hcju/isabelleps huggingface dataset:
prappend_signed_list_map_uniform_snd: fixes b :: "bool"   and xs :: "'a list"   and ys :: "'a list" shows "prappend_signed_list (map (\<lambda>s. (s, b)) xs) (map (\<lambda>s. (s, b)) ys) = map (\<lambda>s. (s, b)) xs @ map (\<lambda>s. (s, b)) ys"
lucas_lehmer_add_assoc: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat"   and z :: "nat \<times> nat" assumes "0 < m" shows "lucas_lehmer_add m x (lucas_lehmer_add m y z) = lucas_lehmer_add m (lucas_lehmer_add m x y) z"
L: shows "(xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs X) \<in> failures P"
uncurry_def: fixes \<tau> :: "'a1 \<Rightarrow> ('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and f :: "'a1 \<times> 'a2" shows "uncurry \<tau> f \<equiv> if A1xA2.arr f then E.map (\<tau> (fst f), snd f) else B.null"
correctCompositionDiffLevelsS13: shows "correctCompositionDiffLevels sS13"
adder_helper: fixes w1 :: "'a list"   and w2 :: "'b list" assumes "0 < max (length w1) (length w2)" shows "2 ^ (length w1 - 1) + 2 ^ (length w2 - 1) \<le> 2 ^ max (length w1) (length w2)"
out_degree_app_iso_eq: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and u :: "'e" assumes "wf_digraph G"   and "pre_digraph.digraph_isomorphism G hom"   and "u \<in> verts G" shows "out_degree (app_iso hom G) (iso_verts hom u) = out_degree G u"
red_union_singleton_zero: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "red (F \<union> {0}) = red F"
less_eq_prod_simps: fixes leq_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and leq_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c"   and x2 :: "'d"   and y1 :: "'c"   and y2 :: "'d" shows "less_eq_prod leq_a less_a leq_b (x1, x2) (y1, y2) = (less_a x1 y1 \<or> leq_a x1 y1 \<and> leq_b x2 y2)"
scalars_zero_if_independent: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "finite A"   and "l.vs2.independent A"   and "(\<Sum>x\<in>A. scaleC (f x) x) = (0::'c)" shows "\<forall>x\<in>A. f x = (0::'a)"
inductive_step_R: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list"   and signs1 :: "rat list list"   and signs2 :: "rat list list"   and subsets1 :: "(nat list \<times> nat list) list"   and subsets2 :: "(nat list \<times> nat list) list" assumes "p \<noteq> 0"   and "0 < length qs1"   and "0 < length qs2"   and "well_def_signs (length qs1) signs1"   and "well_def_signs (length qs2) signs2"   and "distinct signs1"   and "distinct signs2"   and "set (characterize_consistent_signs_at_roots p qs1) \<subseteq> set signs1"   and "set (characterize_consistent_signs_at_roots p qs2) \<subseteq> set signs2"   and "all_list_constr_R subsets1 (length qs1)"   and "all_list_constr_R subsets2 (length qs2)"   and "invertible_mat (M_mat_R signs1 subsets1)"   and "invertible_mat (M_mat_R signs2 subsets2)" shows "satisfy_equation_R p (qs1 @ qs2) (subsets_smash_R (length qs1) subsets1 subsets2) (signs_smash signs1 signs2) \<and> invertible_mat (M_mat_R (signs_smash signs1 signs2) (subsets_smash_R (length qs1) subsets1 subsets2))"
1: shows "((\<lambda>x. \<Sum>b\<in>basis. F (reversepath \<gamma> x) \<bullet> b * (vector_derivative (reversepath \<gamma>) (at x within {0..1}) \<bullet> b)) has_integral - i) {0..1}"
op_image_fst_ivl_def: fixes X :: "('a \<times> 'b) set" shows "op_image_fst_ivl X = SPEC (\<lambda>R. R = fst ` X)"
Variable_def: shows "Variable = (SOME f. inj f \<and> range f \<subseteq> var)"
Cn_same: fixes A :: "'a set"   and B :: "'a set" shows "(Cn A = Cn B) = (\<forall>C. (A \<subseteq> Cn C) = (B \<subseteq> Cn C))"
contlubI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>Y. chain Y \<Longrightarrow> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))" shows "contlub f"
cannot_validate_contradictary_literals: fixes l :: "'b literal"   and t :: "'b trm"   and I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" assumes "l = Neg (Eq t t)"   and "fo_interpretation I" shows "\<not> validate_ground_lit I l"
