes i :: "nat"   and TNs :: "('f inference set \<times> ('f \<times> 'l) set) llist" assumes "enat (Suc i) < llength TNs"   and "lgc_invar TNs (enat i)"   and "lnth TNs i \<leadsto>LGC lnth TNs (Suc i)" shows "lgc_invar TNs (enat (Suc i))"
simps_1: fixes e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" shows "preT SKIP e = e"
fresh_def: fixes a :: "atom"   and x :: "'c" shows "a \<sharp> x \<equiv> a \<notin> supp x"
simps_4: fixes x2 :: "'b"   and x1 :: "'b" shows "Neg x2 \<noteq> Pos x1"
intro: fixes I :: "'c set" assumes "finite I" shows "finite_product_sigma_finite_axioms I"
dmu_array_row_main: fixes mm :: "nat"   and j :: "nat" assumes "mm \<le> length fs"   and "j \<le> mm" shows "dmu_array_row_main (IArray fs) (IArray fs !! mm) mm (IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (if i = mm then Suc j else Suc i)) (Suc mm)) j = IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (Suc i)) (Suc mm)"
rref_condition2: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'b)) = (1::'b)"
simps: fixes a :: "intv" shows "isGreater a = (\<exists>uu_. a = Greater uu_)"
calc_ideal_point_2_unit: fixes A :: "complex"   and B :: "complex"   and z1 :: "complex"   and z2 :: "complex" assumes "is_real A"   and "(cmod A)\<^sup>2 < (cmod B)\<^sup>2"   and "(z1, z2) = calc_ideal_point2_cvec A B" shows "z1 * cnj z1 = z2 * cnj z2"
Int_UNIV_right: fixes A :: "'b set" shows "A \<inter> UNIV = A"
norm_numeral: fixes w :: "num" shows "norm (numeral w) = numeral w"
callee_invariant_on_axioms: fixes \<I> :: "('a, 'b) \<I>" shows "callee_invariant_on run_resource (WT_resource \<I>) \<I>"
simps_5: fixes r :: "'b rexp"   and s :: "'b rexp" shows "lang (Times r s) = lang r \<cdot> lang s"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_mult_hom hom" shows "comm_monoid_mult_hom hom"
evalEffect_cl_inv: fixes g :: "gState\<^sub>I"   and p :: "pState"   and prog :: "program"   and e :: "edgeEffect" assumes "cl_inv (g, p)"   and "program_inv prog"   and "gState_inv prog g"   and "pState_inv prog p" shows "cl_inv (evalEffect e prog g p)"
not_triv_relI: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> int" assumes "a \<in> A"   and "f a \<noteq> 0" shows "f \<noteq> (\<lambda>_\<in>A. 0)"
max_list_Max: fixes xs :: "('x, nat) pp list" assumes "xs \<noteq> []" shows "ordered_powerprod_lin.max_list xs = ordered_powerprod_lin.Max (set xs)"
adds_pp_add_linorder_def: shows "adds_pp_add_linorder = (adds)"
hds: shows "head_\<omega> Ma < head_\<omega> Na"
perm_compose: fixes pi2 :: "(name \<times> name) list"   and pi1 :: "(name \<times> name) list"   and x :: "'a" shows "pi2 \<bullet> pi1 \<bullet> x = (pi2 \<bullet> pi1) \<bullet> pi2 \<bullet> x"
sum_of_squares_nat_bound': fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "x\<^sup>2 + y\<^sup>2 = n" shows "y \<le> n"
executableI_step: fixes p :: "'a"   and execute :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and enabled :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and initial :: "'a \<Rightarrow> bool"   and a :: "'b" assumes "p \<in> transition_system_initial.nodes execute enabled initial"   and "enabled a p" shows "a \<in> transition_system_initial.executable execute enabled initial"
semiring_normalization_rules_17: fixes lx :: "'a"   and ly :: "'a"   and rx :: "'a" shows "lx * ly * rx = lx * (ly * rx)"
fps_shift_nth: fixes n :: "nat"   and f :: "'a fps"   and i :: "nat" shows "fps_nth (fps_shift n f) i = fps_nth f (i + n)"
mplus_fail2: fixes m :: "'r match" shows "m +++ Fixrec.fail = m"
distinct_filter: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "distinct xs" shows "distinct (List.filter P xs)"
Rep_perm_simps_4: fixes a :: "atom"   and b :: "atom" shows "Rep_perm (a \<rightleftharpoons> b) = (if sort_of a = sort_of b then \<lambda>c. if a = c then b else if b = c then a else c else id)"
rev_exhaust: fixes xs :: "'d list" obtains "xs = []"   | ys :: "'d list" and y :: "'d" where "xs = ys @ [y]"
density_def: fixes G :: "nat set \<times> nat set set" shows "Ugraph_Lemmas.density G = real (card (uedges G)) / real (card (uverts G))"
