hcju/isabelleps huggingface dataset:
z_less: fixes i :: "nat" assumes "i < ord" shows "tmf_c prec (map interval_of xs) f i = Some (z ! i)"
sentence_iff: fixes G :: "('c, 'b) labeled_graph"   and e\<^sub>1 :: "'c allegorical_term"   and e\<^sub>2 :: "'c allegorical_term" shows "G \<Turnstile> e\<^sub>1 \<sqsubseteq> e\<^sub>2 = (:G:\<lbrakk>e\<^sub>1\<rbrakk> \<subseteq> :G:\<lbrakk>e\<^sub>2\<rbrakk>)"
AE_all_countable: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b measure" shows "(AE x in M. \<forall>i. P i x) = (\<forall>i. almost_everywhere M (P i))"
d_idem_par: fixes x :: "'a" shows "d x \<parallel> d x = d x"
first_conv: fixes p :: "'a"   and xs :: "('b \<times> 'a) list" shows "first (p, xs) = p"
l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"
linear_poly_root: fixes a :: "'b"   and as :: "'b list" assumes "a \<in> set as" shows "poly (\<Prod>a\<leftarrow>as. [:- a, 1::'b:]) a = (0::'b)"
trancl_into_rtrancl: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a, b) \<in> r\<^sup>+" shows "(a, b) \<in> r\<^sup>*"
map_indets_map_indets: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'd"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "map_indets f (map_indets g p) = map_indets (f \<circ> g) p"
maps_to_n_def: fixes n :: "int"   and f :: "nat \<Rightarrow> int" shows "maps_to_n n f = (\<forall>k. f k \<in> {0..n})"
simps_977: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x23 :: "v" shows "(case C x23 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f23 x23"
summable_hurwitz_zeta_real: fixes x :: "real"   and a :: "real" assumes "1 < x"   and "0 < a" shows "summable (\<lambda>n. (real n + a) powr - x)"
vector_2_1: fixes x :: "'b"   and y :: "'b" shows "vector [x, y] $ (1::'c) = x"
nAct_less_active: fixes n' :: "enat"   and t :: "cnf llist"   and c :: "'id"   and n :: "nat" assumes "n' - 1 < llength t"   and "\<langle>c #\<^bsub>enat n\<^esub>t\<rangle> < \<langle>c #\<^bsub>n'\<^esub>t\<rangle>" shows "\<exists>i\<ge>n. enat i < n' \<and> \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"
proper_implies_overlap: fixes x :: "i"   and y :: "i" assumes "M"   and "PP x y" shows "O x y"
cinner_complex_def: fixes x :: "complex"   and y :: "complex" shows "\<langle>x, y\<rangle> = cnj x * y"
unfold: shows "op' = (\<Lambda> pat v x. Fixrec.run\<cdot>(match_Pair\<cdot>v\<cdot>(\<Lambda> us vs. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>vs then (us :@ [:x:], stail\<cdot>vs) else If snull\<cdot>us then ([::], pat) else sfoldl\<cdot>(op'\<cdot>pat)\<cdot>([::], pat)\<cdot>(stail\<cdot>us :@ [:x:])))))"
simps_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "msort2 f [] = []"
pmf_le_1: fixes p :: "'g pmf"   and x :: "'g" shows "pmf p x \<le> 1"
trm_simps_3: fixes a :: "'a" assumes "ide a" shows "cod \<t>[a] = \<I>"
accp_imp_SN_on: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> Wellfounded.accp g x" shows "SN_on {(y, z). g z y} A"
bij_pointE: fixes f :: "'d \<Rightarrow> 'e"   and y :: "'e" assumes "bij f" obtains x :: "'d" where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
incr_bounds_wellformed: fixes m :: "nat"   and u :: "term"   and k :: "nat"   and n :: "nat" assumes "wellformed' m u" shows "wellformed' (k + m) (incr_bounds (int k) n u)"
aux7aa: fixes a :: "'a set set"   and b :: "'a set set"   and poo :: "'b"   and p :: "('a set set, 'b) Combinators list"   and c :: "'a set set"   and d :: "'a set set"   and po :: "'b" assumes "AllowPortFromTo a b poo \<in> set p"   and "allNetsDistinct (AllowPortFromTo c d po # p)"   and "a \<noteq> c \<or> b \<noteq> d" shows "twoNetsDistinct a b c d"
SigmaE2: fixes a :: "'b"   and b :: "'c"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "(a, b) \<in> Sigma A B" obtains "a \<in> A" and "b \<in> B a"
