hcju/isabelleps huggingface dataset:
sorted_filter: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (list.map f xs)" shows "linorder.sorted less_eq (list.map f (List.filter P xs))"
inline_sel: fixes callee :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b, 'c, 'd) gpv"   and c :: "('a, 'e, 'f) gpv"   and s :: "'b" shows "the_gpv (inline callee c s) = map_spmf (\<lambda>result. case result of Inl xs \<Rightarrow> Pure xs | Inr (out, oracle, rpv) \<Rightarrow> IO out (\<lambda>input. oracle input \<bind> (\<lambda>(x, s'). inline callee (rpv x) s'))) (inline1 callee c s)"
g': fixes y :: "complex" assumes "y \<in> V" shows "g' y = inv ((*) (exp (g y)))"
RHS: shows "\<forall>T. compact T \<and> T \<subseteq> S \<longrightarrow> compact (f ` T)"
inj_on_subgroup_iff_trivial_ker: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and I :: "'a set" assumes "group_hom G H h"   and "subgroup I G" shows "inj_on h I = (kernel (G\<lparr>carrier := I\<rparr>) H h = {\<one>\<^bsub>G\<^esub>})"
modify_tprog_fetch_even: fixes st :: "nat"   and tp :: "(action \<times> nat) list" assumes "st \<le> length tp div 2"   and "0 < st" shows "modify_tprog tp ! (4 * (st - Suc 0)) = action_map (fst (tp ! (2 * (st - Suc 0))))"
labeled_typed_model_axioms: shows "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2"
generate_subgroup_incl: fixes G :: "('e, 'f) monoid_scheme"   and K :: "'e set"   and H :: "'e set" assumes "Group.group G"   and "K \<subseteq> H"   and "subgroup H G" shows "generate G K \<subseteq> H"
fset_simps_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "fset_from_list (x21 # x22) = finsert x21 (fset_from_list x22)"
eval_cases_3: fixes P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains "e' = Val v" and "s' = s"
a_rcos_module_imp: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> carrier G"   and "x' \<in> H +> x" shows "x' \<oplus> \<ominus> x \<in> H"
Torder_def: fixes D :: "('h, 'i) Order_scheme" shows "Torder D \<equiv> Order D \<and> Torder_axioms D"
cases: fixes S :: "'a set"   and a1 :: "'a list set"   and a2 :: "'a list set" assumes "bisimilar S a1 a2" obtains K :: "'a list set" and L :: "'a list set" where "a1 = K" and "a2 = L" and "K \<subseteq> lists S" and "L \<subseteq> lists S" and "([] \<in> K) = ([] \<in> L)" and "\<And>x. x \<in> S \<Longrightarrow> bisimilar S (Deriv x K) (Deriv x L)"
F2_def: shows "F2 = (\<lambda>x. x - C - 1)"
monoid_mult_isom_axioms: shows "monoid_mult_isom (inv hom)"
measure_well_com: fixes dims :: "nat list"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and k :: "nat" assumes "state_sig.well_com dims (Measure n M S)"   and "k < n" shows "state_sig.well_com dims (S ! k)"
iverson_def: fixes b :: "bool" shows "iverson b = (if b then 1 else 0)"
IS_RIGHT_TOTAL_alt: fixes R :: "('a \<times> 'b) set" shows "IS_RIGHT_TOTAL R = (\<forall>x. \<exists>y. (y, x) \<in> R)"
update_convs_1: fixes arrays' :: "(typerep \<Rightarrow> nat \<Rightarrow> nat list) \<Rightarrow> typerep \<Rightarrow> nat \<Rightarrow> nat list"   and arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'b" shows "arrays_update arrays' \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = \<lparr>arrays = arrays' arrays, refs = refs, lim = lim, \<dots> = more\<rparr>"
IH: fixes gpv :: "('b, 'a, 'c) gpv" shows "interaction_bound' (gpv_stop gpv) = interaction_bound'' gpv"
atLeastLessThan_empty_iff2: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.preorder less_eq less" shows "({} = ord.atLeastLessThan less_eq less a b) = (\<not> less a b)"
