hcju/isabelleps huggingface dataset:
lang\<^sub>M\<^sub>2\<^sub>L_FOr: fixes n :: "nat"   and \<phi>\<^sub>1 :: "'a formula"   and \<phi>\<^sub>2 :: "'a formula" assumes "wf_formula n (FOr \<phi>\<^sub>1 \<phi>\<^sub>2)" shows "lang\<^sub>M\<^sub>2\<^sub>L n (FOr \<phi>\<^sub>1 \<phi>\<^sub>2) \<subseteq> (lang\<^sub>M\<^sub>2\<^sub>L n \<phi>\<^sub>1 \<union> lang\<^sub>M\<^sub>2\<^sub>L n \<phi>\<^sub>2) \<inter> {enc (w, I) |w I. length I = n \<and> wf_interp_for_formula (w, I) (FOr \<phi>\<^sub>1 \<phi>\<^sub>2)}"
construct_lhs_vector_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list" shows "w_vec_R p qs signs \<equiv> vec_of_list (map (\<lambda>w. rat_of_int (int (length (filter (\<lambda>v. v = w) (map (consistent_sign_vec qs) (characterize_root_list_p p)))))) signs)"
const_in_iff: fixes c :: "'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "((\<lambda>_. c) \<in> L F f) = ((\<lambda>_. 1::'b) \<in> L F f)"
cring_simprules_22: fixes R :: "('a, 'b) ring_scheme" assumes "cring R" shows "\<ominus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
prems_1: shows "\<forall>y. ListMem y (a # l1) \<longrightarrow> P y"
orthogonal_matrix_def: fixes Q :: "(('b, 'c) vec, 'c) vec" shows "orthogonal_matrix Q = (Finite_Cartesian_Product.transpose Q ** Q = mat (1::'b) \<and> Q ** Finite_Cartesian_Product.transpose Q = mat (1::'b))"
Rep_circline_mat_inverse: fixes x :: "circline_mat" shows "Abs_circline_mat (Rep_circline_mat x) = x"
simps_2: fixes b :: "bit list"   and v :: "bit"   and va :: "bit list"   and A :: "bit list"   and B :: "bit list"   and C :: "bit list"   and D :: "bit list"   and E :: "bit list" shows "sha1block b (v # va) A B C D E = (let H = sha1compressstart 79 b A B C D E; AA = addmod32 A (select H 0 31); BB = addmod32 B (select H 32 63); CC = addmod32 C (select H 64 95); DD = addmod32 D (select H 96 127); EE = addmod32 E (select H 128 159) in sha1block (getblock (v # va)) (delblock (v # va)) AA BB CC DD E)"
last_prog_append_Write\<^sub>s\<^sub>b: fixes p :: "'b"   and sb :: "'b memref list"   and volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "last_prog p (sb @ [Write\<^sub>s\<^sub>b volatile a sop v A L R W]) = last_prog p sb"
weakBisimCoinductAux: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox><(X \<union> weakBisim)> Q \<and> \<Psi> \<rhd> P \<leadsto><(X \<union> weakBisim)> Q \<and> (\<forall>\<Psi>'. (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X \<or> \<Psi> \<otimes> \<Psi>' \<rhd> P \<approx> Q) \<and> ((\<Psi>, Q, P) \<in> X \<or> \<Psi> \<rhd> Q \<approx> P)" shows "\<Psi> \<rhd> P \<approx> Q"
Int_commute: fixes A :: "'d set"   and B :: "'d set" shows "A \<inter> B = B \<inter> A"
atom_Sup: fixes x :: "'a"   and Y :: "'a set" assumes "x \<noteq> \<bottom>"   and "\<forall>y. x \<le> y \<or> x \<le> - y" shows "(\<exists>y\<in>Y. x \<le> y) = (x \<le> \<Squnion> Y)"
dualr_iff: fixes x :: "'a"   and y :: "'a" shows "((x, y) \<in> order (dual cl)) = ((y, x) \<in> order cl)"
diff_zero_right: fixes x :: "'a" assumes "x \<in> V" shows "(0::'a) - x = - x"
Union_member: fixes F :: "'c set set"   and x :: "'c" shows "(\<exists>S\<in>F. x \<in> S) = (x \<in> \<Union> F)"
select_convs_3: fixes NTDom :: "('g, 'i, 'j, 'h, 'k, 'l) Functor"   and NTCod :: "('g, 'i, 'j, 'h, 'k, 'l) Functor"   and NatTransMap :: "'g \<Rightarrow> 'h"   and more :: "'m" shows "NatTransMap \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NatTransMap"
x_as_monom: shows "[:0::'b, 1::'b:] = monom (1::'b) 1"
