hcju/isabelleps huggingface dataset:
poly_of_pair_alt_spp: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "distinct fs"   and "rb_aux_inv (bs, ss, p # ps)" shows "spp_of (poly_of_pair p) = spp_of_pair (app_pair spp_of p)"
monoD: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and n :: "nat"   and A :: "'c set"   and p :: "nat"   and s :: "'c"   and t :: "'c" assumes "SemilatAlg.mono r step n A"   and "p < n"   and "s \<in> A"   and "s \<sqsubseteq>\<^bsub>r\<^esub> t" shows "set (step p s) {\<sqsubseteq>\<^bsub>r\<^esub>} set (step p t)"
iT_Max_2: fixes n :: "nat"   and d :: "nat" shows "Max [n\<dots>,d] = n + d"
rep_eq: shows "Linear_Aggregation_Algebras.unit.Rep_unit top = 1"
simps_2: fixes p :: "fm"   and q :: "fm" shows "d_\<delta> (Or p q) = (\<lambda>d. d_\<delta> p d \<and> d_\<delta> q d)"
le_mask_high_bits: fixes w :: "'b word"   and n :: "nat" shows "(w \<le> mask n) = (\<forall>i\<in>{n..<size w}. \<not> w !! i)"
measure_pmf_single: fixes M :: "'b pmf"   and x :: "'b" shows "measure_pmf.prob M {x} = pmf M x"
measurable_stl: fixes M :: "'b measure" shows "stl \<in> stream_space M \<rightarrow>\<^sub>M stream_space M"
col_per2_cases: fixes B :: "'p"   and C :: "'p"   and B' :: "'p"   and D :: "'p"   and A :: "'p" assumes "B \<noteq> C"   and "B' \<noteq> C"   and "C \<noteq> D"   and "Col B C D"   and "Per A B C"   and "Per A B' C" shows "B = B' \<or> \<not> Col B' C D"
arr_mkArr: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>" shows "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F) = (A \<subseteq> Collect (category.terminal S) \<and> |A| <o \<AA> \<and> B \<subseteq> Collect (category.terminal S) \<and> |B| <o \<AA> \<and> F \<in> A \<rightarrow> B)"
lprefix_code_2: fixes x :: "'c"   and xs :: "'c llist" shows "LCons x xs \<sqsubseteq> LNil = False"
LIMSEQ_const_iff: fixes k :: "'d"   and l :: "'d" shows "(\<lambda>n. k) \<longlonglongrightarrow> l = (k = l)"
moebius_translation_comp: fixes v1 :: "complex"   and v2 :: "complex" shows "moebius_comp (moebius_translation v1) (moebius_translation v2) = moebius_translation (v1 + v2)"
control_within_def: fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'h, 'f, 'i) seqp"   and \<sigma> :: "('j \<times> ('g, 'h, 'f, 'i) seqp) set" shows "control_within \<Gamma> \<sigma> \<equiv> \<forall>(\<xi>, p)\<in>\<sigma>. \<exists>pn. p \<in> subterms (\<Gamma> pn)"
Restrp_UNIV: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "r \<restriction> UNIV \<equiv> r"
assms_2: fixes x :: "'a" shows "(s has_derivative blinfun_apply (Ds x)) (at x)"
vs1: shows "valid_sop (eval_expr t e\<^sub>1)"
eventually_ae_filter: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure" shows "almost_everywhere M P = (\<exists>N\<in>null_sets M. {x \<in> space M. \<not> P x} \<subseteq> N)"
abs_eq: fixes xa :: "'c regular \<times> 'c dense Filters.filter"   and x :: "'c regular \<times> 'c dense Filters.filter" assumes "eq_onp (\<lambda>x. x \<in> stone_phi.pairs) xa xa"   and "eq_onp (\<lambda>x. x \<in> stone_phi.pairs) x x" shows "Abs_stone_phi_pair xa \<squnion> Abs_stone_phi_pair x = Abs_stone_phi_pair (stone_phi.pairs_sup xa x)"