pullback_algebra_def: fixes f :: "'c \<Rightarrow> 'a"   and \<Omega>' :: "'c set" shows "pullback_algebra f \<Omega>' = sigma_sets \<Omega>' {f -` A \<inter> \<Omega>' |A. A \<in> M}"
simps_4: fixes v :: "'b"   and va :: "'b"   and vb :: "'c" shows "first_srcNet (AllowPortFromTo v va vb) = srcNet (AllowPortFromTo v va vb)"
normg: fixes x :: "'n" assumes "x \<in> S" shows "norm (g x) \<le> h x"
assms_8: shows "has_greatest_fixpoint (h \<circ> g)"
1: shows "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
cart2_append1_between: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "z_non_zero p"   and "z_non_zero q"   and "z_non_zero r" shows "(B\<^sub>\<real> cart2_pt p cart2_pt q cart2_pt r) = (\<exists>k\<ge>0. k \<le> 1 \<and> cart2_append1 q = k *\<^sub>R cart2_append1 r + (1 - k) *\<^sub>R cart2_append1 p)"
complex_poly_decompose': fixes p :: "complex poly" obtains root :: "nat \<Rightarrow> complex" where "Polynomial.smult (Polynomial.lead_coeff p) (\<Prod>i<Polynomial.degree p. [:- root i, 1:]) = p"
facet_of_empty: fixes S :: "'b set" shows "\<not> S facet_of {}"
disjnt_iff: fixes A :: "'c set"   and B :: "'c set" shows "disjnt A B = (\<forall>x. \<not> (x \<in> A \<and> x \<in> B))"
mat_mult_eq_lincomb: fixes A :: "'a mat"   and nc :: "nat"   and a :: "'a vec \<Rightarrow> 'a" assumes "A \<in> carrier_mat n nc"   and "distinct (cols A)" shows "A *\<^sub>v vec nc (\<lambda>i. a (col A i)) = lincomb a (set (cols A))"
lemma3_5_d_v: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and \<tau> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>' :: "'a \<times> ('b \<times> 'a) list"   and \<tau>' :: "'a \<times> ('b \<times> 'a) list"   and \<upsilon> :: "'a \<times> ('b \<times> 'a) list"   and \<upsilon>' :: "'a \<times> ('b \<times> 'a) list"   and \<tau>'' :: "'a \<times> ('b \<times> 'a) list" assumes "diagram ars (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "diagram ars (\<tau>', \<upsilon>, \<upsilon>', \<tau>'')" shows "diagram ars (\<tau>, (fst \<sigma>, snd \<sigma> @ snd \<upsilon>), (fst \<sigma>', snd \<sigma>' @ snd \<upsilon>'), \<tau>'')"
Inv_preserves_Can_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Can C (monoidal_language.Inv C t)"
set_eq_subset: fixes A :: "'c set"   and B :: "'c set" shows "(A = B) = (A \<subseteq> B \<and> B \<subseteq> A)"
retraction_map_product_projection: fixes i :: "'b"   and I :: "'b set"   and X :: "'b \<Rightarrow> 'c topology" assumes "i \<in> I" shows "retraction_map (product_topology X I) (X i) (\<lambda>x. x i) = (topspace (product_topology X I) = {} \<longrightarrow> topspace (X i) = {})"
nonneg_bounded: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_clinear f" shows "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
distinct_31: fixes x21 :: "char list"   and x22 :: "expr"   and x4 :: "val" shows "Cast x21 x22 \<noteq> Val x4"
sccs_verts_dv: shows "H.sccs_verts = sccs_verts - {{u}}"
simps_2: fixes a :: "symbol list"   and d :: "nat \<times> symbol \<times> symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" shows "LeftDerivation a (d # D) b = (\<exists>x. LeftDerives1 a (fst d) (snd d) x \<and> LeftDerivation x D b)"
KBasic2_12: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<rightarrow> \<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi> in v]"
inferences_between_def: fixes \<Gamma> :: "'b inference set"   and CC :: "'b literal multiset set"   and C :: "'b literal multiset" shows "inference_system.inferences_between \<Gamma> CC C = {\<gamma> \<in> \<Gamma>. infer_from (CC \<union> {C}) \<gamma> \<and> C \<in># prems_of \<gamma>}"
