hcju/isabelleps huggingface dataset:
up_to_mono: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and J :: "nat set"   and H :: "nat set" assumes "Ionescu_Tulcea P M"   and "J \<subseteq> H"   and "finite H" shows "Ionescu_Tulcea.up_to J \<le> Ionescu_Tulcea.up_to H"
iprev_min_step2: fixes I :: "nat set"   and n :: "nat" shows "\<not> (\<exists>x\<in>I. iprev n I < x \<and> x < n)"
Int_Collect: fixes x :: "'e"   and A :: "'e set"   and P :: "'e \<Rightarrow> bool" shows "(x \<in> A \<inter> {x. P x}) = (x \<in> A \<and> P x)"
disjoint_zero_sym: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.sep_algebra plus zero sep_disj" shows "sep_disj zero x"
u\<^sub>h_def: shows "u\<^sub>h = tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0"
lossless_weight_spmfD: fixes p :: "'b spmf" assumes "lossless_spmf p" shows "weight_spmf p = 1"
domsMT_notND_DD2: fixes a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set" assumes "dom (C (DenyAllFromTo a b)) \<inter> dom (C (DenyAllFromTo c d)) \<noteq> {}" shows "\<not> netsDistinct b d"
subword_length: fixes i :: "nat"   and j :: "nat"   and w :: "'c list" assumes "i + j \<le> length w" shows "length (subword w i j) = j"
perpendicular_3: shows "perpendicular (poincare_line u v) H"
exp_def: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "hfsetcat.exp b c = hf_to_ide (hexp (ide_to_hf b) (ide_to_hf c))"
infinite: fixes A :: "'b sat set" assumes "infinite A" shows "Inf A = top"
l_new_character_data_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_character_data_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<equiv> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
continuation_in_generat_conts: fixes r :: "('d, 'e, 'f) generat" assumes "\<not> is_Pure r" shows "continuation r \<in> generat_conts r"
C_star: shows "C_star = (\<lambda>l. {ps. length ps = l + 1 \<and> adj_path (hd ps) (tl ps) \<and> last ps = hd ps})"
not_eq_order_implies_strict: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<noteq> b"   and "a \<subseteq># b" shows "a \<subset># b"
max_reg_def: fixes g :: "vname aexp" shows "max_reg g = (let regs = enumerate_regs g in if regs = {} then None else Some (Max regs))"
rel_fundefs_def: fixes f :: "'a \<Rightarrow> ('dyn, 'b, 'c, 'd, 'opinl) Inca.instr fundef option"   and g :: "'a \<Rightarrow> ('dyn, 'b, 'c, 'd, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef option" shows "rel_fundefs f g = (\<forall>x. rel_option (rel_fundef (\<lambda>y z. y = norm_instr z)) (f x) (g x))"
intro: fixes f :: "'c \<Rightarrow> 'd" assumes "clinear f"   and "bounded_clinear_axioms f" shows "bounded_clinear f"
simps_1: fixes uu :: "('c \<Rightarrow> real) stream" shows "dur uu 0 = 0"
map_comp0: fixes g1 :: "'n \<Rightarrow> 'k"   and f1 :: "'h \<Rightarrow> 'n"   and g2 :: "'o \<Rightarrow> 'l"   and f2 :: "'i \<Rightarrow> 'o"   and g3 :: "'p \<Rightarrow> 'm"   and f3 :: "'j \<Rightarrow> 'p" shows "Fmap (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) = Fmap g1 g2 g3 \<circ> Fmap f1 f2 f3"
dualization_unl_1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'b" shows "(\<not> (\<box>F) w) = (\<diamond>\<not> F) w"
simps_4: fixes V :: "Vars.var set"   and x :: "Vars.var"   and S :: "stack_elem list" shows "restr_stack V (Upd x # S) = (if x \<in> V then Upd x # restr_stack V S else restr_stack V S)"