borel_measurable_const_scaleR: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and b :: "real" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. b *\<^sub>R f x) \<in> borel_measurable M"
x_y_in_carrier: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq> y" shows "x \<in> carrier \<and> y \<in> carrier"
lc: shows "lc (monom_mult c (0::'a) f) = c * lc f"
legal_\<E>D: fixes is_justification :: "'a prog \<Rightarrow> ('b \<times> ('c, 'b) obs_event action) llist \<times> (nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> ('c, 'b) justifying_execution) \<Rightarrow> bool"   and P :: "'a prog"   and \<E> :: "('b \<times> ('c, 'b) obs_event action) llist set"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "gen_legal_execution is_justification P \<E> (E, ws)" shows "E \<in> \<E>"
map_conv_bind_option: fixes f :: "'f \<Rightarrow> 'e"   and x :: "'f option" shows "map_option f x = Option.bind x (Some \<circ> f)"
induct: fixes P :: "ty_Rep \<Rightarrow> bool"   and ty_Rep :: "ty_Rep" assumes "P TBase_Rep"   and "P TUnit_Rep"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Arrow_Rep x1 x2)" shows "P ty_Rep"
matrix_Tensor_elements: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and M1 :: "'a list list"   and M2 :: "'a list list" assumes "Matrix_Tensor.mult id f" shows "\<forall>i j. (i < mult.row_length M1 * mult.row_length M2 \<and> j < length M1 * length M2) \<and> Matrix_Legacy.mat (mult.row_length M1) (length M1) M1 \<and> Matrix_Legacy.mat (mult.row_length M2) (length M2) M2 \<longrightarrow> mult.Tensor f M1 M2 ! j ! i = f (M1 ! (j div length M2) ! (i div mult.row_length M2)) (M2 ! (j mod length M2) ! (i mod mult.row_length M2))"
inv_equality: fixes G :: "('a, 'b) monoid_scheme"   and y :: "'a"   and x :: "'a" assumes "Group.group G"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> x = y"
execute: fixes p :: "'state"   and a :: "'action" assumes "p \<in> R.nodes"   and "ren a p" shows "ex a p \<in> R.nodes"
gh: shows "src\<^sub>C g = trg\<^sub>C h"
substInp_def: fixes xs :: "'varSort"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and x :: "'var" shows "substInp xs X x \<equiv> psubstInp (idEnv [x \<leftarrow> X]_xs)"
check_eqv_def: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" shows "slow.check_eqv n \<phi> \<psi> = (wf_formula n (FOr \<phi> \<psi>) \<and> slow.check_eqvRE enum_class.enum n (rexp_of'' n (norm \<phi>)) (rexp_of'' n (norm \<psi>)))"
rel_pmf_bindI: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "'h pmf"   and q :: "'i pmf"   and P :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'j pmf"   and g :: "'i \<Rightarrow> 'k pmf" assumes "rel_pmf R p q"   and "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)" shows "rel_pmf P (p \<bind> f) (q \<bind> g)"
zipped_subset_mapped_Elem: fixes xs :: "'b list"   and e :: "nat \<Rightarrow> real"   and XS :: "('b \<times> 'b pdevs) list"   and ys :: "'b list"   and YS :: "('b \<times> 'b pdevs) list" assumes "xs = map (aform_val e) XS"   and "\<And>i. e i \<in> {- 1..1}"   and "length xs = length XS"   and "length ys = length YS"   and "set (zip ys YS) \<subseteq> set (zip xs XS)" shows "ys = map (aform_val e) YS"
Im_Ln_eq_0: fixes z :: "complex" assumes "z \<noteq> 0" shows "(Im (Ln z) = 0) = (0 < Re z \<and> Im z = 0)"
sg2: shows "Suc (3 + k) = 4 + k"
CX3h_range: fixes h :: "H1"   and X :: "X3 set" shows "CX3h h X \<subseteq> {x \<in> X. H = h}"
uminus_add_conv_diff: fixes a :: "'a"   and b :: "'a" shows "- a + b = b - a"
obs: shows "\<forall>i. \<exists>!L. (\<lambda>n. \<c>\<o>\<l> i (X n)) \<longlonglongrightarrow> L"
hypnat_diff_commute: fixes i :: "nat star"   and j :: "nat star"   and k :: "nat star" shows "i - j - k = i - k - j"
conv_complement: fixes x :: "'a" shows "id (- x) = - id x"
enumerate_mono: fixes m :: "nat"   and n :: "nat"   and S :: "'a set" assumes "m < n"   and "infinite S" shows "enum S m < enum S n"
