hcju/isabelleps huggingface dataset:
comm_mult_strict_left_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_comm_semiring_strict plus minus less_eq less zero times"   and "less a b"   and "less zero c" shows "less (times c a) (times c b)"
group_isomorphisms_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'c" shows "group_isomorphisms G H f g \<equiv> f \<in> hom G H \<and> g \<in> hom H G \<and> (\<forall>x\<in>carrier G. g (f x) = x) \<and> (\<forall>y\<in>carrier H. f (g y) = y)"
model_def: fixes C :: "'c set"   and G :: "('c Standard_Constant, 'c + 'd) labeled_graph"   and T :: "('c Standard_Constant allegorical_term \<times> 'c Standard_Constant allegorical_term) set" shows "model C G T \<equiv> standard' C G \<and> (\<forall>S\<in>T. G \<Turnstile> S)"
sg_i_induced1: fixes H :: "'a set"   and K :: "'a set"   and a :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "a \<in> H \<inter> K" shows "\<rho>\<^bsub>\<natural>H \<inter> K\<^esub> a = \<rho>\<^bsub>\<natural>H\<^esub> a"
Ass: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and va :: "Term.var"   and T :: "ty"   and v :: "expr"   and T' :: "ty" assumes "E,dt\<Turnstile>va\<Colon>=T"   and "va \<noteq> LVar This"   and "E,dt\<Turnstile>v\<Colon>-T'"   and "prg E\<turnstile>T'\<preceq>T" shows "E,dt\<Turnstile>va:=v\<Colon>-T'"
image_mset_empty: fixes f :: "'e \<Rightarrow> 'd" shows "image_mset f {#} = {#}"
right_neutral: fixes a :: "'b" shows "a AND - (1::'b) = a"
phiDefNodes_aux_unvisited_monotonic: fixes un :: "'node list"   and un' :: "'node list"   and g :: "'g"   and v :: "'var"   and n :: "'node" assumes "set un \<subseteq> set un'" shows "phiDefNodes_aux g v un n \<subseteq> phiDefNodes_aux g v un' n"
Sup_Sec_def: fixes S :: "Sec set" shows "Sup S = (if High \<in> S then High else Low)"
absorb2: fixes b :: "'b multiset"   and a :: "'b multiset" assumes "b \<subseteq># a" shows "a \<inter># b = b"
preserves_seq: fixes g :: "'j"   and f :: "'j" assumes "J.seq g f" shows "J.seq (J.map g) (J.map f)"
run_properties_2: fixes \<chi> :: "'a ltl"   and \<phi> :: "'a ltl"   and i :: "nat"   and q :: "'a ltl_prop_equiv_quotient" assumes "\<chi> \<in> \<^bold>G \<phi>" shows "the (snd (run (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) w i) \<chi>) q = semi_mojmir_def.state_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)) w q i"
group_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y * z = x * (y * z)"
k: shows "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
simps_1: fixes V :: "char list"   and Vs :: "char list list"   and T :: "ty"   and Ts :: "ty list"   and v :: "val"   and vs :: "val list"   and e :: "char list exp" shows "blocks (V # Vs, T # Ts, v # vs, e) = {V:T; V:=Val v;; blocks (Vs, Ts, vs, e)}"
simps: fixes f :: "'b \<rightarrow> 'a"   and x :: "'b"   and xs :: "'b Stream" shows "smap\<cdot>f\<cdot>(x && xs) = f\<cdot>x && smap\<cdot>f\<cdot>xs"
hd_rev: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "hd (rev xs) = last xs"
is_suffix_eq_proper_suffix: fixes a :: "'b list"   and b :: "'b list" shows "is_suffix a b = (a = b \<or> is_proper_suffix a b)"
insert_bal: fixes t :: "'a btree"   and k :: "nat"   and x :: "'a" assumes "bal t" shows "bal (abs_split.insert k x t)"
major: shows "\<tau>Exec_movet ci P t e h (stk, loc, pc, xcp) (stk'', loc'', pc'', xcp'')"