red_uminus: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.red F p q" shows "punit.red F (- p) (- q)"
le_iff_sup: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x y = (sup x y = y)"
islimpt_UNIV_iff: fixes x :: "'a" shows "x islimpt UNIV = (\<not> open {x})"
sets_Sup_measure'2: fixes M :: "'b measure set" shows "sets (Sup_measure' M) = sigma_sets (\<Union> (space ` M)) (\<Union> (sets ` M))"
basis_row_space_code: fixes A :: "(('a, 'b) vec, 'c) vec" shows "basis_row_space A = (let A' = Gauss_Jordan A in {row i A' |i. row i A' \<noteq> 0})"
cfs_add_int_pow: fixes p :: "nat \<Rightarrow> 'a"   and n :: "int"   and k :: "nat" assumes "p \<in> carrier P" shows "([n] \<cdot>\<^bsub>P\<^esub> p) k = [n] \<cdot> p k"
sin_in_Reals: fixes z :: "'a" assumes "z \<in> \<real>" shows "sin z \<in> \<real>"
sub_branch_image: fixes f :: "'d \<Rightarrow> 'c"   and branch :: "(('b, 'd) fm list \<times> 'd) list" shows "set (sub_branch f branch) = sub_block f ` set branch"
formulaEntailsLiteral_def: fixes formula :: "Literal list list"   and literal :: "Literal" shows "formulaEntailsLiteral formula literal \<equiv> \<forall>valuation. model valuation formula \<longrightarrow> literalTrue literal valuation"
bit_Suc: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit a (Suc n) = bit (divide a (numeral.numeral one plus (num.Bit0 num.One))) n"
H_wf: shows "tm_wf0 (tcopy |+| H)"
deriv_body: shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P Z) the (\<Gamma> p) (Q Z),(A Z)"
snth_Stream: fixes x :: "'b"   and s :: "'b stream"   and i :: "nat" shows "(x ## s) !! Suc i = s !! i"
Singleton_nonEmpty: fixes x :: "ZF" shows "Singleton x \<noteq> Empty"
callee_invariant: fixes y :: "'d"   and s' :: "'s2"   and s :: "'s2"   and x :: "'c" assumes "(y, s') \<in> set_spmf (right s x)"   and "Ir s"   and "x \<in> outs_\<I> \<I>r" shows "Ir s'"
synth_analz_insert: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> synth (analz H)" shows "synth (analz (insert X H)) = synth (analz H)"
simps_8: fixes d :: "'a \<Rightarrow> ('a, 'b, 'c) com" shows "redexes (DynCom d) = {DynCom d}"
from_hma\<^sub>m_def: fixes a :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>m a = Matrix.mat CARD('d) CARD('c) (\<lambda>(i, j). a $h Bij_Nat.from_nat i $h Bij_Nat.from_nat j)"
hyps_4: fixes m :: "'a" assumes "m \<in> F"   and "\<forall>a\<in>F. m \<le> a" shows "Min F = m"
cbigD': fixes o' :: "'state \<times> 'coll"   and P :: "'prog"   and \<sigma> :: "'state" assumes "o' \<in> cbig P \<sigma>" shows "\<exists>n. o' \<in> csmall_nstep P \<sigma> n \<and> fst o' \<in> endset"
RedSCall: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   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 vs :: "val list"   and s :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "length vs = length pns"   and "length Ts = length pns" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(map Val vs),s,True\<rangle> \<rightarrow> \<langle>blocks (pns, Ts, vs, body),s,False\<rangle>"
simulation_silent1: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and s1' :: "'x1 \<times> 'm1" assumes "t \<turnstile> s1 \<approx> s2"   and "r1.silent_move t s1 s1'" shows "\<exists>s2'. r2.silent_moves t s2 s2' \<and> t \<turnstile> s1' \<approx> s2'"
monos_compose: fixes m :: "'a"   and m' :: "'a" assumes "local.mono m"   and "local.mono m'"   and "seq m' m" shows "local.mono (m' \<cdot> m)"
