hcju/isabelleps huggingface dataset:
simps_60: fixes x71 :: "trm"   and x72 :: "char list"   and x6 :: "(char list \<times> trm) list" shows "x71..x72 \<noteq> Rcd x6"
box_ne_empty_1: fixes a :: "'b"   and b :: "'b" shows "(cbox a b \<noteq> {}) = (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
perm_weak_filter': fixes P :: "'b list \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "\<forall>xs ys. xs <~~> ys \<longrightarrow> P xs = P ys"   and "\<forall>x xs. P xs \<longrightarrow> P (x # xs)" shows "\<forall>ys. P (ys @ filter Q xs) \<longrightarrow> P (ys @ xs)"
Union_Un_distrib: fixes A :: "'a set set"   and B :: "'a set set" shows "\<Union> (A \<union> B) = \<Union> A \<union> \<Union> B"
abstract_filter_def: fixes f :: "unit \<Rightarrow> 'b filter" shows "Filter.abstract_filter f = f ()"
main: shows "poly_of_vec (vec_first v n) * p + poly_of_vec (vec_last v m) * q = 0"
closed_nonpos_Ints: shows "closed \<int>\<^sub>\<le>\<^sub>0"
pigeon_cols_4: shows "(col \<circ> a_r) r1 = (col \<circ> a_r) r2"
Sup_subset_mono: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
case_sum_inject: fixes f1 :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g1 :: "'a \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "case_sum f1 f2 = case_sum g1 g2" obtains "f1 = g1" and "f2 = g2"
IH_1: fixes Ts :: "typ list"   and T :: "typ" assumes "typ_of1 Ts t1 = Some T" shows "typ_of1 (map (subst_typ insts) Ts) (subst_typ' insts t1) = Some (subst_typ insts T)"
rotate7: fixes A7 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and A5 :: "fmla"   and A6 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A7 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 H)))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H)))))) \<turnstile> B"
A2: shows "A2 \<in> carrier_mat n (Suc m)"
assms: shows "GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec"
ord_divides': fixes a :: "nat"   and d :: "nat"   and n :: "nat" shows "[a ^ d = Suc 0] (mod n) = (ord n a dvd d)"
map_add_map_of_fold: fixes ps :: "('a \<times> 'b) list"   and m :: "'a \<Rightarrow> 'b option" assumes "distinct (map fst ps)" shows "m ++ map_of ps = fold (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
isCont_difg: fixes m :: "nat"   and x :: "real" assumes "m < n"   and "0 \<le> x"   and "x \<le> h" shows "isCont (difg m) x"
staticSecret_def: fixes A :: "agent" shows "staticSecret A \<equiv> {Key (priEK A), Key (priSK A), Key (shrK A)}"
A_B_step: fixes a :: "'b"   and b :: "'b"   and a' :: "'c" assumes "B a b"   and "sim2 a a'"   and "PB a"   and "PC a'" shows "\<exists>b'. C a' b' \<and> sim2 b b'"
gs: shows "gs = fs[i := fs ! j, j := fs ! i]"
mset_msort: fixes xs :: "'a list" assumes "distinct xs" shows "mset (msort xs) = mset xs"
dgrad_set_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" shows "dgrad_set d m = {t. d t \<le> m}"
subgroup_generated_def: fixes G :: "('e, 'f) monoid_scheme"   and S :: "'e set" shows "subgroup_generated G S = G\<lparr>carrier := generate G (carrier G \<inter> S)\<rparr>"
semiring_normalization_rules_34: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "times x (plus y z) = plus (times x y) (times x z)"
push_bit_push_bit: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> '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 m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit m (push_bit n a) = push_bit (m + n) a"
nonzero_inverse_eq_divide: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse a = divide one a"