TensorDiag_preserves_Ide: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Ide C t"   and "monoidal_language.Ide C u"   and "monoidal_language.Diag C t"   and "monoidal_language.Diag C u" shows "monoidal_language.Ide C (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
reachable_init: fixes s :: "'a"   and A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> reachable A I"
HPhase1or2Write_HInv4d_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase1or2Write s s' p d"   and "HInv4d s p"   and "HInv4a s p" shows "HInv4d s' p"
P1: shows "Per G H I \<and> G H I LtA A B C"
mpoly_base_conv_1: fixes x :: "'c" shows "(0::'c) = poly 0 x"
replace_comp_portfolio: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'c" assumes "portfolio pf1"   and "portfolio pf2" shows "portfolio (qty_replace_comp pf1 x pf2)"
type_definition_Domainp: fixes Rep :: "'b \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and T :: "'d \<Rightarrow> 'b \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "Domainp T = (\<lambda>x. x \<in> A)"
that: fixes s :: "nat \<Rightarrow> real" assumes "\<And>i. P i (s i)"   and "filterlim s at_top sequentially" shows "thesis"
simps: fixes a1 :: "'b set"   and a2 :: "nat" shows "R a1 a2 = ((\<exists>A. a1 = A \<and> a2 = 0 \<and> finite A) \<or> (\<exists>k. a1 = {} \<and> a2 = k) \<or> (\<exists>A x k. a1 = insert x A \<and> a2 = Suc k \<and> finite A \<and> x \<notin> A \<and> R A (Suc k) \<and> R (insert x A) k))"
fconst_on_vempty: fixes c :: "V" shows "fconst_on []\<^sub>\<circ> c = []\<^sub>\<circ>"
choice_iff: fixes Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<forall>x. \<exists>y. Q x y) = (\<exists>f. \<forall>x. Q x (f x))"
conv_has_start_points: fixes x :: "'a" shows "has_start_points x = has_end_points (x\<^sup>T)"
normed_vectorspace_with_fn_norm_def: fixes V :: "'b set"   and norm :: "'b \<Rightarrow> real" shows "normed_vectorspace_with_fn_norm V norm \<equiv> normed_vectorspace V norm"
disj_Un_runiq: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" assumes "runiq P"   and "runiq Q"   and "Domain P \<inter> Domain Q = {}" shows "runiq (P \<union> Q)"
inj_semiring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_semiring_hom hom \<equiv> semiring_hom hom \<and> injective hom"
x: shows "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r"
Gets_Server_message_form: fixes A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Gets A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "evs \<in> bankerb_gets" shows "K \<notin> range shrK \<and> X = Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace> \<or> X \<in> analz (knows Spy evs)"
insert_node_writes: fixes ptr :: "(_) object_ptr"   and new_child :: "(_) node_ptr"   and reference_child_opt :: "(_) node_ptr option"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_child_nodes_locs ptr) (a_insert_node ptr new_child reference_child_opt) h h'"
id: shows "(2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) * (2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) = 1\<^sub>m N"
Shuffle_OneR: fixes r :: "'a language" shows "Shuffle r One = r"
f_expand_take_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'d message_af list" shows "xs \<odot> k \<down> (n * k) = xs \<down> n \<odot> k"
neq_if_length_neq: fixes xs :: "'d list"   and ys :: "'d list" assumes "length xs \<noteq> length ys" shows "xs = ys \<equiv> False"
that: fixes N :: "'a set" assumes "p \<in> N"   and "open N"   and "finite {i \<in> I. U i \<inter> N \<noteq> {}}" shows "thesis"
assms_12: shows "val_Zp (pderiv f \<bullet> \<alpha>) = val_Zp (pderiv f \<bullet> a)"
inst_weakly_step_consistent: fixes u :: "thread_id_t"   and s :: "rstate_t"   and t :: "rstate_t"   and a :: "int_point_t" assumes "rvpeq u s t"   and "rvpeq (rcurrent s) s t"   and "rcurrent s = rcurrent t"   and "rprecondition s (rcurrent s) a"   and "rprecondition t (rcurrent t) a" shows "rvpeq u (rstep s a) (rstep t a)"