index_mat_multrow_2: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and mul :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_multrow_gen mul i a A $$ (i, j) = mul a (A $$ (i, j))"
pair_in_pair_image_iff: fixes s :: "('fun, 'a) Term.term"   and t :: "('fun, 'a) Term.term"   and P :: "(('fun, 'a) Term.term \<times> ('fun, 'a) Term.term) set" shows "(pair (s, t) \<in> pair ` P) = ((s, t) \<in> P)"
cross_refl: fixes x :: "(real, 3) vec" shows "x \<times> x = 0"
continuous_on_eq_continuous_at: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "open s" shows "continuous_on s f = (\<forall>x\<in>s. isCont f x)"
norm_unsigned_nat_to_bv: fixes n :: "nat" shows "norm_unsigned (nat_to_bv n) = nat_to_bv n"
simps_2: fixes v :: "'c" shows "siftDown (T v E E) = T v E E"
exhaust_sel: fixes option :: "'b option" obtains "option = None"   | "option = Some (option.the option)"
f5: fixes cs :: "char list" shows "List.gen_length 0 cs - Suc 0 = List.gen_length 0 (tl cs)"
path_valid_node_2: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n as n'" shows "While_CFG.valid_node prog n'"
preserves_dom: fixes f :: "'a" assumes "C.arr f" shows "C.dom (src\<^sub>C f) = src\<^sub>C (C.dom f)"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and \<eta> :: "'b arrow_of_spans_data"   and \<epsilon> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "adjunction_data_in_bicategory (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) (span_vertical_category.trg C) f g \<eta> \<epsilon>" shows "adjunction_data_in_span_bicategory C prj0 prj1 f g \<eta> \<epsilon>"
ibounded_by_def: fixes "consider" :: "'e \<Rightarrow> bool" shows "ind_cpa_pk.ibounded_by consider = (\<lambda>(\<A>1, \<A>2) q. \<exists>q1 q2. (\<forall>pk. interaction_bounded_by consider (\<A>1 pk) q1) \<and> (\<forall>cipher \<sigma>. interaction_bounded_by consider (\<A>2 cipher \<sigma>) q2) \<and> q1 + q2 \<le> q)"
Zn_neq1_cyclic_group: fixes n :: "nat" assumes "n \<noteq> 1" shows "Finite_And_Cyclic_Groups.cyclic_group (Z n) 1"
fmdom_foldl_add: fixes m :: "('a, 'b) fmap"   and ns :: "('a, 'b) fmap list" shows "fmdom (foldl (++\<^sub>f) m ns) = fmdom m |\<union>| ffUnion (fmdom |`| fset_of_list ns)"
comp_null_1: fixes f :: "'c" shows "C C.null f = C.null"
refBy_order: fixes Q :: "'b upred"   and P :: "'b upred" shows "(P \<sqsubseteq> Q) = `Q \<Rightarrow> P`"
iptables_bigstepD_4: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Reject]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Decision FinalDeny"
client_def: shows "client \<equiv> RETURN (fold (\<lambda>f. f) [Challenge1.insert 1, Challenge1.insert 2, Challenge1.insert 3, Challenge1.insert 5, move_left, Challenge1.insert 4, move_right, Challenge1.insert 6, delete] Challenge1.empty)"
meta_spec: fixes P :: "'b \<Rightarrow> prop"   and x :: "'b" assumes "\<And>x. PROP P x" shows "PROP P x"
mset_ordering_addition: fixes A :: "'c multiset"   and B :: "'c multiset"   and C :: "'c multiset" assumes "A = B + C" shows "B \<subseteq># A"
update_convs_3: fixes set_cod' :: "'b set \<Rightarrow> 'b set"   and set_dom :: "'b set"   and set_func :: "'b \<Rightarrow> 'b"   and set_cod :: "'b set"   and more :: "'c" shows "set_cod_update set_cod' \<lparr>set_dom = set_dom, set_func = set_func, set_cod = set_cod, \<dots> = more\<rparr> = \<lparr>set_dom = set_dom, set_func = set_func, set_cod = set_cod' set_cod, \<dots> = more\<rparr>"
preservation: shows "rel_weakly_preserves_barbs (indRelRTPO TRel) (STCalWB SWB TWB)"