elims_2: fixes x :: "('b \<times> nat) option"   and xa :: "'b"   and xb :: "lock_action" assumes "lock_action_ok x xa xb" obtains l :: "('b \<times> nat) option" and t :: "'b" where "x = la__" and "xa = tb__" and "xb = Lock" and "may_lock la__ tb__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = lb__" and "xa = tc__" and "xb = Unlock" and "has_lock lb__ tc__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = lc__" and "xa = td__" and "xb = UnlockFail" and "\<not> has_lock lc__ td__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = l" and "xa = t" and "xb = ReleaseAcquire"
fds_nth_zero: shows "($) 0 = (\<lambda>_. 0::'b)"
IH: fixes c :: "String.literal" assumes "c \<in> s"   and "c \<notin> normalize_sort cs s"   and "sort_ex cs s" shows "\<exists>c'\<in>normalize_sort cs s. class_les cs c' c"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a' \<Rightarrow> a\<guillemotright>"   and "S.iso (S.UP f)" shows "local.iso f"
supr_least_Above: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> local.Above B" shows "(supr B, a) \<in> r"
totalI1: fixes f :: "recf" assumes "recfn 1 f"   and "\<And>x. eval f [x] \<down>" shows "Partial_Recursive.total f"
simps: fixes P :: "(char list list \<times> char list exp) prog"   and a1 :: "char list exp list"   and a2 :: "heap \<times> sfields \<times> sheap"   and a3 :: "bool"   and a4 :: "char list exp list"   and a5 :: "heap \<times> sfields \<times> sheap"   and a6 :: "bool" shows "redsp P a1 a2 a3 a4 a5 a6 = ((\<exists>e s b e' s' b' es. a1 = e # es \<and> a2 = s \<and> a3 = b \<and> a4 = e' # es \<and> a5 = s' \<and> a6 = b' \<and> redp P e s b e' s' b') \<or> (\<exists>es s b es' s' b' v. a1 = Val v # es \<and> a2 = s \<and> a3 = b \<and> a4 = Val v # es' \<and> a5 = s' \<and> a6 = b' \<and> redsp P es s b es' s' b'))"
incseqD: fixes f :: "nat \<Rightarrow> 'e"   and i :: "nat"   and j :: "nat" assumes "incseq f"   and "i \<le> j" shows "f i \<le> f j"
intro: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "ring_ow U plus zero minus uminus times"   and "zero_neq_one_ow U one zero"   and "monoid_mult_ow U one times" shows "ring_1_ow U one times plus zero minus uminus"
store_sops_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "'c" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"
surj_atom_map: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and Y :: "'a set" assumes "class.complete_boolean_algebra_alt Inf Sup inf less_eq less sup bot top minus uminus"   and "Y \<subseteq> Collect (bounded_lattice.atom less bot)" shows "Y = bounded_lattice.atom_map less_eq less bot (Sup Y)"
closure_complement: fixes A :: "'a set" shows "closure (carrier - A) = carrier - core A"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_singleton: fixes t :: "('d, 'c) Term.term"   and s :: "('e, 'c) Term.term" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s [(t, s)] = fv t \<union> fv s"
ProjInd_aHom1: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and f :: "('c \<Rightarrow> 'd) \<Rightarrow> 'f"   and B :: "'f set" assumes "\<forall>j\<in>I. aGroup (A j)"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to f (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B" shows "\<forall>j\<in>I. ProjInd I A f j \<in> aHom (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) f) (A j)"
countable_or: fixes A :: "'b set" assumes "countable A" shows "(\<exists>f. finite A \<and> inj_on f A) \<or> (\<exists>f. infinite A \<and> bij_betw f A UNIV)"
index_of_c_pair_n_main: fixes n :: "nat" shows "nat_to_pr (index_of_c_pair_n n) = c_pair n"
