hcju/isabelleps huggingface dataset:
has_integral_scaleR_left: fixes f :: "'c \<Rightarrow> real"   and y :: "real"   and S :: "'c set"   and c :: "'d" assumes "(f has_integral y) S" shows "((\<lambda>x. f x *\<^sub>R c) has_integral y *\<^sub>R c) S"
sup2CI: fixes B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<not> B x y \<Longrightarrow> A x y" shows "sup A B x y"
aHom_compos: fixes L :: "('c, 'd) aGroup_scheme"   and M :: "('e, 'f) aGroup_scheme"   and N :: "('g, 'h) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'g" assumes "aGroup L"   and "aGroup M"   and "aGroup N"   and "f \<in> aHom L M"   and "g \<in> aHom M N" shows "compos L g f \<in> aHom L N"
Phi_sur: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_direct_product w x"   and "is_direct_product y z" shows "is_sur (\<Phi> w x y z)"
llast_def: fixes xs :: "'b llist" shows "llast xs = (case llength xs of enat (Suc n') \<Rightarrow> lnth xs n')"
extensional_funcset_fun_upd_restricts_rangeI: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and T :: "'d set" assumes "\<forall>y\<in>S. f x \<noteq> f y"   and "f \<in> insert x S \<rightarrow>\<^sub>E T" shows "f(x := undefined) \<in> S \<rightarrow>\<^sub>E T - {f x}"
ballE: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "\<forall>x\<in>A. P x" obtains "P x"   | "x \<notin> A"
plus_ms_aux: fixes basis :: "(real \<Rightarrow> real) list"   and F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and G :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "is_expansion_aux F f basis"   and "is_expansion_aux G g basis" shows "is_expansion_aux (plus_ms_aux F G) (\<lambda>x. f x + g x) basis"
ge: fixes x :: "'a" shows "f x \<ge>p g x \<and> g x \<ge>p zero_poly"
parts_mult: fixes s :: "(agent \<times> msg) set"   and A :: "nat"   and B :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "A \<otimes> B \<in> parts (used s)" shows "(\<exists>n. A = Auth_PriK n \<and> (Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s) \<or> {PriKey A, PriKey B} \<subseteq> spied s"
emeasure_finite: fixes M :: "'c measure"   and A :: "'c set" assumes "finite_measure M" shows "emeasure M A \<noteq> \<top>"
le_\<gamma>: fixes X :: "'a set" assumes "X \<in> sets M" shows "d X \<le> \<gamma>"
def: shows "mult_sys.sys_block_sizes n = {int (card bl) |bl. bl \<in># multiple_blocks n}"
set_bit_eq: fixes n :: "nat"   and k :: "int" shows "ring_bit_operations_class.set_bit n k = k + of_bool (\<not> bit k n) * 2 ^ n"
ennreal_minus_cancel_iff: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a - b = a - c) = (b = c \<or> a \<le> b \<and> a \<le> c \<or> a = \<top>)"
transC_MtransT: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and cf :: "('a, 'c) com \<times> 'b"   and cf' :: "('a, 'c) com \<times> 'b"   and s'' :: "'b" assumes "PL.transC tval aval cf cf'"   and "PL.MtransT tval aval cf' s''" shows "PL.MtransT tval aval cf s''"
TOTALS: shows "\<forall>a\<in>Field Bonk_Schramm_extension_unfolded_wo. \<forall>b\<in>Field Bonk_Schramm_extension_unfolded_wo. (a, b) \<in> Bonk_Schramm_extension_unfolded_wo \<or> (b, a) \<in> Bonk_Schramm_extension_unfolded_wo"
cod_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "partial_magma.cod (free_monoidal_category.comp C) f = free_monoidal_category.mkarr C (monoidal_language.Cod C (free_monoidal_category.rep C f))"
ofilter_under_Union: fixes A :: "'a set" assumes "local.ofilter A" shows "A = \<Union> {local.under a |a. a \<in> A}"
Lcm_nat_def: fixes M :: "nat set" shows "Lcm M = (if finite M then F M else 0)"
MapsToObj_2: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "Y \<in> obj\<^bsub>C\<^esub>"
3: shows "has_bochner_integral (lebesgue_on {b..b + c}) f (integral\<^sup>L (lebesgue_on {a..a + c}) f)"