simps_17: fixes e :: "char list exp" shows "fv (throw e) = fv e"
ord_Zp_p_int_unit: fixes n :: "int" assumes "n mod p \<noteq> 0" shows "ord_Zp ([n] \<cdot> \<one>) = 0"
paste_subrel: fixes P :: "('a \<times> 'b) set"   and Q :: "('a \<times> 'b) set" assumes "P || Domain Q \<subseteq> Q" shows "P +* Q = P \<union> Q"
union_right_cancel: fixes M :: "'a multiset"   and K :: "'a multiset"   and N :: "'a multiset" shows "(M + K = N + K) = (M = N)"
FDOM_pre_subset_prob_dom: fixes p :: "('d, 'e) fmap"   and e :: "('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set" assumes "(p, e) \<in> PROB" shows "fmdom' p \<subseteq> prob_dom PROB"
SourcesA32_L1: shows "Sources level1 sA32 = {sA11, sA23}"
ncloses_elim: fixes x :: "name"   and B :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and \<theta>_n :: "(name \<times> coname \<times> trm) list" assumes "(x, B) \<in> set \<Gamma>"   and "\<theta>_n ncloses \<Gamma>" shows "\<exists>c P. \<theta>_n nmaps x to Some (c, P) \<and> <c>:P \<in> \<parallel><B>\<parallel>"
variant_variable_fresh: fixes S :: "variable set"   and s :: "variable" assumes "finite S" shows "fst (variant_variable s S) \<notin> S"
groupF_tuple: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "groupF f xs = map (map fst) (groupF snd (map (\<lambda>x. (x, f x)) xs))"
exactly_one_snapshot: fixes t :: "('a, 'b, 'c) event list"   and p :: "'a" assumes "trace init t final" shows "\<exists>!i. \<not> ps (S t i) p \<noteq> None \<and> ps (S t (i + 1)) p \<noteq> None"
B: shows "\<forall>m<n. m \<in> {..<length ys} \<longrightarrow> D (ys ! m) \<notin> sinks I D u (take (Suc m) ys) \<longrightarrow> out (foldl step (foldl step s\<^sub>0 xs) (ipurge_tr I D u (take m ys))) (ys ! m) = out (foldl step (foldl step s\<^sub>0 xs) (take m ys)) (ys ! m)"
degree2_coeffs: fixes p :: "'a poly" assumes "degree p = 2" shows "\<exists>a b c. p = [:c, b, a:] \<and> a \<noteq> (0::'a)"
plan_is_at_least_singleton_plan_if_trace_has_at_least_two_elements: fixes k :: "nat"   and I :: "'c \<Rightarrow> 'd option"   and \<psi> :: "('c, 'd) sas_plus_operator list list" assumes "k < length (trace_parallel_plan_sas_plus I \<psi>) - 1" obtains ops :: "('c, 'd) sas_plus_operator list" and \<psi>' :: "('c, 'd) sas_plus_operator list list" where "\<psi> = ops # \<psi>'"
natural_isomorphism_axioms: shows "natural_isomorphism A B F G \<phi>"
up_one_closed: fixes R :: "('a, 'b) ring_scheme" assumes "ring R" shows "(\<lambda>n. if n = 0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>) \<in> up R"
r_inv: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "x \<otimes> inv x = \<one>"
univ_ce_is_not_comp1: shows "- univ_ce \<notin> ce_sets"
same_append_eq: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "(xs @ ys = xs @ zs) = (ys = zs)"
set_upto: fixes i :: "int"   and j :: "int" shows "set [i..j] = {i..j}"
avg_aux_def: fixes s :: "(nat, unit) HashMap.hashmap" shows "avg_aux s \<equiv> hs.iterate s (\<lambda>x (c, s). (c + 1, s + x)) (0, 0)"
group_l_invI: assumes "\<And>x. x \<in> carrier H \<Longrightarrow> \<exists>y\<in>carrier H. y \<otimes>\<^bsub>H\<^esub> x = \<one>\<^bsub>H\<^esub>" shows "Group.group H"
heap_first_el_alt: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int"   and x :: "'a" assumes "transp cmp"   and "is_heap cmp heap l r"   and "x \<in># arr_mset heap l r"   and "heap l \<noteq> x" shows "cmp (heap l) x"
assms_3: shows "\<forall>n\<in>set (tl ns). v \<notin> allDefs g n"
ACC: shows "set (lasso_cycle L) \<inter> bg_F (degeneralize_ext m) \<noteq> {}"