gFreshClsRev_recAll_reflFreshAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "presWlsAll (rec MOD) (recAbs MOD) MOD"   and "presCons (rec MOD) (recAbs MOD) MOD"   and "presFreshAll (rec MOD) (recAbs MOD) MOD"   and "gFreshClsRev MOD" shows "reflFreshAll (rec MOD) (recAbs MOD) MOD"
single_valuedD: fixes r :: "('a \<times> 'b) set"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "single_valued r"   and "(x, y) \<in> r"   and "(x, z) \<in> r" shows "y = z"
has_integral_alt': fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and s :: "'c set" shows "(f has_integral i) s = ((\<forall>a b. (\<lambda>x. if x \<in> s then f x else (0::'d)) integrable_on cbox a b) \<and> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'c) B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else (0::'d)) - i) < e))"
existence_ivl_empty2: fixes x0 :: "'a" assumes "x0 \<notin> X" shows "existence_ivl t0 x0 = {}"
implies_import_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (x \<sqinter> y \<leadsto> (x \<leadsto> z)) = x \<sqinter> (y \<leadsto> z)"
gen_mutil_not_tiling: fixes t :: "(nat \<times> nat) set"   and i :: "nat"   and j :: "nat"   and m :: "nat"   and n :: "nat" assumes "t \<in> tiling domino"   and "(i + j) mod 2 = 0"   and "(m + n) mod 2 = 0"   and "{(i, j), (m, n)} \<subseteq> t" shows "t - {(i, j)} - {(m, n)} \<notin> tiling domino"
SAStates_ChiPlus: fixes S :: "'a"   and SA :: "('a, 'b, 'c) seqauto"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> States SA"   and "T \<in> States SA"   and "SA \<in> SAs A" shows "(S, T) \<notin> ChiPlus A"
simps_1: fixes F :: "'b formula"   and G :: "'b formula" shows "is_disj (F \<^bold>\<or> G) = (is_lit_plus F \<and> is_disj G)"
read_only_share_all_acquired_in: fixes \<O> :: "nat set"   and S :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and a :: "nat" assumes "\<O> \<inter> read_only S = {}"   and "weak_sharing_consistent \<O> sb"   and "a \<in> read_only (share sb S)" shows "a \<in> read_only (share sb Map.empty) \<or> a \<in> read_only S \<and> a \<notin> all_acquired sb"
assms_3: shows "t \<le> v * v\<^sup>T"
elims_2: fixes x :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" assumes "valid_chain_list x" obtains "x = []"   | oneC :: "int \<times> (real \<Rightarrow> real \<times> real)" where "x = [oneC__]"   | oneC :: "int \<times> (real \<Rightarrow> real \<times> real)" and v :: "int \<times> (real \<Rightarrow> real \<times> real)" and va :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" where "x = oneC # v # va" and "pathfinish (coeff_cube_to_path oneC) = pathstart (rec_join (v # va)) \<and> valid_chain_list (v # va)"
bigo_const1: fixes c :: "'d" shows "(\<lambda>x. c) \<in> O(\<lambda>x. 1::'d)"
alpha_qSkel: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "X #= X'" shows "qSkel X = qSkel X'"
vrange_vinsert: fixes a :: "V"   and b :: "V"   and r :: "V" shows "\<R>\<^sub>\<circ> (vinsert \<langle>a, b\<rangle> r) = vinsert b (\<R>\<^sub>\<circ> r)"
mr: shows "applied_rule_rev C x p = \<lfloor>DenyAllFromTo a b\<rfloor>"
sorted_sort_id: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = xs"
map_tailrec_is_listmap: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b list"   and accs :: "'a list" shows "rev (Efficient_Distinct.map_tailrec f l accs) = rev accs @ map f l"