strict_trans: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry"   and c :: "'b DBMEntry" assumes "b \<prec> a"   and "c \<prec> b" shows "c \<prec> a"
ab_rational: shows "(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) \<in> \<rat>"
hyps: assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l t = One"   and "y 0 = [500]" shows "last (fin_make_untimed (inf_truncate z t)) = One"
diff_add_mset_swap: fixes b :: "'a"   and A :: "'a multiset"   and M :: "'a multiset" assumes "b \<notin># A" shows "add_mset b M - A = add_mset b (M - A)"
DERIV_cmult_Id: fixes c :: "'b"   and x :: "'b"   and s :: "'b set" shows "((*) c has_field_derivative c) (at x within s)"
row_length_mat: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m1 :: "'b list list"   and m2 :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "mult.row_length (mult.Tensor f m1 m2) = mult.row_length m1 * mult.row_length m2"
inv_fn: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "bij f" shows "inv (f ^^ n) = inv f ^^ n"
sig_red_regular_rtrancl_lt: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red (\<prec>\<^sub>t) top_tail F)\<^sup>*\<^sup>* p q" shows "lt q = lt p"
asymp_equiv_add_right: fixes h :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "h \<in> o[F](g)" shows "((\<lambda>x. f x + h x) \<sim>[F] g) = (f \<sim>[F] g)"
fix_eq2: fixes f :: "'d"   and F :: "'d \<rightarrow> 'd" assumes "f \<equiv> fix\<cdot>F" shows "f = F\<cdot>f"
rep_eq: fixes x :: "nat" shows "Rep_linear_poly (Var x) = fun_var x"
time_distrib_bind: fixes tm :: "'b tm"   and f :: "'b \<Rightarrow> 'a tm" shows "time (tm \<bind> f) = time tm + time (f (Time_Monad.val tm))"
submodule_def: fixes R :: "('e, 'f) Ring_scheme"   and A :: "('g, 'e, 'h) Module_scheme"   and H :: "'g set" shows "submodule R A H = (H \<subseteq> carrier A \<and> A +> H \<and> (\<forall>a m. a \<in> carrier R \<and> m \<in> H \<longrightarrow> a \<cdot>\<^sub>s\<^bsub>A\<^esub> m \<in> H))"
lookup_hom_components: fixes q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "q \<in> hom_components p"   and "t \<in> keys q" shows "lookup q t = lookup p t"
aezfun_inject: fixes x :: "('b, 'a) aezfun"   and y :: "('b, 'a) aezfun" shows "(aezfun x = aezfun y) = (x = y)"
assms: shows "S = {m..n} \<inter> {i. odd i}"
flrestrictionE: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<bullet> A" obtains a :: "V" and b :: "V" where "x = [a, b]\<^sub>\<circ>" and "a \<in>\<^sub>\<circ> A" and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
elims: fixes x :: "'b extB"   and y :: "'b extB" assumes "- x = y" obtains "x = extB.Bot" and "y = extB.Val top"   | uu_ :: "'b" where "x = extB.Val uu_" and "y = extB.Bot"
elims: fixes x :: "('a \<times> 'b \<times> 'a) list"   and xa :: "('a, 'b) graph"   and y :: "('a, 'b) graph" assumes "rem_unPath x xa = y" obtains g :: "('a, 'b) graph" where "x = []" and "xa = g__" and "y = g__"   | v :: "'a" and w :: "'b" and v' :: "'a" and ps :: "('a \<times> 'b \<times> 'a) list" and g :: "('a, 'b) graph" where "x = (v, w, v') # ps" and "xa = g" and "y = rem_unPath ps (del_unEdge v w v' g)"
relpow_0_E: fixes x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set" assumes "(x, y) \<in> R ^^ 0" obtains "x = y"
arrE: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.arr f" shows "arr (fst f) \<and> VV.arr (snd f) \<and> src (fst f) = trg (fst (snd f))"
indices_neq_imp_events_neq: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and X :: "'a set"   and i :: "nat"   and j :: "nat" assumes "[f[a .. b ..  c]X]"   and "i \<noteq> j"   and "j < card X"   and "i < card X" shows "f i \<noteq> f j"