whisker_right: fixes f :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.ide f"   and "S.seq \<nu> \<mu>" shows "\<nu> \<cdot>\<^sub>S \<mu> \<star>\<^sub>S f = (\<nu> \<star>\<^sub>S f) \<cdot>\<^sub>S (\<mu> \<star>\<^sub>S f)"
i: shows "\<nexists>L. L \<in> C \<and> complement L \<in> PP1"
lm118: fixes l :: "'a list"   and X :: "'a set" shows "(!) l ` set (concat (map (\<lambda>n. if l ! n \<in> X then [n] else []) [0..<length l])) \<subseteq> X \<inter> set l"
Ex1_rev_wrap_def: fixes P :: "'b list \<Rightarrow> bool" shows "Ex1_rev_wrap P = (\<exists>!x. P (rev x))"
simps_3: fixes \<Gamma> :: "char list set"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" shows "wfe \<Gamma> (Add e\<^sub>1 e\<^sub>2) = (wfe \<Gamma> e\<^sub>1 \<and> wfe \<Gamma> e\<^sub>2)"
NotNotI: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<phi> in v]" shows "[\<^bold>\<not>\<^bold>\<not>\<phi> in v]"
distinct_5: fixes x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list" shows "Z \<noteq> Cn x41 x42 x43"
range_of'_def: fixes \<Psi> :: "('b, 'a) sas_plus_problem"   and v :: "'b" shows "\<R>\<^sub>+ \<Psi> v \<equiv> case range_of \<Psi> v of None \<Rightarrow> {} | Some as \<Rightarrow> set as"
scaleR: fixes r :: "real"   and b :: "'a" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
inj_zero_hom_axioms: shows "inj_zero_hom to_int_mod_ring"
assms: shows "r1\<^sup>*\<^sup>* = r2\<^sup>*\<^sup>*"
proj_psi_minus_1_mult_psi_Suc_l: fixes l :: "nat" shows "(2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) *\<^sub>v psi_l (l + 1) = psi'_l l"
cases: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b list"   and a2 :: "'b list" assumes "lexext gt a1 a2" obtains y :: "'b" and ys :: "'b list" where "a1 = y__ # ys__" and "a2 = []"   | y :: "'b" and x :: "'b" and ys :: "'b list" and xs :: "'b list" where "a1 = y # ysa__" and "a2 = x__ # xs__" and "gt y x__"   | ys :: "'b list" and xs :: "'b list" and x :: "'b" where "a1 = x # ys" and "a2 = x # xs" and "lexext gt ys xs"
r_take_def: shows "r_take \<equiv> Cn 2 r_ifle [recf.Id 2 0, Cn 2 r_length [recf.Id 2 1], Pr 1 Z (Cn 3 r_snoc [recf.Id 3 1, Cn 3 r_nth [recf.Id 3 2, recf.Id 3 0]]), recf.Id 2 1]"
emptyValuationFalsifiesOnlyEmptyClause: fixes clause :: "Literal list" assumes "clause \<noteq> []" shows "\<not> clauseFalse clause []"
A: shows "x \<in> r \<down>s (r \<down>s S)"
assms_2: shows "(\<exists>j'. token_run x n \<noteq> token_run y n \<and> y \<le> n \<and> state_rank (token_run y n) n = Some j') \<or> y = Suc n"
Controller_def: fixes y :: "nat \<Rightarrow> nat list"   and z :: "nat \<Rightarrow> bit list" shows "Controller y z \<equiv> ts y \<longrightarrow> (\<exists>l. Controller_L y (fin_inf_append [Zero] l) l z)"
toplevel_summands_ACI_nPlus: fixes r :: "'b rexp"   and s :: "'b rexp" shows "toplevel_summands (ACI_nPlus r s) = toplevel_summands (Plus r s)"
list_all2_Nil: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "list_all2 P [] ys = (ys = [])"
set_8: fixes x1 :: "('e, 'd) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>x1\<rangle>) = fv x1"
nth_butlast: fixes n :: "nat"   and xs :: "'a list" assumes "n < |butlast xs|" shows "butlast xs ! n = xs ! n"
order_less_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x < y"   and "y < z" shows "x < z"
Pi_I: fixes A :: "'g set"   and f :: "'g \<Rightarrow> 'h"   and B :: "'g \<Rightarrow> 'h set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