bind_pmf_cong: fixes p :: "'d pmf"   and q :: "'d pmf"   and f :: "'d \<Rightarrow> 'e pmf"   and g :: "'d \<Rightarrow> 'e pmf" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
cond_var_subst_right: fixes x :: "bool \<Longrightarrow> 'a"   and P :: "('b, 'a) uexpr"   and Q :: "('b, 'a) uexpr" assumes "vwb_lens x" shows "P \<triangleleft> x \<triangleright> Q\<lbrakk>false/x\<rbrakk> = P \<triangleleft> x \<triangleright> Q"
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c" assumes "bicategory V H \<a> \<i> src trg" shows "self_evaluation_map V H \<a> \<i> src trg"
combine_multiple1: assumes "\<V> = \<V>'"   and "\<B> = \<B>'" shows "\<B>\<^sup>+ = sys1.multiple_blocks 2"
poly_of_pm_zero: fixes x :: "'b" shows "poly_of_pm x 0 = 0"
left_commute: fixes b :: "'c \<Rightarrow> bool"   and a :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool" shows "(b \<and>* a \<and>* c) = (a \<and>* b \<and>* c)"
Interleaves_suffix_aux: fixes ws :: "'b list"   and P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and zs :: "'b list" assumes "\<And>n. n < length ws \<Longrightarrow> P (ws ! n) (drop (Suc n) ws)"   and "x # xs @ ws \<cong> {ws, zs, P}" shows "\<not> P x (xs @ ws)"
rep_eq: fixes x :: "'a st"   and xa :: "'a st" shows "(x \<le> xa) = local.less_eq_st_rep (rep_st x) (rep_st xa)"
code: fixes conv :: "('i, 'j, 'k, 'l) converter"   and res :: "('k, 'l) resource" shows "conv \<rhd> res = \<section>R\<section> (\<lambda>a. map_spmf (\<lambda>((b, conv'), res'). (b, conv' \<rhd> res')) (exec_gpv run_resource (run_converter conv a) res))"
intros_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and f :: "'f" shows "\<Gamma>\<turnstile>c \<down> Fault f"
szip_unfold: fixes a :: "'a"   and s1 :: "'a stream"   and b :: "'b"   and s2 :: "'b stream" shows "a ## s1 ||| b ## s2 = (a, b) ## (s1 ||| s2)"
disc_152: fixes x9 :: "'d" shows "\<not> is_fun_call (Ubx.instr.IOp x9)"
measure_eq_0_null_sets: fixes S :: "'c set"   and M :: "'c measure" assumes "S \<in> null_sets M" shows "Sigma_Algebra.measure M S = 0"
t1: fixes x :: "'a" assumes "x \<in> A" shows "f x = g x"
WT_recv: fixes \<eta> :: "nat"   and s :: "bool list cstate" assumes "pred_cstate (\<lambda>x. length x = \<eta>) s" shows "\<I>_uniform UNIV (insert None (Some ` vld \<eta>)) \<turnstile>c A_CHAN.recv_oracle s \<surd>"
while: shows "\<langle>while (b) c,s1\<rangle> \<rightarrow>* \<langle>Skip,s2\<rangle>"
invh_baldR_invc: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "bheight l = bheight r + 1"   and "invc l" shows "invh (baldR l a r) \<and> bheight (baldR l a r) = bheight l"
AddSA_CompFun_PseudoHA_ran2: fixes SA1 :: "('a, 'b, 'c) seqauto"   and RootSA :: "('a, 'b, 'c) seqauto"   and SA2 :: "('a, 'b, 'c) seqauto"   and S :: "'a"   and D :: "'c data" assumes "States SA1 \<inter> States RootSA = {}"   and "States SA2 \<inter> (States RootSA \<union> States SA1) = {}"   and "S \<in> States RootSA" shows "ran (CompFun (PseudoHA RootSA D [++] (S, SA1) [++] (S, SA2))) = {{}, {SA2, SA1}}"
absorb_def: fixes d :: "('c, 'd) distinguisher"   and conv :: "('c, 'd, 'a, 'b) converter" shows "absorb d conv = map_gpv fst id (inline run_converter d conv)"
diff_strict_right_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less a b" shows "less (minus a c) (minus b c)"
LIM: fixes i :: "'a" assumes "isLim i"   and "i \<noteq> zero"   and "\<And>j. j \<in> local.underS i \<Longrightarrow> P j" shows "P i"
norm_prime_divisor: fixes p :: "nat"   and q :: "gauss_int" assumes "noninert_gauss_int_prime p"   and "prime_elem q"   and "q dvd of_nat p" shows "gauss_int_norm q = p"
upper_triangular_imp_det_eq_0_iff: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "upper_triangular A" shows "(det A = (0::'b)) = ((0::'b) \<in> set (diag_mat A))"
set_def: fixes ba :: "'c \<times> 'c" shows "Hom.set ba \<equiv> (S.UP \<circ> inl) ` C.hom (fst ba) (snd ba)"