this: shows "DA alphabet2 init2 delta2 accept2 wellformed2 Language2 wf2 Lang2"
append_same_eq: fixes ys :: "'g list"   and xs :: "'g list"   and zs :: "'g list" shows "(ys @ xs = zs @ xs) = (ys = zs)"
l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs type_wf known_ptr \<equiv> (l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<and> l_set_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_shadow_root set_shadow_root_locs) \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_shadow_root set_shadow_root set_shadow_root_locs remove_shadow_root remove_shadow_root_locs"
chine_composite_in_hom: shows "\<guillemotleft>chine : \<mu>.leg0 \<down>\<down> \<nu>.leg1 \<rightarrow>\<^sub>C \<mu>.cod.leg0 \<down>\<down> \<nu>.cod.leg1\<guillemotright>"
wfvarsoccs\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('f, 'e) stateful_strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map wfvarsoccs\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
Theorem1: fixes \<rho> :: "nat \<rightarrow> ValD"   and \<rho>' :: "nat \<rightarrow> ('o ValK \<rightarrow> 'o) \<rightarrow> 'o"   and e :: "expr" assumes "\<forall>v. \<eta>: \<rho>\<cdot>v \<mapsto> \<rho>'\<cdot>v" shows "\<eta>: \<lbrakk>e\<rbrakk>\<rho> \<mapsto> evalK e\<cdot>\<rho>'"
computation_swingpos: shows "rat_poly_plus (rat_poly_times B (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) B)) (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) (rat_poly_times A (A - rat_poly_times (rat_poly_times B B) B))) = rat_poly_times A (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) A)"
set_rel: shows "(set ys, zs) \<in> \<langle>S\<rangle>set_rel"
abrupt_if_Some: fixes c :: "bool"   and x :: "abopt"   and y :: "abrupt" shows "abrupt_if c x (Some y) = Some y"
mult_sign_intros_1: fixes a :: "'d"   and b :: "'d" assumes "(0::'d) \<le> a"   and "(0::'d) \<le> b" shows "(0::'d) \<le> a * b"
absorb2: fixes a :: "'b::semilattice_sup"   and b :: "'b::semilattice_sup" assumes "(a::'b::semilattice_sup) \<le> (b::'b::semilattice_sup)" shows "(a::'b::semilattice_sup) \<squnion> (b::'b::semilattice_sup) = b"
sel_5: fixes x3 :: "'b formula" shows "un_Neg (Neg x3) = x3"
extends_to_adjoint_equivalence: shows "\<exists>G \<eta> \<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G F\<^sub>1 \<eta> \<epsilon>"
bicategory_axioms: shows "bicategory (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B"
invert_one_side: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B OS P Q" shows "B A OS P Q"
subword_self: fixes w :: "'c list" shows "subword w 0 (length w) = w"
integrable_const_bound: fixes f :: "'g \<Rightarrow> 'h"   and B :: "real"   and p :: "'g spmf" assumes "AE x in measure_spmf p. norm (f x) \<le> B"   and "f \<in> borel_measurable (measure_spmf p)" shows "integrable (measure_spmf p) f"
intro: fixes smult :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "ModuleHom smult V smult' T" shows "VectorSpaceHom smult V smult' T"
C: fixes x :: "(real, 'n) vec" assumes "x \<in> B" shows "norm x \<le> C"
nsubst_sterm: fixes \<sigma> :: "'a \<Rightarrow> ('sf, 'sz) trm"   and \<theta> :: "('sf + 'a, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "TadmitFFO \<sigma> \<theta>"   and "\<And>i. dsafe (\<sigma> i)" shows "sterm_sem I (TsubstFO \<theta> \<sigma>) (fst \<nu>) = sterm_sem (adjointFO I \<sigma> \<nu>) \<theta> (fst \<nu>)"
RedCastFail: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   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" assumes "hp s a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>Cast C (addr a),s\<rangle> \<rightarrow> \<langle>THROW ClassCast,s\<rangle>"