single_def: shows "Poly_Mapping.single \<equiv> map_fun id (map_fun id Abs_poly_mapping) (\<lambda>k v k'. v when k = k')"
invariantCFalseAfterConflict: fixes conflictFlag :: "bool"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and clause :: "Literal list"   and C' :: "Literal list"   and conflictFlag' :: "bool" assumes "conflictFlag = False"   and "formulaFalse F (elements M)"   and "clause el F"   and "clauseFalse clause (elements M)"   and "C' = clause"   and "conflictFlag' = True" shows "InvariantCFalse conflictFlag' M C'"
exp_le_cancel_iff: fixes x :: "real"   and y :: "real" shows "(exp x \<le> exp y) = (x \<le> y)"
hnorm_minus_commute: fixes a :: "'c star"   and b :: "'c star" shows "hnorm (a - b) = hnorm (b - a)"
subst_compose_def: fixes \<sigma> :: "'e \<Rightarrow> ('f, 'h) Term.term"   and \<tau> :: "'h \<Rightarrow> ('f, 'g) Term.term" shows "\<sigma> \<circ>\<^sub>s \<tau> = (\<lambda>x. \<sigma> x \<cdot> \<tau>)"
mHom_lin: fixes N :: "('i, 'b, 'j) Module_scheme"   and m :: "'a"   and f :: "'a \<Rightarrow> 'i"   and a :: "'b" assumes "R module N"   and "m \<in> carrier M"   and "f \<in> mHom R M N"   and "a \<in> carrier R" shows "f (a \<cdot>\<^sub>s m) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> f m"
trapped_def: fixes x :: "'a"   and K :: "'a set" shows "trapped x K = (trapped_forward x K \<and> trapped_backward x K)"
iso_hcomp: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "D.iso \<mu>"   and "D.iso \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "D.iso (\<nu> \<star>\<^sub>D \<mu>)"
match_4: fixes g :: "'a"   and f :: "'a"   and h :: "'a" assumes "C.seq g f"   and "C.seq h g" shows "C.seq h (g \<cdot> f)"
size_tree_def: fixes t :: "'b tree" shows "Root_Balanced_Tree.size_tree t = Time_Monad.val (size_tree_tm t)"
VDMSkip: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" shows "G \<rhd> Skip : \<lambda>s t. t = s"
input2state_nth: fixes i :: "nat"   and ia :: "value list" assumes "i < length ia" shows "input2state ia $ i = Some (ia ! i)"
\<alpha>_ide_simp: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<alpha> f g h = \<a>[f, g, h]"
that: fixes x :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "x \<notin>\<^sub>\<circ> A"   and "x \<in>\<^sub>\<circ> B" shows "thesis"
collapse: fixes option :: "'f option" assumes "option \<noteq> None" shows "Some (the option) = option"
cos_double_sin: fixes w :: "'a" shows "cos ((2::'a) * w) = (1::'a) - (2::'a) * (sin w)\<^sup>2"
assms_3: shows "f |\<in>| (Hom X Y)"
is_trail_intro: fixes v' :: "kon_node"   and ps :: "(kon_node \<times> kon_bridge \<times> kon_node) list"   and v :: "kon_node" assumes "kon_graph.is_trail v' ps v" shows "kon_graph.is_path v' ps v"
2: shows "happ (hlam A B C F) X = (THE G. \<langle>X, G\<rangle> \<^bold>\<in> hlam A B C F)"
dtermsl_dterms: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" assumes "q \<in> dtermsl p"   and "not_call q"   and "wellformed \<Gamma>" shows "q \<in> dterms \<Gamma> p"
expectation_gpv2_def: shows "expectation_gpv2 \<equiv> expectation_gpv fail \<I> (expectation_gpv fail \<I> f \<circ> g)"
le_option_refl: fixes x :: "'a option" shows "le_option x x"
elims: fixes x :: "'a option"   and xa :: "'a option"   and y :: "'a option" assumes "x + xa = y" obtains uu_ :: "'a option" where "x = None" and "xa = uu_" and "y = None"   | v :: "'a" where "x = Some v" and "xa = None" and "y = None"   | a :: "'a" and b :: "'a" where "x = Some a" and "xa = Some b" and "y = Some (a + b)"