hcju/isabelleps huggingface dataset:
back_edge: fixes s :: "('v, 'es) state_scheme"   and u :: "'v"   and v :: "'v"   and s' :: "('v, 'es) state_scheme" assumes "DFS_invar G param s"   and "I s"   and "cond s"   and "\<not> is_break param s"   and "stack s \<noteq> []"   and "(u, v) \<in> pending s"   and "u = hd (stack s)"   and "v \<in> dom (discovered s)"   and "v \<notin> dom (finished s)"   and "s' = back_edge u v (s\<lparr>pending := pending s - {(u, v)}\<rparr>)" shows "on_back_edge param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
lhs_def: shows "lhs \<equiv> finite_dimensional_vector_space_ow (Collect (Domainp B))"
takeWhile_tail: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and l :: "'a list" assumes "\<not> P x" shows "takeWhile P (xs @ x # l) = takeWhile P xs"
deriv: shows "(\<lambda>h. - ((f (x + h) - f x) / h)) \<midarrow>0::'a\<rightarrow>\<^sub>N\<^sub>S - D"
ivl_disj_un_10: fixes l :: "'c"   and u :: "'c" assumes "l \<le> u" shows "{..<l} \<union> {l..u} = {..u}"
\<eta>_p: shows "\<forall>i<Suc n. \<forall>S. S \<subseteq># (DA0' # CAs0') ! i \<longrightarrow> S \<cdot> (\<eta>0' # \<eta>s0') ! i = S \<cdot> \<eta>"
lprefix_ltake_same: fixes n :: "enat"   and xs :: "'b llist"   and m :: "enat" shows "lprefix (ltake n xs) (ltake m xs) = (n \<le> m \<or> llength xs \<le> m)"
map_append: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and ys :: "'f list" shows "map f (xs @ ys) = map f xs @ map f ys"
qstar_inductr: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z \<squnion> y \<cdot> x \<le> y" shows "z \<cdot> qstar x \<le> y"
simps_2: fixes c :: "char"   and s :: "char list" shows "parse_attributes (c # s) = (if c \<in> set ''/>'' then Inr ([], c # s) else (parse_name \<bind> (\<lambda>k. exactly ''='' \<bind> (\<lambda>_. parse_attribute_value \<bind> (\<lambda>v. parse_attributes \<bind> (\<lambda>atts. Parser_Monad.return ((k, v) # atts)))))) (c # s))"
rquot_dfs_def: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and x :: "nat" shows "rquot_dfs M n x = gen_dfs (rquot_succs M n) rquot_ins rquot_memb (rquot_empt M) [x]"
assms: shows "ibounded_by (\<A>1, \<A>2) q"
gfp_unfold: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "\<mu> f = f (\<mu> f)"
sorted_list_subset_correct: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "ord.sorted_list_subset less (=) xs ys = (set xs \<subseteq> set ys)"
enum_UNIV: fixes enum :: "'c list"   and enum_all :: "('c \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('c \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "set enum = UNIV"
sas_plus_equivalent_to_strips_i_a_VII: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'd option"   and ops' :: "('c \<times> 'd) strips_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "dom s \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom s. the (s v) \<in> \<R>\<^sub>+ \<Psi> v"   and "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "are_all_operators_applicable_in s (map (strips_op_to_sasp \<Psi>) ops') \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent (map (strips_op_to_sasp \<Psi>) ops')" shows "STRIPS_Semantics.are_all_operator_effects_consistent ops'"
mult_0_right: fixes m :: "nat" shows "m \<cdot> 0 = 0"
seq_char: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" shows "seq g f = (arr f \<and> arr g \<and> Dom g = Cod f)"
UNIV_witness: shows "\<exists>x. s2p UNIV x"