power_decreasing: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "nat"   and N :: "nat"   and a :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n \<le> N"   and "less_eq zero a"   and "less_eq a one" shows "less_eq (power.power one times a N) (power.power one times a n)"
hom_boundary_hom: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "hom_boundary p X S \<in> hom (relative_homology_group p X S) (homology_group (p - 1) (subtopology X S))"
andalso_eq_TT_iff: fixes x :: "tr"   and y :: "tr" shows "((x andalso y) = TT) = (x = TT \<and> y = TT)"
K_cfg_rept_aux: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)"   and x :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.valid_cfg"   and "abss s = state cfg"   and "x \<in> set_pmf (rept s (action cfg))" shows "(THE s'. s' \<in> set_pmf (rept s (action cfg)) \<and> s' ~ x) = x"
G_def: shows "G \<equiv> \<lambda>x. \<Sum>i\<in>Basis. g i (x \<bullet> i) *\<^sub>R i"
msf: shows "matrix_seq.lowner_is_lub f (matrix_seq.lowner_lub f)"
intro: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd" assumes "category C"   and "category D"   and "functor C D G"   and "right_adjoint_functor_axioms C D G" shows "right_adjoint_functor C D G"
continuous_on_discrete: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on A f"
arrI: fixes f1 :: "'a1"   and f2 :: "'a2" assumes "A1.arr f1"   and "A2.arr f2" shows "A1xA2.arr (f1, f2)"
assms: fixes i :: "'b" assumes "i \<in> I" shows "((\<lambda>x. f x i) \<longlongrightarrow> (1::'c)) F"
fcomp_comp: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "f ;; g = g \<circ> f"
not_el: shows "\<nexists>e. e \<in> set noteq \<and> e < m"
cod_def: fixes f :: "'b" shows "C'.cod f = (if C'.codomains f \<noteq> {} then SOME b. b \<in> C'.codomains f else C'.null)"
nxt_both: fixes t' :: "'b ttree"   and t :: "'b ttree"   and x :: "'b" shows "nxt (t' \<otimes>\<otimes> t) x = (if possible t' x \<and> possible t x then nxt t' x \<otimes>\<otimes> t \<oplus>\<oplus> t' \<otimes>\<otimes> nxt t x else if possible t' x then nxt t' x \<otimes>\<otimes> t else if possible t x then t' \<otimes>\<otimes> nxt t x else TTree.empty)"
entails_trans: fixes N1 :: "'f set"   and N2 :: "'f set"   and N3 :: "'f set" assumes "N1 \<Turnstile>\<inter>\<G> N2"   and "N2 \<Turnstile>\<inter>\<G> N3" shows "N1 \<Turnstile>\<inter>\<G> N3"
s_top_commute1: fixes H :: "'a set"   and K :: "'a set"   and u :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "K \<diamondop> H = H \<diamondop> K"   and "u \<in> H \<diamondop> K" shows "\<rho> u \<in> H \<diamondop> K"
finite_subset_Union: fixes A :: "'b set"   and \<B> :: "'b set set" assumes "finite A"   and "A \<subseteq> \<Union> \<B>" obtains \<F> :: "'b set set" where "finite \<F>" and "\<F> \<subseteq> \<B>" and "A \<subseteq> \<Union> \<F>"
complex_mod_triangle_ineq2: fixes b :: "complex"   and a :: "complex" shows "cmod (b + a) - cmod b \<le> cmod a"
isTrue_\<phi>G: assumes "consistent" shows "isTrue \<phi>G"
assms: fixes k :: "nat" shows "\<exists>v. v \<le>\<^sub>F\<^sub>I w\<^sub>1 \<and> k < length v \<and> v \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
fls_times_def: shows "(*) = (\<lambda>f g. fls_shift (- (fls_subdegree f + fls_subdegree g)) (fps_to_fls (fls_base_factor_to_fps f * fls_base_factor_to_fps g)))"
norm_fold: fixes q :: "'a"   and \<nu> :: "'b"   and p :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list" shows "norm_fold (q, \<nu>, p) xs = ((q, \<nu>, p) \<in> set xs, q, \<nu>, p)"