keys_plus_eqI: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b" assumes "keys p \<inter> keys q = {}" shows "keys (p + q) = keys p \<union> keys q"
that: fixes X1 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "X1 \<subseteq> X0"   and "flowsto X1 T CX Y"   and "flowsto (X0 - X1) T CX Z" shows "thesis"
sngr_prec: shows "precise (\<lambda>x p. p \<mapsto>\<^sub>r x)"
countable_PiE: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (F i)" shows "countable (Pi\<^sub>E I F)"
measurable_equals: fixes t :: "pdf_type" shows "Measurable.pred (stock_measure t \<Otimes>\<^sub>M stock_measure t) (\<lambda>(x, y). x = y)"
obj_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>C a = a"
path_conv_loc_imps_fw: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and loc1 :: "'a"   and loc2 :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list"   and c :: "('a, 'c) configuration" assumes "dataflow_topology summary results_in"   and "graph.path summary loc1 loc2 xs"   and "distinct xs" shows "\<exists>M. dataflow_topology.loc_imps_fw summary results_in c loc1 loc2 M xs"
ext_surjective: fixes x :: "('b, 'c) sparc_state_scheme" shows "x = \<lparr>cpu_reg = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id)))) x, user_reg = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id)))) x, sys_reg = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id)))) x, mem = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id))))) x, mmu = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id))))) x, cache = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id)))) x, dwrite = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id)))) x, state_var = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id)))) x, traps = (Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id))))) x, undef = (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> (Record.iso_tuple_fst sparc_state_ext_Tuple_Iso \<circ> id))))) x, \<dots> = (Record.iso_tuple_snd sparc_state_ext_Tuple_Iso \<circ> id) x\<rparr>"
is_idirprod_subgroup_suffices: fixes A :: "'a set"   and S :: "'c \<Rightarrow> 'a set"   and I :: "'c set" assumes "A = IDirProds G S I"   and "\<forall>i\<in>I. subgroup (S i) G"   and "compl_fam S I" shows "is_idirprod A S I"
assms_4: shows "nprv (insert \<phi>3 F) \<psi>"
nn_integral_0_iff_AE: fixes u :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (AE x in M. u x = 0)"
v_equiv_def: fixes K :: "('c, 'd) Ring_scheme"   and v1 :: "'c \<Rightarrow> ant"   and v2 :: "'c \<Rightarrow> ant" shows "v_equiv K v1 v2 = (n_val K v1 = n_val K v2)"
insert_null_sets_lebesgue_on_iff: fixes a :: "'a"   and S :: "'a set"   and N :: "'a set" assumes "a \<in> S"   and "S \<in> sets lebesgue" shows "(insert a N \<in> null_sets (lebesgue_on S)) = (N \<in> null_sets (lebesgue_on S))"
steps: fixes i :: "nat" shows "(fst (f i), fst (f (Suc i))) \<in> s \<or> (fst (f i), fst (f (Suc i))) \<in> ns \<and> (snd (f i), snd (f (Suc i))) \<in> s2"
row_eq_zero_iff_pivot_fun: fixes A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "pivot_fun A f (dim_col A)"   and "i < dim_row A" shows "(row A i = 0\<^sub>v (dim_col A)) = (f i = dim_col A)"
num_ni_leN: fixes ts :: "nat \<Rightarrow> thread" shows "num_ni ts \<le> N"
id: shows "of_int = of_real \<circ> real_of_int"