valid_weak_action_modality_strong: fixes X :: "'a"   and P :: "'state"   and \<alpha> :: "'act"   and x :: "('idx, 'pred, 'act) formula" assumes "finite (supp X)" shows "P \<Turnstile> \<langle>\<langle>\<alpha>\<rangle>\<rangle>x = (\<exists>\<alpha>' x' P'. Act \<alpha> x = Act \<alpha>' x' \<and> P \<Rightarrow>\<langle>\<alpha>'\<rangle> P' \<and> P' \<Turnstile> x' \<and> bn \<alpha>' \<sharp>* X)"
permutation_inverse_works_2: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" shows "p \<circ> inv p = id"
run_complete: fixes c :: "'a Heap"   and \<sigma> :: "heap option" obtains \<sigma>' :: "heap option" and r :: "'a" where "run c \<sigma> \<sigma>' r"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "a \<sqinter> b \<le> a"
drop_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'c list" shows "drop n (drop m xs) = drop (n + m) xs"
rep_eq: fixes x :: "'c \<Rightarrow> 'e \<Rightarrow> 'd option"   and xa :: "('c, 'e) mapping" shows "mapping.rep (map_values x xa) = (\<lambda>xb. Option.bind (mapping.rep xa xb) (x xb))"
simps_2: fixes p :: "'b \<Rightarrow> bool" shows "next_free atom\<^sub>p(p) = True"
step_Normal_elim_cases_1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> (Skip, Normal s) \<rightarrow> u" shows "P"
remove_vertex_V': fixes x :: "'a" shows "V\<^bsub>H.remove_vertex x\<^esub> \<subseteq> V\<^bsub>remove_vertex x\<^esub>"
suppI_contra: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "x \<notin> supp f" shows "f x = (0::'b)"
comp_null_1: fixes f :: "'c \<times> 'c" shows "C.VV.comp C.VV.null f = C.VV.null"
multiset_nonemptyE: fixes A :: "'a multiset" assumes "A \<noteq> {#}" obtains x :: "'a" where "x \<in># A"
homotopic_with_id2: fixes X :: "'a topology"   and g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> topspace X \<Longrightarrow> g (f x) = x" shows "homotopic_with (\<lambda>x. True) X X (g \<circ> f) id"
ennreal_indicator: fixes A :: "'h set"   and x :: "'h" shows "ennreal (indicat_real A x) = indicator A x"
monic_prod: fixes as :: "'c set"   and f :: "'c \<Rightarrow> 'd poly" assumes "\<And>a. a \<in> as \<Longrightarrow> monic (f a)" shows "monic (prod f as)"
mult_minus1_right: fixes 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 z :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "times z (uminus one) = uminus z"
par_symmetry: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "C D Par A B"
eq: shows "hd (Dec \<BB>\<^sub>F {u, v} u) = hd (Dec \<BB>\<^sub>F {u, v} v)"
f9: shows "fst (fst (get_curr_win () s2)) = fst (fst (get_curr_win () s1))"
pcompose_assoc: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "p \<circ>\<^sub>p (q \<circ>\<^sub>p r) = p \<circ>\<^sub>p q \<circ>\<^sub>p r"
pathstart_join: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" shows "pathstart (g1 +++ g2) = pathstart g1"
at_left_MInf: shows "at_left (- \<infinity>) = bot"
simps_3: fixes x2 :: "'b" shows "extB.Val x2 \<noteq> extB.Bot"
sum2sq_nat_def: fixes a :: "nat"   and b :: "nat" shows "sum2sq_nat a b \<equiv> a\<^sup>2 + b\<^sup>2"
distinct_remove1: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs" shows "distinct (remove1 x xs)"
rPar2: fixes \<Psi>\<^sub>P :: "'b"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> x\<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* x\<alpha>"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* C"   and "xT = P \<parallel> Q'" shows "Prop"