linorders_finite_nonempty: fixes A :: "'b set" assumes "finite A" shows "linorders_on A \<noteq> {}"
dbl_inc_simps_3: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one one = numeral.numeral one plus (num.Bit1 num.One)"
elims: fixes x :: "('a + 'b) + 'c"   and y :: "'a + 'b + 'c" assumes "rsuml x = y" obtains a :: "'a" where "x = Inl (Inl a)" and "y = Inl a"   | b :: "'b" where "x = Inl (Inr b)" and "y = Inr (Inl b)"   | c :: "'c" where "x = Inr c" and "y = Inr (Inr c)"
atm_of_def: fixes literal :: "'b literal" shows "atm_of literal = (case literal of Pos x1 \<Rightarrow> x1 | Neg x2 \<Rightarrow> x2)"
bounded_linear_add: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f x + g x)"
aff: shows "x + a \<in> affine hull insert x (rel_interior S)"
Agent: fixes A :: "agent"   and H :: "msg set" shows "Messages.Agent A \<in> synth H"
extensional_merge_sub: fixes I :: "'d set"   and J :: "'d set"   and K :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "I \<union> J \<subseteq> K" shows "merge I J (x, y) \<in> extensional K"
irreducible\<^sub>d_multD: fixes a :: "'c poly"   and b :: "'c poly" assumes "irreducible\<^sub>d (a * b)" shows "degree a = 0 \<and> a \<noteq> 0 \<and> irreducible\<^sub>d b \<or> degree b = 0 \<and> b \<noteq> 0 \<and> irreducible\<^sub>d a"
13: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
w_simps_4: shows "trg TTfgh_TfTgh.chine = src ((tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0)"
prems_4: shows "\<forall>x\<in>snd ` set (tx # txs). x \<notin> FvarsT s"
comp_arr_inv: fixes f :: "'b"   and g :: "'b" assumes "B.inverse_arrows f g" shows "B f g = B.dom g"
assms_3: shows "match_iface (oiface r) (p_oiface p)"
naturality: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (\<tau> (partial_magma.cod A f)) (F f) = B (G f) (\<tau> (partial_magma.dom A f))"
f_Exec_Stream_Acc_LocalState_1: fixes localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_LocalState (Suc 0) localState trans_fun xs c = map localState (f_Exec_Comp_Stream trans_fun xs c)"
ext_inject: fixes rec1 :: "'a"   and rec2 :: "'b"   and more :: "'c"   and rec1' :: "'a"   and rec2' :: "'b"   and more' :: "'c" shows "(\<lparr>rec1 = rec1, rec2 = rec2, \<dots> = more\<rparr> = \<lparr>rec1 = rec1', rec2 = rec2', \<dots> = more'\<rparr>) = (rec1 = rec1' \<and> rec2 = rec2' \<and> more = more')"
d_IN_alt_def2: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and y :: "'a" shows "d_IN g y = (\<Sum>\<^sup>+ x\<in>{x. (x, y) \<in> support_flow g}. g (x, y))"
eq: shows "S \<union> V = S \<union> T - T \<union> V"
SourcesA41_L1: shows "Sources level1 sA41 = {sA11, sA22, sA23, sA31, sA32, sA41}"
sg_condition: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "H \<noteq> {}"   and "\<forall>a b. a \<in> H \<and> b \<in> H \<longrightarrow> a \<cdot> \<rho> b \<in> H" shows "G \<guillemotright> H "
r2: shows "ram2 \<in> set (fst (splitAt ram1 vs))"
power_strict_decreasing: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and N :: "nat"   and a :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n < N"   and "less zero a" shows "less a one \<longrightarrow> less (power.power one times a N) (power.power one times a n)"
check_ipresIGSwapAbs: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model" shows "ipresIGSwapAbs check (errMOD MOD) MOD"