ne_SUP_commute: fixes U\<^sub>2 :: "'c set"   and U\<^sub>3 :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'al"   and B :: "'d set"   and A :: "'c set" assumes "U\<^sub>2 \<noteq> {}"   and "U\<^sub>3 \<noteq> {}"   and "\<forall>x\<in>U\<^sub>2. \<forall>y\<in>U\<^sub>3. f x y \<in> U"   and "B \<subseteq> U\<^sub>3"   and "A \<subseteq> U\<^sub>2" shows "\<Squnion>\<^sub>o\<^sub>w ((\<lambda>i. \<Squnion>\<^sub>o\<^sub>w (f i ` B)) ` A) = \<Squnion>\<^sub>o\<^sub>w ((\<lambda>j. \<Squnion>\<^sub>o\<^sub>w ((\<lambda>i. f i j) ` A)) ` B)"
loxodromic_isometryI: fixes f :: "'b \<Rightarrow> 'b" assumes "isometry f"   and "stable_translation_length f \<noteq> 0" shows "loxodromic_isometry f"
map_spmf_conv_ap: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'e spmf" shows "map_spmf f p = Applicative.ap (return_spmf f) p"
sigma_algebra_pullback: fixes \<Omega>' :: "'b set"   and M :: "'c measure"   and f :: "'b \<Rightarrow> 'c" shows "sigma_algebra \<Omega>' (sets.pullback_algebra M f \<Omega>')"
projecting_weaken: fixes C :: "'d program \<Rightarrow> 'd set"   and h :: "'e \<times> 'f \<Rightarrow> 'd"   and F :: "'e program"   and X' :: "'d program set"   and X :: "'e program set"   and U' :: "'d program set"   and U :: "'e program set" assumes "projecting C h F X' X"   and "U' \<subseteq> X'"   and "X \<subseteq> U" shows "projecting C h F U' U"
Max_singleton: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.linorder less_eq less" shows "linorder.Max less_eq {x} = x"
point_abscissa_diff: fixes A :: "point"   and B :: "point" shows "abscissa (A - B) = abscissa A - abscissa B"
FL_valid_def: shows "(\<Turnstile>) \<equiv> (id ---> Rep_formula ---> id) FL_valid_Tree\<^sub>\<alpha>"
get_ancestors_si_impl: shows "get_ancestors_si = local.a_get_ancestors_si"
standard_empty_tiebreaker_equiv: fixes Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Inf_G :: "'b inference set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and Bot_F :: "'a set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option" shows "standard_lifting Inf_F Bot_G Inf_G entails_G Red_I_G Red_F_G Bot_F \<G>_F \<G>_I = tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I (\<lambda>g C C'. False)"
list_emb_Nil2: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list" assumes "list_emb P xs []" shows "xs = []"
simps_2: fixes s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" shows "sat_precond_as s (a # as) = (fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as)"
le_linear: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<preceq> b \<or> b \<preceq> a"
4: shows "subst_domain \<delta> \<inter> range_vars \<delta> = {}"
sup_top_right: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and top :: "'d"   and x :: "'d" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup x top = top"
set_rw: shows "{column i (fst (QR_decomposition A)) |i. i < k} = f ` {column i (Gram_Schmidt_matrix A) |i. i < k}"
i_join_i_shrink_iT_Plus_iT_Div_mod: fixes k :: "nat"   and I :: "nat set"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k"   and "I \<noteq> {}"   and "\<forall>x\<in>I. x mod k = 0" shows "f \<longmapsto> k \<Join>\<^sub> (I \<oplus> k - 1) = f \<div> k \<Join>\<^sub> (I \<oslash> k)"
g_non_zero_ind_mod: fixes k :: "nat" assumes "gauss_sum k \<noteq> 0"   and "\<not> coprime k n"   and "0 < k" shows "induced_modulus (n div gcd k n)"