intro: fixes to_tree_order_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog" assumes "to_tree_order_si = l_to_tree_order_si\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_to_tree_order_si get_child_nodes get_shadow_root" shows "l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order_si get_child_nodes get_shadow_root"
sparsegridI: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and lm :: "nat" assumes "p \<in> sparsegrid' dm"   and "level p < lm" shows "p \<in> sparsegrid dm lm"
f1'f1: shows "1 / ((f1' - f1) * c) \<le> real l0"
poly_mod_type_def: fixes m :: "int" shows "poly_mod_type TYPE('a) m \<equiv> m = int CARD('a)"
A3': fixes a :: "'p"   and b :: "'p"   and c :: "'p" assumes "a b \<equiv> c c" shows "a = b"
SeqQuoteP_imp_OrdP: fixes t :: "tm"   and u :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqQuoteP t u s k} \<turnstile> OrdP k"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.in_hom f a b" shows "S.in_hom (S.\<rr>'.map f) (S.map a) (S.R b)"
assms: shows "r : a \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> b"
\<omega>_dec_replicate: fixes m :: "nat"   and \<alpha> :: "V"   and \<alpha>s :: "V list" shows "sorted_wrt (\<lambda>x y. y \<le> x) (replicate m \<alpha> @ \<alpha>s) = (if m = 0 then sorted_wrt (\<lambda>x y. y \<le> x) \<alpha>s else sorted_wrt (\<lambda>x y. y \<le> x) (\<alpha> # \<alpha>s))"
inj_on_map_eq_map: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and ys :: "'c list" assumes "inj_on f (set xs \<union> set ys)" shows "(map f xs = map f ys) = (xs = ys)"
mem_carrier: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "subgroup H G"   and "x \<in> H" shows "x \<in> carrier G"
Quotient_real_int: shows "Quotient (eq_onp (\<lambda>r. fst r \<le> snd r)) Abs_real_int Rep_real_int cr_real_int"
that: fixes U :: "'a set"   and V :: "'a set" assumes "open U"   and "compact (closure U)"   and "open V"   and "S \<subseteq> U"   and "T \<subseteq> V"   and "U \<inter> V = {}" shows "thesis"
abs_eq: fixes x :: "((real, 'a) vec, 'a) vec" shows "to_mtx x\<^sup>\<dagger> = to_mtx (Finite_Cartesian_Product.transpose x)"
prems_1: shows "set (a # p1) \<subseteq> K"
le_fin_eq_fin_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w\<^sub>3 :: "'a list" assumes "traces ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>2 w\<^sub>3" shows "traces.le_fin ind w\<^sub>1 w\<^sub>3"
UN_least: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and C :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> order.greater_eq C (B x)" shows "order.greater_eq C (\<Union> (B ` A))"
st_vec_of_pmf_pmf_of_st_vec: fixes x :: "'b st_vec" shows "st_vec_of_pmf (pmf_of_st_vec x) = x"
distinct_5: fixes x41 :: "PreSimplyTyped.type"   and x42 :: "PreSimplyTyped.type" shows "TUnit \<noteq> TPair x41 x42"
kgdd1_index_adjoin_group_block: fixes x :: "'a"   and ps :: "'a set" assumes "x \<notin> \<V>"   and "ps \<subseteq> insert x \<V>"   and "card ps = 2" shows "(\<B> + mset_set {insert x g |g. g \<in> \<G>}) index ps = 1"
hmsetmset_empty_iff: fixes n :: "hmultiset" shows "(hmsetmset n = {#}) = (n = 0)"
cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "class.linorder less_eq less" obtains "xb__ = []"   | x :: "'b" and ys :: "'b list" where "xb__ = x # ys"
step1: shows "j < length M2 \<longrightarrow> (i < row_length (v # M) * row_length M2 \<and> j < length (v # M) * length M2) \<and> mat (row_length (v # M)) (length (v # M)) (v # M) \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> (v # M \<otimes> M2) ! j ! i = (v # M) ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)"