tail_algebra_subset: fixes n :: "nat" shows "sets (tail_algebra n) \<subseteq> sigma_sets (space P) (\<Union>i\<in>{n..}. sets.pullback_algebra M (\<lambda>f. f i) (space P))"
function_mult_eval_car: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier (function_ring (carrier R) R)"   and "g \<in> carrier (function_ring (carrier R) R)" shows "(f \<otimes>\<^bsub>function_ring (carrier R) R\<^esub> g) a = f a \<otimes> g a"
elementLevelLeqCurrentLevel: fixes a :: "'a"   and M :: "('a \<times> bool) list" shows "elementLevel a M \<le> currentLevel M"
fqf: shows "\<forall>p. qfree p \<longrightarrow> qfree (f p)"
cyclic_on_perm_swap: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "cyclic_on f S" shows "cyclic_on (perm_swap x y f) ((x \<rightleftharpoons>\<^sub>F y) ` S)"
hyps_2: fixes c2 :: "('a, 'b, 'c) com"   and c :: "('a, 'b, 'c) com"   and s :: "'a"   and n :: "nat" assumes "(a2 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>a2,Normal s\<rangle> =n\<Rightarrow> Fault f \<or> \<Gamma>\<turnstile> \<langle>c2,Normal s\<rangle> =n\<Rightarrow> Fault f"
i_append_i_take_eq1: fixes n :: "nat"   and xs :: "'d list"   and f :: "nat \<Rightarrow> 'd" assumes "n \<le> length xs" shows "(xs \<frown> f) \<Down> n = xs \<down> n"
simps_1: fixes s :: "nat \<Rightarrow> 'a" shows "inf_truncate s 0 = [s 0]"
augment_flow_presv: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and f' :: "nat \<times> nat \<Rightarrow> 'a" assumes "NFlow c s t f"   and "Flow (residualGraph c f) s t f'" shows "Flow c s t (NFlow.augment c f f')"
On_CTRL_Delta_def: shows "On_CTRL_Delta = {(''TunerMode'', (And (En ''Src'') (In ''CDFull''), defaultguard, defaultaction), ''CDMode''), (''CDMode'', (En ''Src'', defaultguard, defaultaction), ''TunerMode''), (''CDMode'', (En ''CDEject'', defaultguard, defaultaction), ''TunerMode''), (''CDMode'', (En ''EndOfTitle'', \<lambda>d. (d !P! 0) = (d !P! 1), defaultaction), ''TunerMode'')}"
PowD: fixes A :: "'b set"   and B :: "'b set" assumes "(A::'b set) \<in> Pow (B::'b set)" shows "(A::'b set) \<subseteq> (B::'b set)"
intersect_mult_set_block_with_point_exists: fixes blv :: "'a set"   and p :: "'a set" assumes "blv \<in># \<B>"   and "p \<subseteq> blv"   and "2 \<le> \<Lambda>"   and "card p = 2" shows "\<exists>x\<in>#remove1_mset blv \<B>. p \<in># mset_set {y. y \<subseteq> blv \<and> y \<subseteq> x \<and> card y = 2}"
suffixes_conv_prefixes: fixes xs :: "'c list" shows "suffixes xs = map rev (prefixes (rev xs))"
Inf_lower: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and x :: "'b::type"   and A :: "'b::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(x::'b::type) \<in> (A::'b::type set)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Inf::'b::type set \<Rightarrow> 'b::type) (A::'b::type set)) (x::'b::type)"
sel_1: fixes x1 :: "'a" shows "projl (Inl x1) = x1"
assms_3: shows "\<Psi>\<^sub>F \<turnstile> \<phi>"
dom_abc: shows "\<D>\<^sub>\<circ> abc = 3\<^sub>\<nat>"
X: shows "distributed M MX X (\<lambda>x. ennreal (indicat_real A x / Sigma_Algebra.measure MX A))"
integral_subalgebra2: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "subalgebra M F"   and "f \<in> borel_measurable F" shows "integral\<^sup>L (restr_to_subalg M F) f = integral\<^sup>L M f"
condRel: shows "\<forall>(P, Q)\<in>Rel. Pred (P, Q)"
ide_dom: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.ide (\<F>C.dom f)"
o2: fixes t1 :: "lam"   and t2 :: "lam"   and s1 :: "lam"   and s2 :: "lam" assumes " t1 \<longrightarrow>\<^sub>1 t2"   and " s1 \<longrightarrow>\<^sub>1 s2" shows " App t1 s1 \<longrightarrow>\<^sub>1 App t2 s2"
string_to_nat_def: fixes s :: "char list" shows "string_to_nat s \<equiv> if is_nat_string s then Some (string_to_nat_aux 0 s) else None"
elims: 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 x :: "'a list"   and xa :: "'a list"   and y :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.upd_rev_cs kind x xa = y" obtains cs :: "'a list" where "x = cs__" and "xa = []" and "y = cs__"   | cs :: "'a list" and as :: "'a list" and a :: "'a" where "x = cs" and "xa = as @ [a]" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> case cs of [] \<Rightarrow> CFG.upd_rev_cs kind cs as | c' # cs' \<Rightarrow> CFG.upd_rev_cs kind cs' as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> CFG.upd_rev_cs kind (a # cs) as | _ \<Rightarrow> CFG.upd_rev_cs kind cs as)"