SBRMWWrite: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "'d"   and \<O> :: "'e"   and \<R> :: "'f"   and \<S> :: "'g" assumes "cond (\<theta>(t \<mapsto> m a))" shows "(RMW a t (D, f) cond ret A L R W # is, \<theta>, [], m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>(t \<mapsto> ret (m a) (f (\<theta>(t \<mapsto> m a)))), [], m(a := f (\<theta>(t \<mapsto> m a))), \<D>, \<O>, \<R>, \<S>)"
refine: fixes D :: "nat"   and Xi :: "(real list \<times> real list) list"   and X :: "'b set"   and sctni :: "real list sctn"   and sctn :: "'b sctn" assumes "DIM_precond TYPE('b) D"   and "(Xi, X) \<in> clw_rel lvivl_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(nres_of (local.split_ivls_at_halfspace_impl D sctni Xi), split_ivls_at_halfspace $ sctn $ X) \<in> \<langle>clw_rel lvivl_rel\<rangle>nres_rel"
rem_comps_spec_def: fixes bs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "punit.rem_comps_spec bs data = (card (local.punit.component_of_term ` Keys (fst ` set bs)) = fst data + card (punit.const_lt_component ` (fst ` set bs - {0}) - {None}))"
infDecr_def: shows "infDecr \<equiv> alw (ev shd)"
strict_mono_ordinal_of_nat: shows "OrdinalInduct.strict_mono ordinal_of_nat"
assms_1: shows "\<theta> \<preceq>\<^sub>\<circ> \<delta>"
dynmethd_Some_cases: fixes G :: "prog"   and statC :: "qtname"   and dynC :: "qtname"   and sig :: "sig"   and dynM :: "qtname \<times> methd" assumes "dynmethd G statC dynC sig = Some dynM"   and "is_class G dynC"   and "ws_prog G" obtains "methd G statC sig = Some dynM"   | statM :: "qtname \<times> methd" where "methd G statC sig = Some statM" and "dynM \<noteq> statM" and "G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM"
nonneg_vel: shows "0 \<le> other.s' \<delta>"
plus_left_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c"   and z :: "'c" assumes "class.ordered_ab_semigroup plus zero less_eq"   and "less_eq y x" shows "less_eq (plus y z) (plus x z)"
weak_ranking_unique: fixes As :: "'a set list" assumes "is_weak_ranking As"   and "of_weak_ranking As = map_relation (inv f) le" shows "As = weak_ranking (map_relation (inv f) le)"
nth'_list_augment_diff: fixes i :: "nat"   and j :: "nat"   and \<sigma> :: "'a list"   and v :: "'a" assumes "i \<noteq> j" shows "nth' (list_augment \<sigma> i v) j = nth' \<sigma> j"
pos_ord_trans: fixes y :: "'a trm" shows "\<forall>x. trans (pos_ord x y)"
\<tau>red1r_inj_\<tau>reds1r: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and es :: "'addr expr1 list" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>reds1gr uf P t h (e # es, xs) (e' # es, xs')"
lead_coeff_prod: fixes f :: "'b \<Rightarrow> 'a poly"   and A :: "'b set" shows "lead_coeff (prod f A) = (\<Prod>x\<in>A. lead_coeff (f x))"
set_disconnected_nodes_get_child_nodes_is_l_set_disconnected_nodes_get_child_nodes: shows "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs Shadow_DOM.get_child_nodes_locs"
set_diff_eq: fixes A :: "'g set"   and B :: "'g set" shows "A - B = {x \<in> A. x \<notin> B}"
lcoeff_nonzero2: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "p \<noteq> \<zero>\<^bsub>P\<^esub>" shows "lcoeff p \<noteq> \<zero>"
restrict_relation_carrier: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "Order_Predicates.preorder_on carrier le" shows "restrict_relation carrier le = le"
rel_fset_alt_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fset R = (\<lambda>A B. (\<forall>x. \<exists>y. x |\<in>| A \<longrightarrow> y |\<in>| B \<and> R x y) \<and> (\<forall>y. \<exists>x. y |\<in>| B \<longrightarrow> x |\<in>| A \<and> R x y))"
WN_onD: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and a :: "'b" assumes "WN_on r A"   and "a \<in> A" shows "\<exists>b. (a, b) \<in> r\<^sup>!"