LCons_lprefix_LCons: fixes x :: "'c"   and xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "lprefix (LCons x xs) (LCons y ys) = (x = y \<and> lprefix xs ys)"
gt_same: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t = head s"   and "chksubs (>\<^sub>t) t s"   and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)" shows "t >\<^sub>t s"
monadic_FOREACH_def: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and \<Phi> :: "'e set \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'e set"   and c :: "'d \<Rightarrow> bool nres"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd nres"   and \<sigma>0 :: "'d" shows "monadic_FOREACH R \<Phi> S c f \<sigma>0 \<equiv> ASSERT (finite S) \<bind> (\<lambda>_. it_to_sorted_list R S \<bind> (\<lambda>xs0. REC\<^sub>T (\<lambda>W (xs, \<sigma>). ASSERT (FOREACH_inv xs0 \<Phi> (xs, \<sigma>)) \<bind> (\<lambda>_. if xs \<noteq> [] then c \<sigma> \<bind> (\<lambda>b. if b then FOREACH_body f (xs, \<sigma>) \<bind> W else RETURN (xs, \<sigma>)) else RETURN (xs, \<sigma>))) (xs0, \<sigma>0) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)))"
statEqBisim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<sim> Q"   and "\<Psi> \<simeq> \<Psi>'" shows "\<Psi>' \<rhd> P \<sim> Q"
P_def: fixes x :: "int"   and y :: "int" shows "local.P x y = ([x * y = a] (mod int p) \<and> y \<in> local.S1)"
mem_conc_forms: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and a :: "'b" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "(a |\<in>| Abstract_Task.conc_forms conclusions) = (a \<in> set conclusions)"
card_nlists: fixes A :: "'b set"   and n :: "nat" shows "card (nlists A n) = card A ^ n"
set_union: fixes xs :: "'g list"   and ys :: "'g list" shows "set (List.union xs ys) = set xs \<union> set ys"
assms_2: shows "\<forall>i j. P i \<and> Q j \<longrightarrow> R i j"
encL: fixes S :: "'procS" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<sim>\<lbrakk>\<cdot>\<rbrakk> SourceTerm S"
abs_minus_cancel: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (uminus a) = abs a"
shEx_cong: fixes P :: "'a \<Rightarrow> 'b upred"   and Q :: "'a \<Rightarrow> 'b upred" assumes "\<And>x. P x = Q x" shows "ushEx P = ushEx Q"
app_insf_Seq_if: fixes s :: "hf"   and k :: "hf"   and y :: "hf"   and k' :: "hf" assumes "Seq s k" shows "app (insf s k y) k' = (if k' = k then y else app s k')"
innermost_intvl: fixes f :: "atom fm"   and xs :: "'a list"   and l :: "'a"   and x :: "'a"   and u :: "'a"   and y :: "'a" assumes "nqfree f"   and "nolub f xs l x u"   and "l < x"   and "x < u"   and "x \<notin> EQ f xs"   and "DLO.I f (x # xs)"   and "l < y"   and "y < u" shows "DLO.I f (y # xs)"
algebraic_mult_cancel_left: fixes x :: "'a"   and y :: "'a" assumes "algebraic x"   and "x \<noteq> (0::'a)" shows "algebraic (x * y) = algebraic y"
noDA1eq: fixes p :: "('b, 'c) Combinators list" assumes "noDenyAll p" shows "noDenyAll1 p"
measurement_le_one_mat: fixes d :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and j :: "nat" assumes "measurement d n f"   and "j < n" shows "adjoint (f j) * f j \<le>\<^sub>L 1\<^sub>m d"
arctan_upper_11_def: shows "arctan_upper_11 \<equiv> \<lambda>x. - (pi / 2) - 1 / x"
p_dvd_q: shows "CARD('p) dvd CARD('q)"
sg1: shows "cycleLength (nC i) = cycleLength (nC j)"
forget_subst_fm: fixes a :: "name"   and A :: "fm"   and x :: "tm" assumes "atom a \<sharp> A" shows "A(a::=x) = A"