token_run_enter_final_states: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and x :: "nat"   and n :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n \<in> F" shows "\<exists>m\<ge>x. semi_mojmir_def.token_run \<delta> q\<^sub>0 w x m \<notin> F - {q\<^sub>0} \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc m) \<in> F"
InitBlockRedsFinal: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and V :: "char list"   and v :: "val"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V \<mapsto> v))\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>"   and "final e'" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l'(V := l V))\<rangle>"
writerT_abs_def: shows "writerT_abs \<equiv> PRJ('e\<cdot>('g, 'f) writerT) oo EMB('e\<cdot>'f writer\<cdot>'g)"
elims_1: fixes x :: "'b \<Rightarrow> bool"   and xa :: "bool"   and xb :: "'b match_expr"   and y :: "bool" assumes "has_disc_negated x xa xb = y" obtains uu_ :: "'b \<Rightarrow> bool" and uv_ :: "bool" where "x = uu_" and "xa = uv_" and "xb = MatchAny" and "\<not> y"   | disc :: "'b \<Rightarrow> bool" and neg :: "bool" and a :: "'b" where "x = disc__" and "xa = neg__" and "xb = Match a" and "y = (if disc__ a then neg__ else False)"   | disc :: "'b \<Rightarrow> bool" and neg :: "bool" and m :: "'b match_expr" where "x = disca__" and "xa = nega__" and "xb = MatchNot m" and "y = has_disc_negated disca__ (\<not> nega__) m"   | disc :: "'b \<Rightarrow> bool" and neg :: "bool" and m1 :: "'b match_expr" and m2 :: "'b match_expr" where "x = disc" and "xa = neg" and "xb = MatchAnd m1 m2" and "y = (has_disc_negated disc neg m1 \<or> has_disc_negated disc neg m2)"
ratpolysI: fixes p :: "complex poly" assumes "\<And>i. poly.coeff p i \<in> \<rat>" shows "p \<in> ratpolys"
CondReds2T: fixes e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and e' :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>"
COMP: 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"   and c1 :: "OBJ"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and c2 :: "OBJ"   and \<Psi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "G \<rhd> c1 : Sec \<Phi>"   and "G \<rhd> c2 : Sec \<Psi>" shows "G \<rhd> Comp c1 c2 : Sec (\<lambda>(s, t, \<beta>). \<exists>r. \<Phi> (r, t, \<beta>) \<and> (\<forall>w \<gamma>. r \<equiv>\<^sub>\<gamma> w \<longrightarrow> \<Psi> (s, w, \<gamma>)))"
assert_simplex_unsat: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "'a simplex_state"   and j :: "'a"   and I :: "'a list" assumes "invariant_simplex cs J s"   and "assert_simplex j s = Inl I" shows "set I \<subseteq> insert j J \<and> minimal_unsat_core (set I) cs"
opt_no_defection_imp_no_defection: fixes sc :: "opt_v_state"   and round_votes :: "process \<Rightarrow> val option"   and sa :: "v_state"   and r :: "nat" assumes "opt_no_defection sc round_votes"   and "(sa, sc) \<in> flv_ref_rel"   and "sa \<in> Vinv1"   and "sa \<in> Vinv2" shows "no_defection sa round_votes r"
inext_predicate_change_exists: fixes a :: "nat"   and b :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "a \<in> I"   and "b \<in> I"   and "\<not> P a"   and "P b" shows "\<exists>n\<in>I. a \<le> n \<and> n < b \<and> \<not> P n \<and> P (inext n I)"
upper_adjoint_def: fixes A :: "('d, 'e) gorder_scheme"   and B :: "('f, 'g) gorder_scheme"   and g :: "'f \<Rightarrow> 'd" shows "upper_adjoint A B g \<equiv> \<exists>f. galois_connection \<lparr>orderA = A, orderB = B, lower = f, upper = g\<rparr>"
val_1_nonzero: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x = 1" shows "x \<noteq> \<zero>"
all_set_conv_nth: fixes l :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>set l. P x) = (\<forall>i<length l. P (l ! i))"
map_def: fixes f :: "'a \<times> 'a" shows "cmp\<^sub>U\<^sub>P f = (if B.VV.arr f then \<Phi>\<^sub>o (B.VV.cod f) \<cdot> HoUP_UP.map f else null)"
top: shows "\<Omega> \<in> sigma_sets \<Omega> G"
DOM_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "arr f"   and "arr g" shows "DOM (f \<otimes> g) = DOM f \<^bold>\<otimes> DOM g"
map_def: fixes f :: "'b" shows "\<tau>.B.map f = (if \<tau>.B.arr f then f else \<tau>.B.null)"