keys_restrict_indets_subset: fixes p :: "('b option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "keys (restrict_indets p) \<subseteq> restrict_indets_pp ` keys p"
prob_eq_AE: fixes M :: "'d measure"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "prob_space M"   and "AE x in M. P x = Q x"   and "{x \<in> space M. P x} \<in> sets M"   and "{x \<in> space M. Q x} \<in> sets M" shows "\<P>(x in M. P x) = \<P>(x in M. Q x)"
Units_closed: fixes x :: "'a" assumes "x \<in> Units R" shows "x \<in> carrier R"
fast_dice_roll_n0: fixes v :: "nat"   and c :: "nat" assumes "n = 0" shows "local.fast_dice_roll v c = return_pmf None"
ifte_thms_1: fixes e1 :: "'b"   and e2 :: "'b" shows "If \<bottom> then e1 else e2 = \<bottom>"
exhaust: fixes y :: "value" obtains x1 :: "int" where "y = Num x1"   | x2 :: "String.literal" where "y = Str x2"
assms_2: fixes queue :: "'a list" shows "P (t0 # queue)"
monoid_axioms: shows "monoid (\<circ>\<^sub>s) Var"
symmetric_def: fixes G :: "('c, 'd) pre_digraph" shows "symmetric G \<equiv> sym (arcs_ends G)"
unl_lift: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'d \<Rightarrow> 'c"   and w :: "'d" shows "(f<x>) w \<equiv> f (x w)"
rel_simps_76: fixes n :: "num" shows "(0::'b) \<noteq> numeral n"
insert_second_part: fixes ref :: "'b option"   and r :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and oid :: "'b" assumes "ref = Some r"   and "r \<notin> set xs"   and "r \<in> set ys" shows "insert_spec (xs @ ys) (oid, ref) = xs @ insert_spec ys (oid, ref)"
simps_3: fixes obs' :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and init :: "'a set"   and trans :: "('a \<times> 'a) set"   and obs :: "'a \<Rightarrow> 'b"   and more :: "'c" shows "obs_update obs' \<lparr>init = init, trans = trans, obs = obs, \<dots> = more\<rparr> = \<lparr>init = init, trans = trans, obs = obs' obs, \<dots> = more\<rparr>"
is_unit_hom_if: fixes hom :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "unit_preserving_hom hom"   and "hom x dvd (1::'b)" shows "x dvd (1::'a)"
pairwise_by_adjoining_point: fixes x :: "'a" assumes "x \<notin> \<V>" shows "pairwise_balance (add_point x) (\<B> + mset_set {insert x g |g. g \<in> \<G>}) 1"
pirreducibleE_1: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "pirreducible\<^bsub>K [X]\<^esub> K p" shows "p \<noteq> []"
pp_dist_comp: fixes 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 uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.stone_relation_algebra inf less_eq less sup bot top uminus one times conv" shows "uminus (uminus (times x y)) = times (uminus (uminus x)) (uminus (uminus y))"
intros_3: fixes \<tau> :: "'a OCL_Types.type" shows "element_type (OrderedSet \<tau>) \<tau>"
liminf_PInfty: fixes X :: "nat \<Rightarrow> ereal" shows "X \<longlonglongrightarrow> \<infinity> = (liminf X = \<infinity>)"
eval_nembed_false: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "\<not> P s" shows "\<guillemotleft> \<N> P \<guillemotright> s = 1"
inject: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c"   and x23 :: "'d"   and y21 :: "'c"   and y22 :: "'c"   and y23 :: "'d" assumes "concrete_category Obj Hom Id Comp" shows "(concrete_category.MkArr x21 x22 x23 = concrete_category.MkArr y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