upper_iso: shows "isotone \<Y>\<^bsub>F\<^esub> \<X>\<^bsub>F\<^esub> \<pi>\<^sub>*\<^bsub>F\<^esub>"
Ugp_const_gbij: fixes G :: "('c, 'd) Group_scheme"   and E :: "('e, 'f) Group_scheme" assumes "Ugp G"   and "Ugp E" shows "gbij\<^bsub>G,E\<^esub> (\<lambda>x\<in>carrier G. \<one>\<^bsub>E\<^esub>)"
hurwitz_zeta_neg_of_nat: assumes "0 < a" shows "hurwitz_zeta a (- of_nat 0) = complex_of_real (- bernpoly (Suc 0) a / real (Suc 0))"
list_of_oalist_update_by_fun: fixes k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and xs :: "'x" shows "list_of_oalist (update_by_fun k f xs) = update_by_fun_raw k f (list_of_oalist xs)"
set_insert: fixes x :: "'a"   and xs :: "'a list" shows "set (List.insert x xs) = insert x (set xs)"
comp_assoc: fixes h :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "H.VV.comp (H.VV.comp h g) f = H.VV.comp h (H.VV.comp g f)"
wm_to_state_works_for_I_closed: fixes x :: "object atom formula" assumes "x \<in> close_world I" shows "close_eq (wm_to_state I) \<Turnstile> x"
eq_omega_sum_eq_iff: fixes ms :: "nat list"   and ns :: "nat list" assumes "length ms = length ns" shows "(omega_sum ms = omega_sum ns) = (ms = ns)"
eval_red2_naturality: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "\<lbrace>Cod t \<^bold>\<Down> Cod u\<rbrace> \<cdot> (\<lbrace>t\<rbrace> \<star> \<lbrace>u\<rbrace>) = \<lbrace>t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u\<rbrace> \<cdot> \<lbrace>Dom t \<^bold>\<Down> Dom u\<rbrace>"
converse_iff: fixes a :: "'a"   and b :: "'c"   and r :: "('c \<times> 'a) set" shows "((a, b) \<in> r\<inverse>) = ((b, a) \<in> r)"
fls_X_inv_times_conv_shift_1: fixes f :: "'a fls" shows "fls_X_inv * f = fls_shift 1 f"
Next'_preserves_invs_2: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration" assumes "cri.InvMsgInGlob (exchange_config c0)"   and "inv_init_imp_prop_safe c0"   and "InvGlobPointstampsEq c0"   and "Next' c0 c1"   and "\<forall>p. prop_invs (prop_config c0 p)" shows "\<forall>p. prop_invs (prop_config c1 p)"
fc_implies_wfc: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and I :: "('b \<times> 'b) set" assumes "future_consistent P D R" shows "weakly_future_consistent P I D R"
is_class_is_subcls: fixes m :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal" assumes "wf_prog m P" shows "is_class P C = P \<turnstile> C \<preceq>\<^sup>* Object"
right_options_def: fixes g :: "game" shows "right_options g \<equiv> zimage Abs_game (zexplode (Snd (Rep_game g)))"
src_cod: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src (B.cod \<mu>) = src \<mu>"
path_start_le_path_end: shows "pathstart \<gamma> \<bullet> i \<le> pathfinish \<gamma> \<bullet> i"
threeway_partition_def: fixes x :: "'b"   and R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "threeway_partition x R xs = (filter (\<lambda>y. (y, x) \<in> R \<and> (x, y) \<notin> R) xs, filter (\<lambda>y. (x, y) \<in> R \<and> (y, x) \<in> R) xs, filter (\<lambda>y. (x, y) \<in> R \<and> (y, x) \<notin> R) xs)"
"2_2": shows "e \<in> UNIV \<rightarrow> I"
cindex_poly_ubd_code: fixes q :: "real poly"   and p :: "real poly" shows "cindex_poly_ubd q p = changes_R_smods p q"