class_les_trans: fixes x :: "String.literal"   and y :: "String.literal"   and z :: "String.literal" assumes "class_les cs x y"   and "class_les cs y z" shows "class_les cs x z"
pp_1: shows "set pp = {(i, p i) |i. i < nr \<and> p i \<noteq> nc}"
swap_involutive: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good X" shows "X #[x \<and> y]_zs #[x \<and> y]_zs = X"
distinct_sorted_list_of_fmap: fixes m :: "('c, 'd) fmap" shows "distinct (map fst (sorted_list_of_fmap m))"
Cs0'_def: shows "Cs0' = Cs0 \<cdot>\<cdot>cl \<rho>s"
Mf: shows "(\<lambda>(x, y). f x y) \<in> borel_measurable borel"
strict_part_mono_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "strict_part_mono S f \<equiv> \<forall>A\<in>S. \<forall>B\<in>S. A < B \<longrightarrow> f A < f B"
SUP_ereal_add_right: fixes I :: "'a set"   and c :: "ereal"   and f :: "'a \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "c \<noteq> - \<infinity>" shows "(SUP i\<in>I. c + f i) = c + Sup (f ` I)"
set_rotate1: fixes xs :: "'d list" shows "set (rotate1 xs) = set xs"
inv_inv: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> (inv\<^bsub>G\<^esub> x) = x"
\<alpha>: fixes M :: "nat \<Rightarrow> 'a" assumes "incseq M"   and "\<And>i. P (M i)" shows "\<alpha> (Sup (range M)) = (SUP i. \<alpha> (M i))"
simps_1: fixes p :: "linear_poly"   and c :: "'a" shows "is_leq_ns (LEQ_ns p c) = True"
represents_def: fixes p :: "int poly"   and x :: "'a" shows "p represents x = (ipoly p x = (0::'a) \<and> p \<noteq> 0)"
preserves_dom: fixes f :: "'a" assumes "A.arr f" shows "inv.B.dom (\<sigma>' f) = G (A.dom f)"
R2I: fixes r :: "recf"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "recfn 2 r"   and "Partial_Recursive.total r"   and "\<And>x y. eval r [x, y] = f x y" shows "f \<in> \<R>\<^sup>2"
A'_def: fixes rs :: "nat list"   and y :: "nat"   and shared_weights :: "bool"   and ws :: "nat \<Rightarrow> real" assumes "deep_model_correct_params_y rs y" shows "deep_model_correct_params_y.A' shared_weights rs y ws = ten2mat (deep_model_correct_params_y.A shared_weights rs y ws)"
simps_1: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> (atom list \<times> atom fm list \<times> nat) list \<Rightarrow> atom fm"   and \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "QE_dnf' opt step (And \<phi>\<^sub>1 \<phi>\<^sub>2) = and (QE_dnf' opt step \<phi>\<^sub>1) (QE_dnf' opt step \<phi>\<^sub>2)"
Lev2_0: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and x :: "'e" shows "Lev2 s1 s2 0 x = {[]}"
regular_event_preserves_process_snapshots: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and r :: "'a" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "regular_event ev" shows "ps c r = ps c' r"
yr': shows "y \<circ> r' \<in> complex_rotations"
cnj_I: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A"   and "H \<turnstile> B" shows "H \<turnstile> A AND B"
set_intros_1: fixes x1 :: "'a"   and x2 :: "'a stream" shows "x1 \<in> sset (x1 ## x2)"
Nset_Suc0: shows "{i. i \<le> Suc 0} = {0, Suc 0}"
intros: fixes uu :: "nat" shows "isIntv (Intv uu)"
assms_1: shows "h \<turnstile> map_M g xs \<rightarrow>\<^sub>r ys"
Min_Inf: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top"   and "finite A"   and "A \<noteq> {}" shows "linorder.Min less_eq A = Inf A"
assms_3: shows "(f \<longlongrightarrow> b) F"
entry_return_same_proc: fixes p :: "'b"   and p' :: "'b" assumes "entry fg p = return fg p'" shows "p = p'"