fst_o_inv_eq_inv: fixes h :: "'c \<Rightarrow> 'd"   and s :: "'d" assumes "bij h" shows "fst (inv (\<lambda>(x, u). h x) s) = inv h s"
urD_2: shows "real_of_1 (p, l, r) \<le> real_of_rat r"
abs_of_nat: fixes 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (semiring_1.of_nat one plus zero n) = semiring_1.of_nat one plus zero n"
inf_sup_ord_1: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) x"
elims: fixes x :: "int poly \<times> rat \<times> rat"   and y :: "rat" assumes "rai_lb x = y" obtains uu_ :: "int poly" and l :: "rat" and uv_ :: "rat" where "x = (uu_, l, uv_)" and "y = l"
comp_mapping_complement: fixes x :: "'a"   and y :: "'a" assumes "coreflexive (x[[x]]) \<and> times_top_class.total x" shows "x * - y = - (x * y)"
A_6: shows "region_set' (R \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of A l'} \<noteq> {}"
dim_col_mat_of_rows_list: fixes nr :: "nat"   and ls :: "'a list list" shows "dim_col (mat_of_rows_list nr ls) = nr"
inf_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "inf f g = (\<lambda>x. inf (f x) (g x))"
af_\<nu>LTL: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<in> \<nu>LTL" shows "w \<Turnstile>\<^sub>n \<phi> = (\<forall>i. \<not> af \<phi> (prefix i w) \<sim> false\<^sub>n)"
sup_multiset_def: fixes A :: "'b multiset"   and B :: "'b multiset" shows "sup A B = (if A < B then B else A)"
simps_1: fixes f :: "int \<Rightarrow> int \<Rightarrow> int"   and x :: "int"   and y :: "int" shows "maybe_arith_int f (Some (Num x)) (Some (Num y)) = Some (Num (f x y))"
lowner_le_trans_positiveI: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat" assumes "A \<in> carrier_mat n n"   and "positive A"   and "A \<le>\<^sub>L B" shows "positive B"
le_fininf_word: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream"   and p :: "'state" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"   and "run w\<^sub>2 p" shows "path w\<^sub>1 p"
assms_3: shows "\<forall>n. f n \<le> g n"
1: shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) G p'"
g_def: fixes s0 :: "'a list"   and rs :: "'a list" shows "g s0 rs \<equiv> real (T_off Aoff s0 rs)"
conj_def: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "P \<and> Q \<equiv> \<lambda>A. P A \<and> Q A"
assms_2: shows "is_GB_cofactor_bound (set fs) b"
le: fixes x :: "'n" assumes "x \<in> S" shows "f x \<le> g x"
preorder_on_restrict: fixes A :: "'a set" shows "Order_Predicates.preorder_on ((Aa__ - M) \<inter> A) (restrict_relation A (of_weak_ranking (weak_ranking_aux (Aa__ - M))))"
index: shows "I = {U. is_open U \<and> x \<in> U}"
widen_st_def: fixes F1 :: "'b Abs_State.st"   and F2 :: "'b Abs_State.st" shows "F1 \<nabla> F2 = FunDom (\<lambda>x. fun F1 x \<nabla> fun F2 x) (inter_list (Abs_State.dom F1) (Abs_State.dom F2))"
associatedE2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" obtains u :: "'a" where "dvd.dvd times u one" and "b = times u a"
hyps_2: fixes U :: "POPLmark.type"   and U' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> S' <: U \<rightarrow> U'" obtains S' :: "POPLmark.type" where "\<Gamma> \<turnstile> U <: S" and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'" and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'"
step: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "pre C I l s" shows "I l (postQ C s)"
basic_3: shows "\<exists>!x. root_cond (p2, l2, r2) x"
l_coset_swap: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <#\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> H"   and "x \<in> carrier (G\<lparr>carrier := H__\<rparr>)"   and "subgroup H (G\<lparr>carrier := H__\<rparr>)" shows "x \<in> y <#\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> H"
space_distr: fixes M :: "'d measure"   and N :: "'c measure"   and f :: "'d \<Rightarrow> 'c" shows "space (distr M N f) = space N"