share_all_until_volatile_write_share_commute: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and sb' :: "'b memref list" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "\<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> (all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<union> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)) \<inter> (all_shared sb' \<union> all_unshared sb' \<union> all_acquired sb') = {}" shows "share_all_until_volatile_write ts (share sb' \<S>) = share sb' (share_all_until_volatile_write ts \<S>)"
safe_lpd_fv_regex: fixes fv :: "'b \<Rightarrow> 'c set"   and safe :: "safety \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b regex"   and s :: "'b regex"   and test :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'d" assumes "Regex.safe_regex fv safe Futu Strict r"   and "s \<in> lpd test i r" shows "fv_regex fv s = fv_regex fv r"
foldli_set: fixes list :: "'b list"   and xs :: "'b list" shows "set (foldli list (\<lambda>_. True) (#) xs) = set xs \<union> set list"
chain_on_path: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" assumes "ch_by_ord f X" shows "\<exists>P\<in>\<P>. X \<subseteq> P"
Gram_Schmidt_matrix_iarrays_efficient_def: fixes A :: "real iarray iarray" shows "Gram_Schmidt_matrix_iarrays_efficient A = foldl Gram_Schmidt_column_k_iarrays_efficient A [0..<ncols_iarray A]"
bv_to_nat_nzero_imp_nempty: fixes w :: "bit list" assumes "bv_to_nat w \<noteq> 0" shows "norm_unsigned w \<noteq> []"
GF_advice_subfrmlsn: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "subfrmlsn (\<phi>[X]\<^sub>\<nu>) \<subseteq> {\<psi>[X]\<^sub>\<nu> |\<psi>. \<psi> \<in> subfrmlsn \<phi>}"
plossless_gpv_lossless_spmfD: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "lossless_spmf (the_gpv gpv)"
axioms_1: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "abelian_subgroup H G" shows "additive_subgroup H G"
n_mult_degree_bound: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "deg R (n_mult f) \<le> deg R f"
bal_tree_def: fixes n :: "nat"   and t :: "'b tree" shows "bal_tree n t = bal_list n (inorder t)"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<preceq> a"   and "c \<preceq> b" shows "c \<preceq> a"
assms_1: shows "run_one_step d i (s, vs, ves, $b_e) = (s', vs', RSNormal es')"
cfs_add_nat_pow: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and n :: "nat"   and k :: "nat" assumes "UP_cring R"   and "p \<in> carrier (UP R)" shows "([n] \<cdot>\<^bsub>UP R\<^esub> p) k = [n] \<cdot>\<^bsub>R\<^esub> p k"
dset_def: fixes p :: "perm"   and q :: "perm" shows "dset p q = {a. p \<bullet> a \<noteq> q \<bullet> a}"
ins_rbt_sorted: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'b"   and x :: "'c" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "ord.rbt_sorted less (ord.rbt_ins less f k x t)"
simps_3: fixes i :: "int"   and c :: "int"   and e :: "MIR.num" shows "d_\<delta> (Dvd i (CN 0 c e)) = (dvd) i"
kat_loop_inv_def: fixes x :: "'a"   and i :: "'a" shows "loop x inv i  = x\<^sup>\<star>"
rrr: shows "round_robin_reschedule t0 queue n0 s = None"
entt_disjI1_direct: fixes A :: "assn"   and B :: "assn" shows "A \<Longrightarrow>\<^sub>t A \<or>\<^sub>A B"
rep_eq_iff: fixes x :: "'a"   and y :: "'a" shows "(rep_nat x = rep_nat y) = (x = y)"
fresh_4: fixes a :: "atom"   and var :: "Vars.var"   and exp :: "exp" shows "a \<sharp> Lam [var]. exp = (a \<sharp> exp \<or> a \<in> set [atom var])"