distinct_1: fixes x21 :: "'a"   and x22 :: "'a llist" shows "LNil \<noteq> LCons x21 x22"
Skolem_list_nth: fixes k :: "nat"   and P :: "nat \<Rightarrow> 'm \<Rightarrow> bool" shows "(\<forall>i<k. \<exists>x. P i x) = (\<exists>xs. length xs = k \<and> (\<forall>i<k. P i (xs ! i)))"
nres_of_def: fixes r :: "'b dres" shows "nres_of r \<equiv> case r of dSUCCEEDi \<Rightarrow> SUCCEED | dFAILi \<Rightarrow> FAIL | dRETURN x \<Rightarrow> RETURN x"
cases: fixes a1 :: "(C, BD) sequent list"   and a2 :: "(C, BD) sequent" assumes "g3mod2p a1 a2" obtains A :: "(C, BD) form" where "a1 = [ \<Empt> \<Rightarrow>* \<LM> Aa__  \<RM>]" and "a2 = ( \<Empt> \<Rightarrow>* \<LM> \<diamond> Aa__  \<RM>)"   | A :: "(C, BD) form" where "a1 = [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>]" and "a2 = ( \<LM> \<box> A  \<RM> \<Rightarrow>* \<Empt>)"
valid_refs_invD_7: fixes x :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and y :: "'a" assumes "grey x s"   and "(x reaches y) s"   and "valid_refs_inv s" shows "obj_at (\<lambda>s. True) y s"
Lb_def: shows "Lb = span ((+) (- a) ` (B - {a}))"
comp_res_aux: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<smile> ; - (x ; y) \<cdot> y = 0"
disj2: shows "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> Xin i \<inter> Xin j = {}"
isFault_def: fixes S :: "('d, 'e) xstate" shows "isFault S = (\<exists>f. S = Fault f)"
nonneg: shows "\<forall>\<^sub>F x in at_top. 0 \<le> f x"
reduced_letter_set_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set" assumes "class.monoid_add plus zero" shows "\<Union> (set ` Collect (monoid_add.reduced_word_for plus zero A zero)) = {}"
rep_eq: shows "fls_nth 0 = (\<lambda>_. 0::'b)"
vector_fraction_eq_iff: fixes u :: "real"   and v :: "real"   and a :: "'c"   and x :: "'c" shows "((u / v) *\<^sub>R a = x) = (if v = 0 then x = (0::'c) else u *\<^sub>R a = v *\<^sub>R x)"
stable_on_no_blocking_on: fixes ds :: "'d set"   and X :: "'x set" assumes "stable_on ds X" shows "stable_no_blocking_on ds X"
orthogonal_transformation_matrix: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec" shows "orthogonal_transformation f = (linear f \<and> orthogonal_matrix (matrix f))"
4: shows "\<forall>a\<in>A. F' a \<subseteq> Field r \<and> |F' a| <o r"
lower_asymptotic_density_finite: fixes A :: "nat set" assumes "finite A" shows "lower_asymptotic_density A = 0"
sorted_iff_nth_Suc: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq xs = (\<forall>i. Suc i < length xs \<longrightarrow> less_eq (xs ! i) (xs ! Suc i))"
Abinds_join_fresh: fixes ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "ae' ` domA \<Delta> \<subseteq> {\<bottom>}" shows "ABinds \<Delta>\<cdot>(ae \<squnion> ae') = ABinds \<Delta>\<cdot>ae"
mset_get_min_rest: fixes ts :: "'b tree list"   and t' :: "'b tree"   and ts' :: "'b tree list" assumes "get_min_rest ts = (t', ts')"   and "ts \<noteq> []" shows "mset ts = {#t'#} + mset ts'"
less_eq_literal_def: shows "(\<le>) \<equiv> map_fun literal.explode (map_fun literal.explode id) (ord.lexordp_eq (\<lambda>c d. of_char c < of_char d))"
pdo_close_under_measurement: fixes M :: "complex mat"   and n :: "nat"   and \<rho> :: "complex mat" assumes "M \<in> carrier_mat n n"   and "\<rho> \<in> carrier_mat n n"   and "partial_density_operator \<rho>"   and "adjoint M * M \<le>\<^sub>L 1\<^sub>m n" shows "partial_density_operator (M * \<rho> * adjoint M)"