vec_hom_inj: fixes v :: "'a vec"   and w :: "'a vec" assumes "vec_hom v = vec_hom w" shows "v = w"
hom_one: shows "hom_boundary p (discrete_topology {()}) {()} \<one>\<^bsub>relative_homology_group p (discrete_topology {()}) {()}\<^esub> = \<one>\<^bsub>homology_group (p - 1) (discrete_topology {()})\<^esub>"
maximal_path_base': fixes v :: "'a" assumes "deadend v" shows "maximal_path (LCons v LNil)"
simps_14: fixes \<C> :: "t_context"   and ts :: "checker_type" shows "check_single \<C> Select ts = type_update_select ts"
inf_isotone: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and s :: "'a" assumes "- p \<le> - q"   and "- r \<le> - s" shows "- p \<sqinter> - r \<le> - q \<sqinter> - s"
hadd: fixes x :: "hf"   and y :: "hf" shows "x @+ y = x \<squnion> RepFun y ((@+) x)"
max: shows "maximum i + 1 = minimum (j \<squnion> k)"
concat_take_is_prefix: fixes n :: "nat"   and ws :: "'b list list" shows "concat (take n ws) \<le>p concat ws"
new_tv_subst_comp_2: fixes n :: "nat"   and S :: "nat \<Rightarrow> typ"   and R :: "nat \<Rightarrow> typ" assumes "new_tv n S"   and "new_tv n R" shows "new_tv n (\<lambda>v. $ R (S v))"
PBit: fixes bin :: "int"   and bit :: "bool" assumes "P bin" shows "P (bin BIT bit)"
skip_il_neq_i: fixes i :: "nat"   and l :: "nat" shows "skip i l \<noteq> i"
headconst_zero: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "(headconst p = (0::'b)) = (p = 0\<^sub>p)"
degree_aform_num_aform: fixes x :: "'a" shows "degree_aform (num_aform x) = 0"
adjuster_in_span: fixes w :: "'b vec"   and n :: "nat"   and us :: "'b vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set us)"
float_divr_pos_less1_lower_bound: fixes x :: "float"   and prec :: "nat" assumes "0 < x"   and "x \<le> 1" shows "1 \<le> float_divr prec 1 x"
reflp_on_def: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "reflp_on P A = (\<forall>a\<in>A. P a a)"
mapi_block_id: fixes v :: "nat"   and f :: "('b, 'c) fm \<Rightarrow> ('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and i :: "'c" shows "mapi_block (mapper f {} v) (ps, i) = (ps, i)"
declclassf_simp: fixes n :: "vname"   and c :: "qtname" shows "declclassf (n, c) = c"
PO_refines_implies_R_image_init: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" assumes "PO_refines R Ta Tc" shows "init Tc \<subseteq> R `` init Ta"
Diff_single_insert: fixes A :: "'a set"   and x :: "'a"   and B :: "'a set" assumes "A - {x} \<subseteq> B" shows "A \<subseteq> insert x B"
sum_ereal: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set" shows "(\<Sum>x\<in>A. ereal (f x)) = ereal (sum f A)"
StatImp: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "\<Psi>' \<rhd> S \<lessapprox><Rel> T"
always_imp_mp: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(\<box>(\<lambda>s. P s \<longrightarrow> Q s)) \<sigma>"   and "P \<sigma>" shows "Q \<sigma>"
le_spvec_empty1_sparse_row: fixes b :: "(nat \<times> 'a) list" assumes "sorted_spvec b" shows "le_spvec [] b = (0 \<le> sparse_row_vector b)"
vertFrom_f': shows "verticesFrom f' v = v # vs1 @ a # vs2"
Ops: fixes f :: "'trm \<Rightarrow> 'trm"   and t :: "'trm" assumes "f \<in> Ops"   and "t \<in> trm" shows "f t \<in> trm"
mult_imp_less_multiset\<^sub>H\<^sub>O: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "(M, N) \<in> mult {(x, y). x < y}" shows "less_multiset\<^sub>H\<^sub>O M N"
ptensor_mat_def: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "st1.ptensor_mat m1 m2 = tensor_mat st1.dims0 st1.vars1' m1 m2"
twiddle_mkIDs_compose: fixes b :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and a :: "Var \<Rightarrow> Val"   and ab :: "Var \<Rightarrow> Val"   and bb :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and aa :: "Var \<Rightarrow> Val"   and ba :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "(a, b) \<equiv>\<^sub>mkId b (ab, bb)"   and "(ab, bb) \<equiv>\<^sub>mkId bb (aa, ba)" shows "(a, b) \<equiv>\<^sub>mkId b (aa, ba)"