rel_S_neg_distr_cond_eq: fixes tytok :: "('g \<times> 'h \<times> 'i \<times> 'j \<times> 'k \<times> 'l \<times> 'm) itself" shows "rel_S_neg_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) tytok"
has_derivative_power_int: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and f' :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and n :: "int" assumes "f x \<noteq> (0::'a)"   and "(f has_derivative f') (at x within S)" shows "((\<lambda>x. f x powi n) has_derivative (\<lambda>h. f' h * (of_int n * f x powi (n - 1)))) (at x within S)"
prime_implicates_of_saturated_sets: fixes S :: "'at Literal set set" assumes "saturated_binary_rule resolvent S"   and "all_fulfill finite S"   and "non_redundant S" shows "S = prime_implicates S"
axioms_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "identity_arrow_of_spans C \<mu>"
strict_trans1: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a \<le>p b"   and "b <p c" shows "a <p c"
indicator_simps_2: fixes x :: "'b"   and S :: "'b set" assumes "x \<notin> S" shows "indicator S x = (0::'c)"
init: fixes i :: "nat" shows "{uu_. \<exists>\<sigma> \<zeta> s. uu_ = (\<sigma>, \<zeta>) \<and> s \<in> init (np i) \<and> (\<sigma> i, \<zeta>) = sr s \<and> (\<forall>j. j \<noteq> i \<longrightarrow> \<sigma> j \<in> (fst \<circ> sr) ` init (np j))} \<subseteq> init (onp i)"
vlrestriction_vdomain: fixes A :: "V" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A) \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A) = r \<restriction>\<^sup>l\<^sub>\<circ> A"
exec_move_WhileI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e1 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1" assumes "exec_move ci P t e1 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (while (e) e1) h (stk, loc, Suc (length (compE2 e) + pc), xcp) ta h' (stk', loc', Suc (length (compE2 e) + pc'), xcp')"
sorted_repeat_poly_list_rel_wrt_def: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "('a \<times> char list multiset) set" shows "sorted_repeat_poly_list_rel_wrt S R = {(xs, ys). (xs, ys) \<in> \<langle>R \<times>\<^sub>r int_rel\<rangle>list_rel O list_mset_rel \<and> sorted_wrt S (map fst xs) \<and> 0 \<notin># snd `# ys}"
ifbd_iffb_dual: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "bd\<^sup>-\<^sub>\<F> \<phi> = (op\<^sub>K \<circ> fb\<^sup>-\<^sub>\<F> \<circ> \<partial>\<^sub>F) \<phi>"
split_cong: fixes q :: "'e \<times> 'f"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and p :: "'e \<times> 'f" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
bit_0_eq: 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" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit zero = bot"
S4_excl: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> S4 rmhist p \<longrightarrow> S4 rmhist p \<and> \<not> S1 rmhist p \<and> \<not> S2 rmhist p \<and> \<not> S3 rmhist p"
mon_loc_single: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and e :: "('c, 'f) label list el_step" shows "mon_loc fg [e] = (case e of LOC a \<Rightarrow> mon_w fg a | ENV a \<Rightarrow> {})"
tree_path_disc: fixes v :: "'v"   and w :: "'v" assumes "(v, w) \<in> (tree_edges s)\<^sup>+" shows "\<delta> s v < \<delta> s w"
eventually_filtermap: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c filter" shows "eventually P (filtermap f F) = (\<forall>\<^sub>F x in F. P (f x))"
toy_labels_not_empty: fixes p :: "(state, msg, pseqp, pseqp label) seqp" shows "labels \<Gamma>\<^sub>T\<^sub>O\<^sub>Y p \<noteq> {}"
act_ref: fixes s :: "'a v_state_scheme"   and v_f :: "process \<Rightarrow> val option" assumes "s \<in> Vinv1" shows "map_option snd \<circ> process_mru ((votes s)(v_state.next_round s := v_f)) = (map_option snd \<circ> process_mru (votes s)) ++ v_f"
infnorm_cart: fixes x :: "(real, 'a) vec" shows "infnorm x = Sup {\<bar>x $ i\<bar> |i. i \<in> UNIV}"
is_functor: shows "functor DopxC.comp (\<odot>) FopxC.map"