echelon_form_of_column_k_iarrays_def: fixes A' :: "'c iarray iarray \<times> nat \<times> ('c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c)"   and k :: "nat" shows "echelon_form_of_column_k_iarrays A' k = (let (A, i, bezout) = A'; nrows_A = nrows_iarray A; column_Ak = column_iarray k A; all_zero_below_i = vector_all_zero_from_index (i + 1, column_Ak) in if i = nrows_A \<or> A !! i !! k = (0::'c) \<and> all_zero_below_i then (A, i, bezout) else if all_zero_below_i then (A, i + 1, bezout) else let n = least_non_zero_position_of_vector_from_index column_Ak i; interchange_A = interchange_rows_iarray A i n in (bezout_iterate_iarrays interchange_A (nrows_A - 1) i k bezout, i + 1, bezout))"
assms: shows "summable (\<lambda>n. f n * z ^ n)"
new_document_ptr: shows "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"
not_less_real: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero \<or> a = zero \<or> less zero a"   and "less b zero \<or> b = zero \<or> less zero b" shows "(\<not> less b a) = less_eq a b"
strict_trans1: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "prefix b a"   and "strict_prefix c b" shows "strict_prefix c a"
le_supI2: fixes x :: "'b"   and b :: "'b"   and a :: "'b" assumes "x \<le> b" shows "x \<le> a \<squnion> b"
M_Rel_def: fixes m :: "int"   and x :: "int"   and x' :: "'a mod_ring" assumes "poly_mod_type TYPE('a) m" shows "poly_mod_type.M_Rel m x x' \<equiv> poly_mod.M m x = to_int_mod_ring x'"
restrict_count_space_subset: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "restrict_space (count_space B) A = count_space A"
compact_uniformly_continuous: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "compact S" shows "uniformly_continuous_on S f"
simps: fixes V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set" shows "New_max.filterQuery V Q = Set.filter (\<lambda>(s, uu_). \<not> Set.is_empty (s \<inter> V)) Q"
complete_real: fixes S :: "real set" assumes "\<exists>x. x \<in> S"   and "\<exists>z. \<forall>x\<in>S. x \<le> z" shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
regular_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "regular x"   and "regular y" shows "regular (x \<sqinter> y)"
opt_correct_aux: fixes e :: "exp"   and k :: "nat" shows "E e = E (opt e k)"
map_of_zip_upto2_length_eq_nth: fixes B :: "'d list"   and i :: "nat" assumes "distinct B"   and "i < length B" shows "map_of (zip B [0..<length B]) (B ! i) = Some i"
ta_satisfiable: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "s \<in> wf_state"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" shows "\<exists>s'. actions_ok s' t ta"
that: fixes f :: "nat set cset \<Rightarrow> nat set" assumes "bij f" shows "thesis"
simps_1: fixes \<delta> :: "'c"   and uu :: "'c arctic_delta" shows "gt_arctic_delta \<delta> uu MinInfty_delta = True"
REP_in_hhom_2: fixes F :: "('a, 'a set) arr"   and A :: "('a, 'a set) arr"   and B :: "('a, 'a set) arr" assumes "\<guillemotleft>F : A \<rightarrow> B\<guillemotright>" shows "B.is_left_adjoint (REP F)"
distinct_45: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "Stmt.If x51 x52 x53 \<noteq> While x61 x62"
Or_append_syntactic: fixes xs :: "'c ltl list"   and ys :: "'c ltl list" assumes "xs \<noteq> []" shows "Or (xs @ ys) = Or (Or xs # ys)"
simps_8: fixes x2 :: "'d" shows "\<infinity> \<noteq> Lt x2"
test1: fixes x :: "'a"   and X :: "'a set"   and Y :: "'a set" shows "(x \<in> X \<union> Y) = (x \<in> X \<or> x \<in> Y)"