distinct_21: fixes x12 :: "qtname" shows "Skip \<noteq> Init x12"
allVars_def: fixes g :: "'a" shows "i.allVars g \<equiv> \<Union>n\<in>set (\<alpha>n2 g). i.allDefs g n \<union> i.allUses g n"
emsapss_encode_help6_def: fixes DB :: "bit list"   and dbMask :: "bit list"   and H :: "bit list"   and emBits :: "nat" shows "emsapss_encode_help6 DB dbMask H emBits = (if dbMask = [] then [] else emsapss_encode_help7 (bvxor DB dbMask) H emBits)"
ne: shows "cs\<^bsup>\<pi> \<guillemotleft> k\<^esup> n \<noteq> cs\<^bsup>\<pi>' \<guillemotleft> k'\<^esup> n'"
hom_mult: fixes x :: "'a"   and y :: "'a" shows "to_fract (x * y) = to_fract x * to_fract y"
add_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b"   and "less_eq c d" shows "less_eq (plus a c) (plus b d)"
legal_subst_zpassign: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A"   and "ground_heads.wary_subst arity_sym arity_var ground_heads_var \<rho>" shows "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.subst_zpassign ground_heads_var \<delta> arity_sym wt_sym coef_sym \<rho> A)"
eval_fds_altdef: fixes f :: "'a fds"   and s :: "'a" assumes "fds_abs_converges f s" shows "eval_fds f s = (\<Sum>\<^sub>an. fds_nth f n / nat_power n s)"
quot_fm_fresh: fixes s :: "atom"   and A :: "fm" shows "s \<sharp> \<guillemotleft>A\<guillemotright>"
rPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P P' A\<^sub>P \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (P' \<parallel> Q) (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
AE_space: fixes M :: "'c measure" shows "AE x in M. x \<in> space M"
make_affine_infinity: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "field R" shows "field.make_affine R (x, y, \<zero>\<^bsub>R\<^esub>) = Infinity"
a_eq: shows "a = [mod_type_class.to_nat i..<IArray.length (vec_to_iarray A)] ! ia"
minus_plus_minus_cancel: fixes u :: "'a"   and t :: "'a"   and s :: "'a" assumes "u adds t"   and "s adds u" shows "t - u + (u - s) = t - s"
ideD_1: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and a :: "'ag" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.arr C a"
simps_2: fixes OP :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b"   and r :: "'b" shows "rexp_of_list OP N [r] = r"
to_hma\<^sub>m_def: fixes a :: "'d mat" shows "HMA_Connect.to_hma\<^sub>m a = (\<chi>i j. a $$ (Bij_Nat.to_nat i, Bij_Nat.to_nat j))"
wset_thread_okD: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'d" assumes "wset_thread_ok ws ts"   and "ts t = None" shows "ws t = None"
fresh_ty: fixes a :: "name"   and \<tau> :: "ty" shows "a \<sharp> \<tau>"
intro: fixes M :: "'b measure"   and F :: "'b measure" assumes "subalgebra M F"   and "sigma_finite_measure (restr_to_subalg M F)" shows "sigma_finite_subalgebra M F"
conv_dist_comp: fixes x :: "'a"   and y :: "'a" shows "(x * y)\<^sup>T = y[[x\<^sup>T]]"
lm099: fixes aa :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set"   and bids :: "'a \<times> 'b set \<Rightarrow> nat" assumes "aa \<in> allAllocations N G" shows "sum (summedBidVector (real \<circ> bids) N G) aa = real (sum (summedBidVector bids N G) aa)"
wls_swap_preserves_fresh_distinct: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and x :: "'var"   and y1 :: "'var"   and y2 :: "'var" assumes "wls s X"   and "xs \<noteq> ys \<or> x \<notin> {y1, y2}" shows "fresh xs x (X #[y1 \<and> y2]_ys) = fresh xs x X"
E_2: fixes n :: "nat" shows "A \<inter> C n \<subseteq> E n"
is_binqueue_simp_1: fixes l :: "nat" shows "is_binqueue l [] = True"
cnf_semantics_list_def: fixes \<A> :: "'a \<Rightarrow> bool"   and S :: "'a literal list list" shows "cnf_semantics_list \<A> S = (\<forall>s\<in>set S. \<exists>l\<in>set s. lit_semantics \<A> l)"