second_countable_iff_dense_countable_subset: shows "(\<exists>B. countable B \<and> topological_basis B) = (\<exists>A. countable A \<and> closure A = UNIV)"
IH: assumes "inj_on f (set xs)"   and "y \<notin> f ` set xs" shows "foldr (\<lambda>x \<A>. \<A>(f x := g x)) xs \<A> y = \<A> y"
ideal_eq_UNIV_iff_contains_one: fixes B :: "'c set" shows "(ideal B = UNIV) = ((1::'c) \<in> ideal B)"
F1: shows "\<forall>x\<^sub>2 x\<^sub>1. x\<^sub>1 \<subseteq> x\<^sub>2 \<and> x\<^sub>2 \<subseteq> x\<^sub>1 \<longrightarrow> x\<^sub>1 = x\<^sub>2"
set_subset_Univ: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "HomC.set (b, a) \<subseteq> S.Univ"
divide_divide_eq_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a (divide b c) = divide (times a c) b"
val_type_cexpr_sem: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and V :: "nat set"   and \<sigma> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile>\<^sub>c e : t"   and "free_vars e \<subseteq> V"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "val_type (cexpr_sem \<sigma> e) = t"
inverse_unity_root: fixes k :: "nat"   and m :: "int" shows "inverse (unity_root k m) = cnj (unity_root k m)"
exenormalized_sort_def: fixes cs :: "(String.literal \<times> String.literal) list"   and s :: "String.literal set" shows "exenormalized_sort cs s \<equiv> exenormalize_sort cs s = s"
vec_times_mat_leq_def: fixes y :: "'b vec"   and A :: "'b mat"   and c :: "'b vec" shows "[y\<^sub>v*A]\<le>c = ((\<forall>i<dim_vec c. (A\<^sup>T *\<^sub>v y) $ i \<le> c $ i) \<and> dim_col A\<^sup>T = dim_vec y \<and> dim_row A\<^sup>T = dim_vec c)"
MyContext_def: fixes ops :: "(nat, 'a, 'b) set_ops_scheme" shows "MyContext ops \<equiv> StdSet ops"
NextPerformOp'_def: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and \<Delta>neg :: "('loc \<times> 't) multiset"   and \<Delta>mint_msg :: "('b \<times> 'loc \<times> 't) multiset"   and \<Delta>mint_self :: "('loc \<times> 't) multiset" shows "NextPerformOp' c0 c1 p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self = (cri.next_performop' (exchange_config c0) (exchange_config c1) p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self \<and> prop_config c1 = prop_config c0 \<and> init c1 = init c0)"
weight_spmf_parametric: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun (rel_spmf A) (=) weight_spmf weight_spmf"
derivation_shift_empty: fixes left :: "nat"   and right :: "nat" shows "derivation_shift [] left right = []"
unit_mult_left_cancel: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "is_unit a" shows "(a * b = a * c) = (b = c)"
fset_eq_alt: fixes x :: "'a fset"   and y :: "'a fset" shows "(x = y) = (x |\<subseteq>| y \<and> size x = size y)"
1: shows "inj_on (\<lambda>ia. (ia div y, ia mod y)) {0..<x * y}"
fps_hensel_aux_axioms: shows "fps_hensel_aux g h"
assms_2: shows "\<exists>x\<in>A. l \<le> f x"
elims_2: fixes x :: "'b heap" assumes "pheap x" obtains "x = Empty"   | xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "\<forall>h\<in>set hs. Multiset.Ball (mset_heap h) ((\<le>) xa) \<and> pheap h"
ipt_iprange_compress_normalized_IpAddrNetmask: fixes as :: "'a ipt_iprange negation_type list" shows "\<forall>a'\<in>set (ipt_iprange_compress as). normalized_cidr_ip a'"
seq_if_composable: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> B.null" shows "src \<nu> = trg \<mu>"