col_ABC_ABD_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" assumes "A \<noteq> B"   and "col A B C"   and "col A B D" shows "col B C D"
refl_symm_trans_closure_is_symm_refl_trans_closure: fixes Rel :: "('c \<times> 'c) set" shows "(symcl (Rel\<^sup>=))\<^sup>+ = (symcl Rel)\<^sup>*"
beta_l_real: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "grover_state.beta_l n f l \<in> \<real>"
removeAll_filter_not_eq: fixes x :: "'b" shows "removeAll x = filter ((\<noteq>) x)"
IdNatTrans_def: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor" shows "IdNatTrans F \<equiv> MakeNT (IdNatTrans' F)"
Image_Domain: fixes R :: "('d \<times> 'e) set"   and A :: "'d set" assumes "Domain R = A" shows "R `` A = Range R"
CYK_def: fixes G :: "('c \<times> ('c, 'd) RHS) list"   and w :: "'d list"   and i :: "nat"   and j :: "nat" shows "CYK G w i j = {S. subword w i j \<in> Lang G S}"
convex_hull_insert: fixes S :: "'a set"   and a :: "'a" assumes "S \<noteq> {}" shows "convex hull insert a S = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. u + v = 1 \<and> b \<in> convex hull S \<and> x = u *\<^sub>R a + v *\<^sub>R b}"
Init: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and s0 :: "st"   and s3 :: "state"   and s1 :: "state"   and s2 :: "state" assumes "the (class G C) = c"   and "if inited C (globs s0) then s3 = Norm s0 else G\<turnstile>Norm (init_class_obj G C s0) \<midarrow>(if C = Object then Skip else Init (super c))\<rightarrow> s1 \<and> G\<turnstile>set_lvars Map.empty s1 \<midarrow>init c\<rightarrow> s2 \<and> s3 = set_lvars (locals (snd s1)) s2" shows "G\<turnstile>Norm s0 \<midarrow>Init C\<rightarrow> s3"
R_def: shows "R \<equiv> \<lambda>t. \<exists>u\<in>C. is_wt_instance_of_cond \<Gamma> t u"
a_very_costrict': fixes x :: "'a" shows "(ad x = (1::'a)) = (x = (zero_class.zero::'a))"
check_sat_terminates: fixes s :: "('i, 'a) state" assumes "\<nabla> s"   and "\<triangle> (\<T> s)"   and "\<diamond> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s" shows "check_dom s"
CollectionSemantics_axioms: shows "CollectionSemantics small endset combine collect_id"
point_simps_2: fixes n :: "nat" shows "right (point n) = enat n"
intros: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and conv1 :: "('a, 'c, 'd, 'e) converter"   and conv2 :: "('b, 'f, 'g, 'h) converter" assumes "rel_fun A (rel_gpv'' (rel_prod B (rel_converter A B C R)) C R) (run_converter conv1) (run_converter conv2)" shows "rel_converter A B C R conv1 conv2"
monom_add: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and b :: "'b" shows "MPoly_Type.monom m (a + b) = MPoly_Type.monom m a + MPoly_Type.monom m b"
topological_basis_open: fixes B :: "'a set set"   and X :: "'a set" assumes "topological_basis B"   and "X \<in> B" shows "open X"
Healthy_def: fixes P :: "'b upred"   and H :: "'b upred \<Rightarrow> 'b upred" shows "P is H \<equiv> H P = P"
return: shows "spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S1 m1 o1 \<bind> D)) True = spmf (funct m1 m2 \<bind> (\<lambda>(o1, o2). S1 m1 o1 \<bind> (\<lambda>view. D view \<bind> return_spmf))) True"
Int_insert_left_if1: fixes a :: "'e"   and C :: "'e set"   and B :: "'e set" assumes "a \<in> C" shows "insert a B \<inter> C = insert a (B \<inter> C)"
dirichlet_prod_altdef2: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "dirichlet_prod f g = (\<lambda>n. \<Sum>(r, d)\<in>{(r, d). r * d = n}. f r * g d)"