fls_divide_convert_times_inverse: fixes f :: "'a fls"   and g :: "'a fls" shows "f / g = f * inverse g"
odd_card_imp_not_empty: fixes A :: "'b set" assumes "odd (card A)" shows "A \<noteq> {}"
no_loose_bvar_imp_no_subst_bv1: fixes t :: "Core.term"   and lev :: "nat"   and u :: "Core.term" assumes "\<not> loose_bvar t lev" shows "subst_bv1 t lev u = t"
agreement_is_local: fixes \<sigma> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and \<tau> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and dec :: "'g \<Rightarrow> 'h option"   and p :: "'f"   and v :: "'h"   and q :: "'f"   and w :: "'h"   and m :: "nat"   and n :: "nat" assumes "locally_similar \<sigma> \<tau>"   and "\<And>m n. \<lbrakk>dec (\<sigma> m p) = Some v; dec (\<sigma> n q) = Some w\<rbrakk> \<Longrightarrow> v = w"   and "dec (\<tau> m p) = Some v"   and "dec (\<tau> n q) = Some w" shows "v = w"
RID\<^sub>S_def: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option" shows "RID\<^sub>S \<sigma> \<equiv> \<Union> (RID\<^sub>V ` ran \<sigma>)"
surf: shows "homeomorphism (S - rel_interior S) (sphere (0::'a) 1 \<inter> affine hull S) proj surf"
list_nonempty_induct: fixes xs :: "'a list"   and P :: "'a list \<Rightarrow> bool" assumes "xs \<noteq> []"   and "\<And>x. P [x]"   and "\<And>x xs. \<lbrakk>xs \<noteq> []; P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P xs"
prob_eq_AE: fixes P :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream \<Rightarrow> bool"   and Q :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream \<Rightarrow> bool"   and s :: "('test, 'atom, 'choice) cmd \<times> 'state" assumes "AE x in T s. P x = Q x"   and "{x \<in> space (T s). P x} \<in> T.events s"   and "{x \<in> space (T s). Q x} \<in> T.events s" shows "T.prob s {x \<in> space (T s). P x} = T.prob s {x \<in> space (T s). Q x}"
proj_arcs_H_simps_3: shows "proj_arcs_H vw = rev_G uv"
card_of_nat: shows "ordIso2 (card_of UNIV) natLeq"
ipurge_tr_del_traces: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and ys :: "'c list"   and zs :: "'c list"   and xs :: "'c list" assumes "secure P I D"   and "\<And>v z. \<lbrakk>v \<in> sinks I D u ys; z \<in> set zs\<rbrakk> \<Longrightarrow> (v, D z) \<notin> I"   and "xs @ ipurge_tr I D u ys @ zs \<in> traces P"   and "xs @ ys \<in> traces P" shows "xs @ ys @ zs \<in> traces P"
simps: fixes p1 :: "float poly"   and e1 :: "float interval"   and p2 :: "float poly"   and e2 :: "float interval" shows "tm_add (TaylorModel p1 e1) (TaylorModel p2 e2) = TaylorModel (p1 +\<^sub>p p2) (e1 + e2)"
disjunctive_def: shows "Comp.disjunctive = {x. \<forall>y z. x \<circ> y \<squnion> z = (x \<circ> y) \<squnion> (x \<circ> z)}"
simps_1: fixes f :: "'b \<rightarrow> 'c \<rightarrow> 'b"   and z :: "'b" shows "foldl\<cdot>f\<cdot>z\<cdot>[] = z"
a1: shows "snd (rett_instr i s) = False \<and> s' = snd (fst (rett_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
assms: shows "extend S ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) = ( \<Gamma> \<Rightarrow>* \<Delta>)"
simps_1: fixes e :: "('a, 'b) Expression" shows "sub e in Vx = e"
subo: shows "Subobjs P C (Cs @ C' # Cs')"
pow_subgroup_indicator: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" assumes "finite_comm_group G"   and "subgroup H G"   and "a \<in> carrier G" shows "a [^]\<^bsub>G\<^esub> subgroup_indicator G H a \<in> H"