abs_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup_pp (PP x) = lookup x"
term_of_anything: fixes x :: "'a"   and t :: "term" shows "term_of x \<equiv> t"
carrier_vec_dim_vec: fixes v :: "'a vec" shows "v \<in> carrier_vec (dim_vec v)"
setPair1: fixes a :: "'b"   and b :: "'b"   and d :: "'b" assumes "{a, b} = {d, a}" shows "b = d"
1: shows "s \<star> rs = \<bottom> \<star> rs'"
length_FDERIV_floatariths: fixes fas :: "floatarith list"   and xs :: "nat list"   and ds :: "floatarith list" shows "length (FDERIV_floatariths fas xs ds) = length fas"
TableInv_def: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real"   and l0 :: "nat" shows "TableInv f1 f2 f1' f2' e c l0 \<equiv> Table0 f1 f2 f1' f2' e c \<and> TableInv_axioms f1 f2 e c l0"
disjoint_then_null: fixes A :: "'a set" assumes "A \<in> sets M"   and "\<And>n. 0 < n \<Longrightarrow> A \<inter> (T ^^ n) -` A = {}" shows "A \<in> null_sets M"
index_map_mat_2: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e mat" shows "dim_row (map_mat f A) = dim_row A"
wp_under_wlp_SetDC: fixes S :: "'a \<Rightarrow> 'b set"   and f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> S s \<Longrightarrow> wp_under_wlp (f a)"   and "\<And>s a. a \<in> S s \<Longrightarrow> healthy (wp (f a))"   and "\<And>s. S s \<noteq> {}" shows "wp_under_wlp (SetDC f S)"
topological_space_generate_topology: fixes S :: "'b set set" shows "class.topological_space (generate_topology S)"
simps_1: fixes uu :: "nat"   and n :: "nat" shows "ok uu (MSkip n) = True"
K3_imp_Gets: fixes A :: "agent"   and authTicket :: "msg"   and authK :: "nat"   and T2 :: "nat"   and B :: "agent"   and evs :: "event list" assumes "Says A Tgs \<lbrace>authTicket, Crypt authK \<lbrace>Agent A, Number T2\<rbrace>, Agent B\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "\<exists>Ta. Gets A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"
memo_nat: fixes low :: "nat"   and up :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "memo_nat low up f = f"
simps_8: fixes f :: "'s \<Rightarrow> 'r" shows "map_option f None = None"
sat_n_imp: fixes N :: "'f set" assumes "saturated N" shows "ground.Inf_from (\<G>_Fset N) \<subseteq> {\<iota>. \<exists>\<iota>'\<in>Inf_from N. \<G>_I \<iota>' \<noteq> None \<and> \<iota> \<in> the (\<G>_I \<iota>')} \<union> Red_I_G (\<G>_Fset N)"
fcompare_def: fixes a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "fcompare a b = (if is_nan a \<or> is_nan b then Und else if is_infinity a \<and> sign a = 1 then if is_infinity b \<and> sign b = 1 then ccode.Eq else ccode.Lt else if is_infinity a \<and> sign a = 0 then if is_infinity b \<and> sign b = 0 then ccode.Eq else ccode.Gt else if is_infinity b \<and> sign b = 1 then ccode.Gt else if is_infinity b \<and> sign b = 0 then ccode.Lt else if valof a < valof b then ccode.Lt else if valof a = valof b then ccode.Eq else ccode.Gt)"
le_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and e\<^sub>1 :: "'c err"   and e\<^sub>2 :: "'c err" shows "Err.le r e\<^sub>1 e\<^sub>2 = (case e\<^sub>2 of Err \<Rightarrow> True | OK y \<Rightarrow> case e\<^sub>1 of Err \<Rightarrow> False | OK x \<Rightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
less_option_Some: fixes x :: "'b"   and y :: "'b" shows "(Some x < Some y) = (x < y)"
T: shows "Sigma_Algebra.measure lebesgue T \<le> e / 2 / (2 * B * real DIM('M)) ^ DIM('N)"