ev\<^sub>A_eq: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and P :: "'BExp set"   and e :: "'AExp"   and t :: "'BExp set"   and v :: "'Var" assumes "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"   and "pred P mem\<^sub>1"   and "\<Gamma> \<turnstile>\<^sub>a e \<in> t"   and "t \<le>:\<^sub>P dma_type v"   and "dma mem\<^sub>1 v = Low" shows "ev\<^sub>A mem\<^sub>1 e = ev\<^sub>A mem\<^sub>2 e"
wt: shows "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
induct: fixes P :: "iface list \<Rightarrow> bool"   and a0 :: "iface list" assumes "P []"   and "\<And>i. P [i]"   and "\<And>i1 i2 is. (\<And>x2. iface_conjunct i1 i2 = Some x2 \<Longrightarrow> P (x2 # is)) \<Longrightarrow> P (i1 # i2 # is)" shows "P a0"
pseudo_mod_list_def: fixes p :: "'b list"   and q :: "'b list" shows "pseudo_mod_list p q = (if q = [] then p else let rq = rev q in Let (pseudo_mod_main_list (hd rq) (rev p) rq (1 + length p - length q)) rev)"
h: fixes x :: "'a" assumes "x \<in> A" shows "sleq x z"
base: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f" shows "a \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"
Zp_x_is_UP_domain: shows "UP_domain Zp"
power_Suc: fixes a :: "'a"   and n :: "nat" shows "a ^ Suc n = a * a ^ n"
order_trans_type: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type" assumes "\<tau> \<le> \<sigma>"   and "\<sigma> \<le> \<rho>" shows "\<tau> \<le> \<rho>"
InitFinal\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap"   and C :: "char list"   and b :: "bool" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>INIT C ([],b) \<leftarrow> e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
diagonal_functor_def: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah" shows "diagonal_functor J C \<equiv> category C \<and> category J \<and> functor_category J C"
case_2: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "(case \<lfloor>x2\<rfloor> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
relto_mono: fixes R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and S' :: "('a \<times> 'a) set" assumes "R \<subseteq> R'"   and "S \<subseteq> S'" shows "S\<^sup>* O R O S\<^sup>* \<subseteq> S'\<^sup>* O R' O S'\<^sup>*"
not_prime_0: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "\<not> normalization_semidom.prime zero times one normalize zero"
Int_Collect: fixes x :: "'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "(x \<in> A \<inter> {x. P x}) = (x \<in> A \<and> P x)"
Dualo_undef: fixes \<alpha> :: "game option" shows "(Dualo \<alpha> = undefg) = (\<alpha> = undefg)"
vfsequence_vcons_ex: assumes "dg_Rel \<alpha> \<noteq> []\<^sub>\<circ>" obtains xs' :: "V" and x :: "V" where "dg_Rel \<alpha> = xs' #\<^sub>\<circ> x" and "vfsequence xs'"
has_lock_has_lock_acquire_locks: fixes l :: "('c \<times> nat) option"   and T :: "'c"   and t :: "'c"   and n :: "nat" assumes "has_lock l T" shows "has_lock (acquire_locks l t n) T"
simps_1: fixes x :: "variable"   and \<phi> :: "fml" shows "Existso x (Afml \<phi>) = Afml (Exists x \<phi>)"
rel_refl_strong: fixes x :: "'h pmf"   and Ra :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>z. z \<in> set_pmf x \<Longrightarrow> Ra z z" shows "rel_pmf Ra x x"
adjunction_preserved_by_iso_left: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a"   and \<epsilon> :: "'a"   and \<phi> :: "'a"   and f' :: "'a" assumes "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"   and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>"   and "local.iso \<phi>" shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f' g ((g \<star> \<phi>) \<cdot> \<eta>) (\<epsilon> \<cdot> (local.inv \<phi> \<star> g))"
cong_mat: fixes nr :: "nat"   and nr' :: "nat"   and nc :: "nat"   and nc' :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and f' :: "nat \<times> nat \<Rightarrow> 'a" assumes "nr = nr'"   and "nc = nc'"   and "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> f (i, j) = f' (i, j)" shows "Matrix.mat nr nc f = Matrix.mat nr' nc' f'"