ereal_diff_add_assoc2: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" shows "x + y - z = x - z + y"
implements_represents: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'b) Protocol" assumes "implements jp" shows "represents (jpTraces jp)"
relcomp_Image: fixes X :: "('f \<times> 'g) set"   and Y :: "('g \<times> 'e) set"   and Z :: "'f set" shows "(X O Y) `` Z = Y `` X `` Z"
oth_class_taut_10_d: fixes v :: "i"   and \<phi> :: "\<o>"   and \<chi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<chi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<chi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<or> \<psi> \<^bold>\<rightarrow> \<chi>)) in v]"
nat_neq_iff: fixes m :: "nat"   and n :: "nat" shows "(m \<noteq> n) = (order.greater n m \<or> order.greater m n)"
zero_compare_double_copysign: fixes x :: "double"   and y :: "double" assumes "compare_double (copysign_double 1 x) (copysign_double 1 y) \<le> 0"   and "Code_Double.is_zero x"   and "Code_Double.is_zero y" shows "compare_double x y \<le> 0"
assms: shows "dbm_entry_val u (Some c) None (min a b)"
general_U_is_linear_function: fixes p :: "'a pmf"   and q :: "'a pmf"   and \<alpha> :: "real" assumes "p \<in> local.\<P>"   and "q \<in> local.\<P>"   and "\<alpha> \<in> {0..1}" shows "local.general_U (mix_pmf \<alpha> p q) = \<alpha> * local.general_U p + (1 - \<alpha>) * local.general_U q"
f3: shows "length (w # ws) = length (r1 # r1s)"
split_int: fixes t :: "MIR.num"   and tv :: "MIR.num"   and ti :: "MIR.num"   and bs :: "real list" assumes "split_int t = (tv, ti)" shows "Inum bs (Add tv ti) = Inum bs t \<and> isint ti bs"
lift_def: fixes h :: "'f \<Rightarrow> 'e"   and inp :: "'d \<Rightarrow> 'f option" shows "lift h inp \<equiv> \<lambda>i. case inp i of None \<Rightarrow> None | Some v \<Rightarrow> Some (h v)"
add_pos_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less zero b" shows "less zero (plus a b)"
simps_2: fixes v :: "'c" shows "Num_arc_delta v * MinInfty_delta = MinInfty_delta"
composition_invertible: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and x :: "'c"   and y :: "'c" assumes "monoid M composition unit"   and "monoid.invertible M composition unit x"   and "monoid.invertible M composition unit y"   and "x \<in> M"   and "y \<in> M" shows "monoid.invertible M composition unit (composition x y)"
hyps_3: fixes r :: "V" assumes "F = \<D>\<^sub>\<circ> r"   and "Limit \<alpha>"   and "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "vsv r" shows "\<R>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>"
finite_UnI: fixes F :: "'e set"   and G :: "'e set" assumes "finite F"   and "finite G" shows "finite (F \<union> G)"
simps_2: fixes f :: "'a" shows "\<lbrace>\<^bold>\<langle>f\<^bold>\<rangle>\<rbrace> = (if arr f then f else null)"
embed_bool_def: fixes P :: "'c \<Rightarrow> bool" shows "\<guillemotleft> P \<guillemotright> \<equiv> \<lambda>s. if P s then 1 else 0"
H: shows "filter (\<lambda>x. key x = k) (nths xs (A - {mini xs key})) = filter (\<lambda>x. key x = k) (nths vs {i. Suc i + mini xs key \<in> A})"
user_accessible_mod_trap: fixes new_traps :: "Trap set"   and s :: "'b sparc_state"   and pa :: "word36" shows "user_accessible (s\<lparr>traps := new_traps\<rparr>) pa = user_accessible s pa"
enumerateable_def: fixes T :: "'b set"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "enumerateable T less \<equiv> (finite T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. x \<noteq> y \<longrightarrow> less x y \<or> less y x)) \<and> (\<forall>x\<in>T. \<forall>y\<in>T. \<forall>z\<in>T. less x y \<longrightarrow> less y z \<longrightarrow> less x z) \<and> (\<forall>x\<in>T. \<forall>y\<in>T. less x y \<longrightarrow> \<not> less y x)"