Cons_in_subseqsD: fixes y :: "'b"   and ys :: "'b list"   and xs :: "'b list" assumes "y # ys \<in> ssubseqs xs" shows "ys \<in> ssubseqs xs"
guessed_frame_Init_xpkE: fixes R :: "fid_t" assumes "role (guessed_runs R) = Init" shows "guessed_frame R xpkE = Some (epubKF (R $ kE))"
simps: fixes t :: "name"   and v :: "tm"   and tm :: "tm"   and y :: "tm"   and y' :: "tm"   and t' :: "name"   and u :: "name"   and u' :: "name" assumes "atom t \<sharp> (v, tm, y, y', t', u, u')"   and "atom t' \<sharp> (v, tm, y, y', u, u')"   and "atom u \<sharp> (v, tm, y, y', u')"   and "atom u' \<sharp> (v, tm, y, y')" shows "SubstAtomicP v tm y y' = SyntaxN.Ex t (SyntaxN.Ex u (SyntaxN.Ex t' (SyntaxN.Ex u' (SubstTermP v tm (Var t) (Var t') AND SubstTermP v tm (Var u) (Var u') AND (y EQ Q_Eq (Var t) (Var u) AND y' EQ Q_Eq (Var t') (Var u') OR y EQ Q_Mem (Var t) (Var u) AND y' EQ Q_Mem (Var t') (Var u'))))))"
N: fixes n :: "nat" assumes "N \<le> n" shows "\<bar>Fourier_coefficient f n\<bar> < e / 4"
RPDs_ok: fixes m :: "nat"   and mr :: "mregex"   and ms :: "mregex" assumes "ok m mr"   and "ms \<in> RPDs mr" shows "ok m ms"
Gp_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" shows "\<natural>\<^bsub>G\<^esub>H \<equiv> G\<lparr>carrier := H, top := (\<cdot>\<^bsub>G\<^esub>), iop := iop G, one := \<one>\<^bsub>G\<^esub>\<rparr>"
prv_imp_exi: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<phi>"   and "bprv (exi x (imp \<phi> \<chi>))" shows "bprv (imp \<phi> (exi x \<chi>))"
at_within_open: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S"   and "open S" shows "at a within S = at a"
finiteE: fixes G :: "'c graph" assumes "wf_graph G" shows "finite (edges G)"
\<pi>0: shows "\<pi>' 0 = \<pi> 0"
enum_ChildAct_def: shows "enum_class.enum = [SayIKnow, SayNothing]"
not_cong_4312: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Cong A B C D" shows "\<not> Cong D C A B"
vals_pqueue: fixes xs :: "('d, 'c) bintree option list" shows "set (vals xs) = set |pqueue xs|"
subresultant_swap: fixes J :: "nat"   and f :: "'b poly"   and g :: "'b poly" shows "subresultant J f g = Polynomial.smult ((- (1::'b)) ^ ((degree f - J) * (degree g - J))) (subresultant J g f)"
unit: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dgrp prod one" shows "Dmonoid.unit prod one x"
the_brp: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_unique R" shows "brp (the_brp_\<alpha> R) (Domainp R) = R"
simps_17: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and e1' :: "'a aexp"   and e2' :: "'a aexp" shows "local.less_aexp_aux (Minus e1 e2) (Times e1' e2') = True"
next_match: fixes s :: "'b sparc_state" assumes "snd (execute_instruction () s) = False" shows "NEXT s = Some (snd (fst (execute_instruction () s)))"
standardize_apart_3: shows "eval\<^sub>c\<^sub>s F G Cs"
mult_eq: fixes x :: "nat"   and y :: "nat" shows "x \<otimes>\<^bsub>units_of R\<^esub> y = x * y mod p"
S_intersections_defs_1: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection1 p q \<equiv> proj2_abs (S_intersection1_rep p q)"
less_asym': fixes a :: "'b word"   and b :: "'b word"   and P :: "bool" assumes "a <s b"   and "b <s a" shows "P"