simps_1: 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 A :: "'a"   and uu :: "'a"   and uv :: "'b" assumes "concrete_category Obj Hom Id Comp" shows "concrete_category.Dom (concrete_category.MkArr A uu uv) = A"
idd: fixes a :: "rat" shows "(ipoly p a = 0) = False"
AddSA_InitConf2: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates A = {}"   and "S \<notin> InitConf A"   and "S \<in> HAStates A" shows "InitConf (A [++] (S, SA)) = InitConf A"
rat_lepoll_nat: shows "UNIV \<lesssim> UNIV"
inf_concat_bounds: fixes n :: "nat \<Rightarrow> nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "\<exists>\<^sub>\<infinity>i. 0 < n i"   and "inf_concat n k = (i, j)" shows "j < n i"
simps_2: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "RBTi2 TYPE('b) bal_i e" shows "RBTi2.\<Phi> e \<langle>l, x, r\<rangle> = 6 * e * real (imbal \<langle>l, x, r\<rangle>) + RBTi2.\<Phi> e l + RBTi2.\<Phi> e r"
flatdom_strict2cont: fixes f :: "'b \<Rightarrow> 'a" assumes "f \<bottom> = \<bottom>" shows "cont f"
nearly_healthy_wlp_PC: fixes f :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "nearly_healthy (wlp f)"   and "nearly_healthy (wlp g)"   and "unitary P" shows "nearly_healthy (wlp (f \<^bsub>P\<^esub>\<oplus> g))"
WT_gpv_inline_invar: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('d, 'e) \<I>"   and s :: "('a, 'b, 'd, 'e) converter" assumes "\<I> \<turnstile>g gpv \<surd>"   and "\<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "\<I>' \<turnstile>g inline run_converter gpv s \<surd>"
r_contains_prim: fixes xs :: "nat list" shows "prim_recfn 1 (r_contains xs)"
sup: fixes x :: "'a" shows "(SUP i. U i x) = ennreal (u x)"
prod_le_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x) \<and> less_eq (f x) one" shows "less_eq (comm_monoid_mult.prod times one f A) one"
that: assumes "x \<in> RF T"   and "\<not> essential \<Gamma> (B \<Gamma>) T x" shows "thesis"
meta_spec: fixes P :: "'c \<Rightarrow> prop"   and x :: "'c" assumes "\<And>x. PROP P x" shows "PROP P x"
m_o_h: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and X :: "char list set"   and opt :: "(char list \<Rightarrow> 'a) option" assumes "Measure1_fun m h"   and "finite X" shows "Measure1_fun.m_o m h opt X \<le> h * card X + 1"
normalfilters_def: shows "normalfilters = {F \<in> filters. \<forall>a b. (a l\<rightarrow> b \<in> F) = (a r\<rightarrow> b \<in> F)}"
simps: fixes a1 :: "'a sequent list \<times> 'a sequent"   and a2 :: "'a form" shows "rightPrincipal a1 a2 = (\<exists>C F Fs Ps. a1 = (Ps, C) \<and> a2 = Compound F Fs \<and> C = ( \<Empt> \<Rightarrow>* \<LM> Compound F Fs  \<RM>))"
left_mult_one_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "1\<^sub>m nr * A = A"
forest_CCs: fixes E' :: "'edge set" assumes "forest E'" shows "card (CCs E') + card E' = card V"
select_convs_2: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and more :: "'a" shows "getM \<lparr>getF = getF, getM = getM, \<dots> = more\<rparr> = getM"
assms_2: shows "suffix i w \<Turnstile>\<^sub>n \<psi>[\<G>\<F> \<psi> (suffix i w)]\<^sub>\<Sigma>\<^sub>2 = suffix i w \<Turnstile>\<^sub>n \<psi>"
\<rho>\<^sub>2_def: shows "\<rho>\<^sub>2 \<equiv> run\<^sub>t \<delta>\<^sub>2 q\<^sub>2 w"
map_M_pure_E2: fixes h :: "'m"   and g :: "'p \<Rightarrow> ('m, 'n, 'o) prog"   and xs :: "'p list"   and ys :: "'o list"   and y :: "'o" assumes "h \<turnstile> map_M g xs \<rightarrow>\<^sub>r ys"   and "y \<in> set ys"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (g x) h" obtains x :: "'p" where "h \<turnstile> g x \<rightarrow>\<^sub>r y" and "x \<in> set xs"