of_int_vec_eq_iff: fixes a :: "(int, 'b) vec"   and b :: "(int, 'b) vec" shows "(of_int_vec a = of_int_vec b) = (a = b)"
tape_of_ex1: fixes a :: "nat"   and rn :: "nat" shows "\<exists>rna ml. Oc \<up> a @ Bk \<up> rn = <ml> @ Bk \<up> rna \<or> Oc \<up> a @ Bk \<up> rn = Bk # <ml> @ Bk \<up> rna"
less_option_def: fixes x :: "'a option"   and y :: "'a option" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
3: shows "taq {x. poly p x = 0} q \<equiv> \<Sum>y\<in>{x. poly p x = 0}. Sturm_Tarski.sign (poly q y)"
seqr_iter_def: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> 'b hrel" shows "seqr_iter xs P = foldr (\<lambda>i. (;;) (P i)) xs II"
word_add_not: fixes x :: "'a word" shows "x + NOT x = - 1"
span_clauses_4: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and S :: "'e set"   and c :: "'d" assumes "module scale"   and "x \<in> module.span scale S" shows "scale c x \<in> module.span scale S"
Call: fixes 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"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and ps :: "char list exp list"   and vs :: "val list"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and l\<^sub>2' :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h\<^sub>3 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>3 :: "char list \<Rightarrow> val option" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = (pns, body) in D"   and "length vs = length pns"   and "l\<^sub>2' = [this \<mapsto> Addr a, pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2')\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>3)\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>2)\<rangle>"
thetaZOCh_ZOretr: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and ch :: "'a"   and c1 :: "('c, 'd, 'a) cmd"   and d1 :: "('c, 'd, 'a) cmd"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and c2 :: "('c, 'd, 'a) cmd"   and d2 :: "('c, 'd, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.compatCh cval indis ch"   and "(c1, d1) \<in> PL_Indis.ZObis aval tval cval indis"   and "(c2, d2) \<in> PL_Indis.ZObis aval tval cval indis" shows "thetaZOCh ch c1 c2 d1 d2 \<subseteq> PL_Indis.ZOretr aval tval cval indis (thetaZOCh ch c1 c2 d1 d2 \<union> PL_Indis.ZObis aval tval cval indis)"
seteq: shows "{p. p partitions n} = (\<Union>k\<le>n. {p. p partitions n \<and> sum p {..n} = k})"
lt2p_lem: fixes n :: "nat"   and w :: "'a Word.word" assumes "LENGTH('a) \<le> n" shows "uint w < 2 ^ n"
Diag_TensorE_3: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "Diag t"
same_pmod_iff_pdivides: fixes K :: "'a set"   and a :: "'a list"   and b :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])"   and "q \<in> carrier (K [X])" shows "(a pmod q = b pmod q) = q pdivides (a \<ominus>\<^bsub>K [X]\<^esub> b)"
abs_eq: fixes xa :: "'state"   and x :: "('b, 'pred, 'act) Tree" shows "valid_Tree\<^sub>\<alpha> xa (abs_Tree\<^sub>\<alpha> x) = valid_Tree xa x"
rbt_max_opt_in_set: fixes t :: "('b, unit) RBT_Impl.rbt" assumes "t \<noteq> rbt.Empty" shows "rbt_max_opt t \<in> set (RBT_Impl.keys t)"
emeasure_density: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and A :: "'a set" assumes "f \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure (density M f) A = set_nn_integral M A f"
simps: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" shows "ord.lexordp_eq less a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>x y xs ys. a1 = x \<Colon> xs \<and> a2 = y \<Colon> ys \<and> less x y) \<or> (\<exists>x y xs ys. a1 = x \<Colon> xs \<and> a2 = y \<Colon> ys \<and> \<not> less x y \<and> \<not> less y x \<and> ord.lexordp_eq less xs ys))"
exhaust: fixes y :: "'d iarray" obtains x :: "'d list" where "y = IArray x"