atom_sup: fixes x :: "'a" assumes "\<forall>y. x \<le> y \<or> x \<le> - y" shows "\<forall>y z. (x \<le> y \<or> x \<le> z) = (x \<le> y \<squnion> z)"
card_UNIV_int_def: shows "card_UNIV_class.card_UNIV = Phantom(int) 0"
acc_iff: fixes x :: "'b"   and y :: "'b" shows "((x, y) \<in> K2.acc) = (\<exists>n. 0 < K2.p x y n)"
S0: fixes x :: "'a" assumes "x \<in> U" shows "(setdist {x} S = 0) = (x \<in> S)"
intro: fixes Inf :: "'b Calculus.inference set"   and Bot :: "'b set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and I_of :: "'b set \<Rightarrow> 'b set" assumes "calculus_with_standard_redundancy Inf Bot entails"   and "Standard_Redundancy_Criterion.counterex_reducing_inference_system Bot entails Inf I_of" shows "counterex_reducing_calculus_with_standard_redundancy Bot entails I_of Inf"
simps_4: fixes ux :: "'a \<Rightarrow> 'b tm list \<Rightarrow> bool"   and H :: "('b, 'a) fm \<Rightarrow> bool"   and p :: "('b, 'a) fm" shows "boolean ux H (\<^bold>\<forall> p) = H (\<^bold>\<forall> p)"
ptr_kinds_ptr_kinds_M: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "(ptr \<in> set |h \<turnstile> object_ptr_kinds_M|\<^sub>r) = (ptr |\<in>| object_ptr_kinds h)"
vifunion_vsingleton_eq_vrange: fixes I :: "V"   and f :: "V \<Rightarrow> V" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>I. set {f a}) `\<^sub>\<circ> I) = \<R>\<^sub>\<circ> (VLambda I f)"
assms_4: shows "(i, Add i e1) \<in> set (node_deliver_messages xs)"
oalist_inv_raw_map_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'c" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "\<And>a b. comp (fst (f a)) (fst (f b)) = comp (fst a) (fst b)" shows "comparator.oalist_inv_raw comp (map_pair f xs)"
dgrad_set_le_def: fixes d :: "'c \<Rightarrow> nat"   and S :: "'c set"   and T :: "'c set" shows "dgrad_set_le d S T = (\<forall>s\<in>S. \<exists>t\<in>T. d s \<le> d t)"
topeqI: fixes T :: "'b set set"   and S :: "'b set set" assumes "\<And>m. m open\<^bsub>T\<^esub> \<Longrightarrow> m open\<^bsub>S\<^esub>"   and "\<And>m. m open\<^bsub>S\<^esub> \<Longrightarrow> m open\<^bsub>T\<^esub>" shows "T = S"
satisfies_sumlist_le_constraints: fixes cs :: "'b le_constraint list"   and v :: "nat \<Rightarrow> 'b" assumes "\<And>c. c \<in> set cs \<Longrightarrow> v \<Turnstile>\<^sub>l\<^sub>e c" shows "v \<Turnstile>\<^sub>l\<^sub>e sum_list cs"
simps_2: fixes x :: "'e"   and y :: "'e"   and xs :: "'e list" shows "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
select_convs_1: fixes orderA :: "('d, 'e) gorder_scheme"   and orderB :: "('f, 'g) gorder_scheme"   and lower :: "'d \<Rightarrow> 'f"   and upper :: "'f \<Rightarrow> 'd"   and more :: "'h" shows "\<X>\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = orderA"
card_Un_disjnt: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "disjnt A B" shows "card (A \<union> B) = card A + card B"
lm_iff_empty: fixes lm :: "nat list" shows "(<lm> = []) = (lm = [])"
getNeg_Pos_empty: fixes X :: "'b \<Rightarrow> 'a"   and src :: "'b list" shows "getNeg (NegPos_map X (map Pos src)) = []"
comp_associative: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "f \<in> Ar"   and "g \<in> Ar"   and "h \<in> Ar"   and "Cod h = Dom g"   and "Cod g = Dom f" shows "f \<bullet> (g \<bullet> h) = f \<bullet> g \<bullet> h"