trss_find_call: fixes sp :: "'n"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and v :: "'n"   and r' :: "'n list"   and c' :: "'n list multiset" assumes "(([sp], c), w, v # r', c') \<in> trcl (trss fg)"   and "r' \<noteq> []" shows "\<exists>rh ch p wa wb. w = wa @ LCall p # wb \<and> proc_of fg v = p \<and> (([sp], c), wa, rh, ch) \<in> trcl (trss fg) \<and> ((rh, ch), LCall p, entry fg p # r', ch) \<in> trss fg \<and> (([entry fg p], ch), wb, [v], c') \<in> trcl (trss fg)"
less_irrefl: fixes ko :: "'c key_order"   and x :: "'c" shows "\<not> ko.lt ko x x"
image_monic_is_monic_set: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_monic_set (monic ` A)"
inject: fixes x1 :: "int option"   and x2 :: "int option"   and y1 :: "int option"   and y2 :: "int option" shows "(I x1 x2 = I y1 y2) = (x1 = y1 \<and> x2 = y2)"
polynate: fixes bs :: "'b list"   and p :: "'b poly" shows "Ipoly bs (polynate p) = Ipoly bs p"
ldl1: shows "LeftDerivation (ladder_stepdown_\<alpha>_0 \<alpha> D L) (drop (ladder_stepdown_diff L) D) \<gamma>"
in_set_conv_nth: fixes x :: "'c"   and xs :: "'c list" shows "(x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)"
inf_top_left: fixes x :: "'a upred" shows "(true \<and> x) = x"
free_no_res: shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot) \<longrightarrow> (\<nexists>x. 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1)"
fin: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. U \<inter> V \<noteq> {}}"
measure: fixes i :: "nat" shows "emeasure M (A i) \<noteq> \<infinity>"
d: fixes i :: "'a" assumes "i \<in> I" shows "d i \<noteq> 0 \<and> x i ^ d i = 1"
suminf_bound_add: fixes f :: "nat \<Rightarrow> ereal"   and y :: "ereal"   and x :: "ereal" assumes "\<forall>N. sum f {..<N} + y \<le> x"   and "\<And>n. 0 \<le> f n"   and "y \<noteq> - \<infinity>" shows "suminf f + y \<le> x"
SourcesA5_L0: shows "Sources level0 sA5 = {sA1, sA2, sA3, sA4}"
tllist_ord_def: shows "local.tllist_ord \<equiv> map_fun (\<lambda>ys. (llist_of_tllist ys, terminal ys)) (map_fun (\<lambda>ys. (llist_of_tllist ys, terminal ys)) id) (\<lambda>(xs1, b1) (xs2, b2). if lfinite xs1 then b1 = b \<and> lprefix xs1 xs2 \<or> xs1 = xs2 \<and> flat_ord b b1 b2 else xs1 = xs2)"
power_ms_aux: fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and abort :: "bool"   and n :: "nat" assumes "basis_wf basis"   and "is_expansion_aux F f basis"   and "trimmed_ms_aux F" shows "is_expansion_aux (power_ms_aux abort F n) (\<lambda>x. f x ^ n) basis"
the_apartment_isoD_1: fixes A :: "'a set set"   and A' :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> A'" shows "ChamberComplexIsomorphism A A' (the_apartment_iso A A')"
outgoing_outside: fixes x :: "nat" assumes "x \<notin> cf.V" shows "cf.outgoing x = {}"
emeasure_eq_measure: fixes p :: "'d spmf"   and A :: "'d set" shows "emeasure (measure_spmf p) A = ennreal (Sigma_Algebra.measure (measure_spmf p) A)"
Tag_LtK: fixes X :: "tag"   and X' :: "ltkey" shows "Tag X \<noteq> LtK X'"
gcd_neg2_int: fixes x :: "int"   and y :: "int" shows "gcd x (- y) = gcd x y"
Abs_eq: fixes eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "Quotient_Type.quotient eq Rep Abs" shows "(Abs x = Abs y) = eq x y"
assms_3: shows "A, n \<turnstile> (ps @ ps', a) # branch"
simps_2: fixes psubst :: "variable \<Rightarrow> object"   and c :: "object" shows "subst_term psubst (term.CONST c) = c"
foundation7: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> not (\<delta> x)) = (\<tau> |\<noteq> \<delta> x)"
fml_equiv_def: fixes \<phi> :: "fml"   and \<psi> :: "fml" shows "fml_equiv \<phi> \<psi> \<equiv> \<forall>I. fml_sem I \<phi> = fml_sem I \<psi>"