inorder_tree: fixes t :: "'a bro" shows "inorder (tree t) = inorder t"
all_1: shows "\<forall>x\<in>X. h x = h g"
succ_nonempty: shows "succ AB w (p1, p2) \<noteq> {}"
single_of_int: fixes k :: "int" shows "single (0::'b) (of_int k) = of_int k"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "x \<preceq> y"
S_compatible_le_scheme_lists: fixes A' :: "type_scheme list"   and A :: "type_scheme list"   and S :: "nat \<Rightarrow> typ" assumes "A' \<le> A" shows "$ S A' \<le> $ S A"
that: fixes p' :: "'a list" assumes "set p' = set p"   and "path E v p' v" shows "thesis"
simps: fixes a1 :: "'b list"   and a2 :: "'b list"   and R :: "('b \<times> 'b) set" shows "((a1, a2) \<in> lprod R) = ((\<exists>a b. a1 = [a] \<and> a2 = [b] \<and> (a, b) \<in> R) \<or> (\<exists>ah at bh bt a b. a1 = ah @ a # at \<and> a2 = bh @ b # bt \<and> (ah @ at, bh @ bt) \<in> lprod R \<and> ((a, b) \<in> R \<or> a = b)))"
countable_INT: fixes A :: "'c \<Rightarrow> 'b set"   and X :: "'c set"   and M :: "'b measure" assumes "A ` X \<subseteq> sets M"   and "X \<noteq> {}" shows "\<Inter> (A ` X) \<in> sets M"
point_indices_alt_def: fixes t :: "int" shows "point_indices t = {int (\<B> index ps) |ps. int (card ps) = t \<and> ps \<subseteq> \<V>}"
noS: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "norm x = 1"
dim_def: fixes V :: "'b set" shows "dim V = (if \<exists>b. independent b \<and> span b = span V then card (SOME b. independent b \<and> span b = span V) else 0)"
low_indistinguishable_def: fixes mds :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and c\<^sub>1 :: "'Com\<^sub>C"   and c\<^sub>2 :: "'Com\<^sub>C" shows "conc.low_indistinguishable mds c\<^sub>1 c\<^sub>2 = (\<forall>mem\<^sub>1 mem\<^sub>2. conc.low_mds_eq mds mem\<^sub>1 mem\<^sub>2 \<longrightarrow> conc.mm_equiv_abv \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>C \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>C)"
set_zero_plus2: fixes A :: "'b set"   and B :: "'b set" assumes "(0::'b) \<in> A" shows "B \<subseteq> A + B"
rel_Grp: fixes A1 :: "'i set"   and f1 :: "'i \<Rightarrow> 'l"   and A2 :: "'j set"   and f2 :: "'j \<Rightarrow> 'm" shows "rel_gpv (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. results'_gpv x \<subseteq> A1 \<and> outs'_gpv x \<subseteq> A2} (map_gpv f1 f2)"
cs_length_one_1: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "length (cs\<^bsup>\<pi>\<^esup> i) = 1" shows "cs\<^bsup>\<pi>\<^esup> i = [\<pi> i]"
A: shows "A \<in> sets (Pi\<^sub>M {0..<Suc n} M)"
pt_set_finite_ineq: fixes pi :: "('b \<times> 'b) list"   and X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "finite (pi \<bullet> X) = finite X"
assms_4: fixes C :: "char list"   and sobj :: "sfields \<times> init_state" assumes "sh C = \<lfloor>sobj\<rfloor>" shows "C \<noteq> C"
bind_spmf_cong: fixes p :: "'a spmf"   and q :: "'a spmf"   and f :: "'a \<Rightarrow> 'b spmf"   and g :: "'a \<Rightarrow> 'b spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
unit_simps_6: fixes a :: "'a" assumes "C.obj a" shows "D.cod (unit a) = map\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0 a"
acute_bet__obtuse: fixes A :: "'p"   and B :: "'p"   and A' :: "'p"   and C :: "'p" assumes "Bet A B A'"   and "A' \<noteq> B"   and "Acute A B C" shows "Obtuse A' B C"
a: shows "u \<cdot> v \<cdot> x \<le> v \<cdot> x \<cdot> z"
that_def: shows "that \<equiv> map_fun (map_fun id eval\<o>) make\<kappa> (\<lambda>\<phi>. if \<exists>!x. \<phi> x dj dw then Some (THE x. \<phi> x dj dw) else None)"
rlfm_I: fixes p :: "fm"   and bs :: "real list" assumes "qfree p" shows "Ifm bs (rlfm p) = Ifm bs p \<and> isrlfm (rlfm p)"
prj_simps_18: shows "C.cod fgf.Prj\<^sub>0\<^sub>0 = f.apex"
x_c_U_split2: fixes x :: "'a" shows "d x \<cdot> nc = (x \<sqinter> nc) \<cdot> nc + x \<cdot> (0::'a) \<parallel> nc"