sums_alternating_upper_lower: fixes a :: "nat \<Rightarrow> real" assumes "\<And>n. a (Suc n) \<le> a n"   and "\<And>n. 0 \<le> a n"   and "a \<longlonglongrightarrow> 0" shows "\<exists>l. ((\<forall>n. (\<Sum>i<2 * n. (- 1) ^ i * a i) \<le> l) \<and> (\<lambda>n. \<Sum>i<2 * n. (- 1) ^ i * a i) \<longlonglongrightarrow> l) \<and> (\<forall>n. l \<le> (\<Sum>i<2 * n + 1. (- 1) ^ i * a i)) \<and> (\<lambda>n. \<Sum>i<2 * n + 1. (- 1) ^ i * a i) \<longlonglongrightarrow> l"
simps_3: fixes r :: "'form" shows "outPorts (Conclusion r) = {||}"
independent_is_basis: fixes B :: "('a, 'b) vec set" shows "(vec.independent B \<and> card B = CARD('b)) = is_basis B"
in_trancl_closure_iff_in_trancl_fun: fixes a :: "'e"   and b :: "'e"   and TI :: "('e \<times> 'e) list" shows "((a, b) \<in> (set TI)\<^sup>+) = in_trancl TI a b"
simps: fixes Def :: "'f \<Rightarrow> 'g set"   and E :: "'f"   and X :: "'f"   and H :: "'g \<Rightarrow> bool"   and L :: "'g set"   and a1 :: "'f LDCFG_node"   and a2 :: "'g" shows "lift_Def_setp Def E X H L a1 a2 = ((\<exists>V n. a1 = Node n \<and> a2 = V \<and> V \<in> Def n) \<or> (\<exists>V. a1 = Node E \<and> a2 = V \<and> H V))"
coinduct: fixes X :: "('a, 'b, 'c, 'd) converter \<Rightarrow> bool"   and x :: "('a, 'b, 'c, 'd) converter"   and \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>conv. x = conv \<and> (\<forall>x. x \<in> outs_\<I> \<I> \<longrightarrow> \<I>' \<turnstile>g run_converter conv x \<surd>) \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> results_gpv \<I>' (run_converter conv x) \<longrightarrow> xa \<in> responses_\<I> \<I> x \<and> (X xb \<or> \<I>, \<I>' \<turnstile>\<^sub>C xb \<surd>))" shows "\<I>, \<I>' \<turnstile>\<^sub>C x \<surd>"
compl_fam_imp_triv_finprod: fixes S :: "'d \<Rightarrow> 'a set"   and I :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "compl_fam S I"   and "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (S i) G"   and "finprod G f I = \<one>"   and "f \<in> Pi I S" shows "\<forall>i\<in>I. f i = \<one>"
meq_def: fixes s :: "'a"   and Z :: "'a" shows "s__ may_not_modify_globals Z__ = (s = Z)"
safe_reach: shows "safe_reach_direct safe_delayed (ts, m, \<S>)"
F_def: shows "local.F = {f \<in> {0..<n} \<rightarrow> {0..<m}. \<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i}"
len_def: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "len v ts c \<equiv> if right (ext v) < left (space ts v c) then Abs_real_int (right (ext v), right (ext v)) else if right (space ts v c) < left (ext v) then Abs_real_int (left (ext v), left (ext v)) else Abs_real_int (max (left (ext v)) (left (space ts v c)), min (right (ext v)) (right (space ts v c)))"
norms_mus_def: fixes fs :: "'a vec list" shows "norms_mus fs = (let (n_gs, mus) = norms_mus' fs [] [] in (rev n_gs, rev mus))"
HoVH_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "B.HoVH \<mu> \<equiv> if B.VVV.arr \<mu> then fst \<mu> \<star>\<^sub>B fst (snd \<mu>) \<star>\<^sub>B snd (snd \<mu>) else B.null"
g0: shows "g0 = (fv \<phi> \<subseteq> {0..<b})"
all_trms_irreducible_def: fixes E :: "'c trm set"   and f :: "'c trm \<Rightarrow> 'c trm" shows "all_trms_irreducible E f = (\<forall>x y. x \<in> E \<longrightarrow> occurs_in y x \<longrightarrow> f y = y)"
supp_4: fixes fm :: "fm" shows "supp (Neg fm) = supp fm"
representation_zero: fixes basis :: "'b set" shows "representation basis (0::'b) = (\<lambda>b. 0::'a)"
shiftl1_is_mult: fixes x :: "'b word" shows "x << 1 = x * 2"
min_step_inext2: fixes x :: "nat"   and y :: "nat"   and I :: "nat set" assumes "x < y"   and "x \<in> I"   and "y \<in> I"   and "\<not> (\<exists>k\<in>I. x < k \<and> k < y)" shows "inext x I = y"
Spec: fixes A :: "fmla"   and H :: "fmla set" assumes "A \<in> special_axioms" shows "H \<turnstile> A"
const: fixes c :: "'b"   and F :: "'c filter" assumes "c \<noteq> (0::'b)" shows "O[F](\<lambda>_. c) = O[F](\<lambda>_. 1::'b)"
subset_AboveS_UnderS: fixes B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "B \<subseteq> Field r" shows "B \<subseteq> AboveS r (UnderS r B)"