inject_5: fixes x61 :: "'b"   and x62 :: "'b fm"   and y61 :: "'b"   and y62 :: "'b fm" shows "(K x61 x62 = K y61 y62) = (x61 = y61 \<and> x62 = y62)"
rep_eq: fixes x :: "'a"   and xa :: "('a, 'b) alist" shows "impl_of (delete x xa) = AList.delete x (impl_of xa)"
frechet_derivative_works: fixes f :: "'c \<Rightarrow> 'd"   and net :: "'c filter" shows "(f differentiable net) = (f has_derivative frechet_derivative f net) net"
pairwise_mono: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "pairwise P A"   and "\<And>x y. P x y \<Longrightarrow> Q x y"   and "B \<subseteq> A" shows "pairwise Q B"
insert_DiffM2: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "remove1_mset x M + {#x#} = M"
CastNull: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"
eclose_facts_2: fixes X :: "hf" shows "X \<le> eclose X"
not_empty_eq_Iic_eq_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'a" assumes "class.preorder less_eq less" shows "{} \<noteq> ord.atMost less_eq h"
Least_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and k :: "'c" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
succ_imp_graph_def: fixes cnf :: "lit uprod set"   and l :: "lit" shows "succ_imp_graph cnf l = set_fold_cfi (succs_loop_body l) {} cnf"
insert_weights_cong: fixes s :: "bool"   and m :: "(nat \<times> nat) convnet"   and w1 :: "nat \<Rightarrow> real"   and w2 :: "nat \<Rightarrow> real" assumes "\<And>i. i < count_weights s m \<Longrightarrow> w1 i = w2 i" shows "insert_weights s m w1 = insert_weights s m w2"
induct: fixes P :: "'b word \<Rightarrow> 'b wordinterval \<Rightarrow> bool"   and a0 :: "'b word"   and a1 :: "'b wordinterval" assumes "\<And>el s e. P el (WordInterval s e)"   and "\<And>el r1 r2. \<lbrakk>P el r1; P el r2\<rbrakk> \<Longrightarrow> P el (RangeUnion r1 r2)" shows "P a0 a1"
dfree_Const_simps: fixes r :: "real" shows "dfree (Const r) = True"
distincts_remove1: fixes xss :: "'b list list"   and xs :: "'b list" assumes "distincts xss" shows "distincts (remove1 xs xss)"
atUV_single: fixes U :: "'a set"   and V :: "'a set"   and s :: "'a list" shows "\<not> atUV U V {#s#}"
Entry_no_predecessor: fixes g :: "'g" shows "predecessors g (Entry g) = []"
mult_isor: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set"   and z :: "('b \<times> 'b) set" assumes "x \<subseteq> y" shows "x O z \<subseteq> y O z"
end_correct: fixes n :: "nat" assumes "0 < n" shows "{inv_end1 n} tcopy_end {inv_end0 n}"
nat_pow_zero: fixes n :: "nat" assumes "n \<noteq> 0" shows "(0::'a) ^ n = (0::'a)"
commutative_squareI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "category C"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g"   and "partial_magma.arr C h \<and> partial_magma.arr C k \<and> partial_magma.dom C h = partial_magma.dom C k"   and "partial_magma.dom C f = partial_magma.cod C h"   and "C f h = C g k" shows "category.commutative_square C f g h k"
join_least: fixes x :: "'c"   and z :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> z"   and "y \<sqsubseteq> z" shows "x \<squnion> y \<sqsubseteq> z"
free_liftn: fixes n :: "nat"   and t :: "dB"   and k :: "nat"   and i :: "nat" shows "free (liftn n t k) i = (i < k \<and> free t i \<or> k + n \<le> i \<and> free t (i - n))"
simps_3: fixes a :: "'b" shows "\<guillemotleft>Atom a\<guillemotright> = Atom a"
neg_le_iff_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus b) (uminus a) = less_eq a b"