simps_4: fixes c :: "int"   and e :: "MIR.num" shows "plusinf (NEq (CN 0 c e)) = T"
append_def: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "Execute.append xs ys zs = (zs = xs @ ys)"
bottom_def: fixes P :: "'a set" shows "bottom P \<equiv> THE z. z \<in> P \<and> (\<forall>x\<in>P. x \<succeq> z)"
B_def: shows "B \<equiv> of_int_hom.mat_hom (mat_of_rows n fs_init)"
invert_opposite_sides_of_square: fixes f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "D.seq f g"   and "f \<cdot>\<^sub>D g = h \<cdot>\<^sub>D k"   and "D.iso f"   and "D.iso k" shows "D.seq g (D.inv k) \<and> D.seq (D.inv f) h \<and> g \<cdot>\<^sub>D D.inv k = D.inv f \<cdot>\<^sub>D h"
NextRecvCapD_3: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('d, 'c, 'a) Combined.configuration"   and c1 :: "('d, 'c, 'a) Combined.configuration"   and p :: "'d"   and t :: "'a \<times> 'c" assumes "dataflow_topology summary results_in"   and "dataflow_topology.NextRecvCap' c0 c1 p t" shows "init c1 = init c0"
If_transC_invert: fixes tst :: "'test"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(if tst then c1 else c2, s) \<rightarrow>c (c', s')" obtains "tval tst s" and "c' = c1" and "s' = s"   | "\<not> tval tst s" and "c' = c2" and "s' = s"
not_sum_power2_lt_zero: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less (plus (power.power one times x 2) (power.power one times y 2)) zero"
urel_integer_inj': fixes x :: "integer"   and z :: "int"   and y :: "integer" assumes "urel_integer x z"   and "urel_integer y z" shows "x = y"
rbt_del_rbt_greater: fixes v :: "'a"   and lt :: "('a, 'c) rbt"   and x :: "'a" assumes "v \<guillemotleft>| lt" shows "v \<guillemotleft>| rbt_del x lt"
ev_order: fixes x :: "complex" assumes "cmod x = 1" shows "order x (char_poly A) \<le> d"
units_of_inv: fixes x :: "nat" assumes "x \<in> Units (add_monoid R)" shows "inv\<^bsub>units_of (add_monoid R)\<^esub> x = inv\<^bsub>add_monoid R\<^esub> x"
sylow_def: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" shows "sylow G p a m \<equiv> Group.group G \<and> sylow_axioms G p a m"
intro: fixes G :: "'c graph" assumes "fst ` edges G \<subseteq> nodes G"   and "snd ` edges G \<subseteq> nodes G"   and "finite (edges G)"   and "finite (nodes G)" shows "wf_graph G"
non_trivial_combine_imp_dependent: fixes Ks :: "'a list"   and Us :: "'a list" assumes "set Ks \<subseteq> K"   and "combine Ks Us = \<zero>"   and "\<not> set (take (length Us) Ks) \<subseteq> {\<zero>}" shows "local.dependent K Us"
split_rflow_incoming: fixes u :: "nat" shows "(\<Sum>v\<in>cf.E\<inverse> `` {u}. f' (v, u)) = (\<Sum>v\<in>E `` {u}. f' (v, u)) + (\<Sum>v\<in>E\<inverse> `` {u}. f' (v, u))"
iTL_defs_6: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iWeakSince I P Q \<equiv> (\<box> t I. P t) \<or> (\<diamond> t I. Q t \<and> (\<box> t' I \<down>> t. P t'))"
simps_11: fixes n :: "nat"   and p :: "fm"   and q :: "fm" shows "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
h'_def: fixes x :: "'a" shows "h' x = (let (y, a) = SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi)"
contour_integrable_joinD1: fixes f :: "complex \<Rightarrow> complex"   and g1 :: "real \<Rightarrow> complex"   and g2 :: "real \<Rightarrow> complex" assumes "f contour_integrable_on g1 +++ g2"   and "valid_path g1" shows "f contour_integrable_on g1"