hcju/isabelleps huggingface dataset:
homotopic_loops_trans: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b"   and r :: "real \<Rightarrow> 'b" assumes "homotopic_loops s p q"   and "homotopic_loops s q r" shows "homotopic_loops s p r"
C': shows "Collect \<phi>.A.arr = {0..<n} \<and> invertible_functor C' (\<cdot>) (\<lambda>i. if \<phi>.A.arr i then \<phi> i else null)"
equivclp_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "equivclp r = (symclp r)\<^sup>*\<^sup>*"
image_mset_subseteq_mono: fixes A :: "'c multiset"   and B :: "'c multiset"   and f :: "'c \<Rightarrow> 'd" assumes "A \<subseteq># B" shows "image_mset f A \<subseteq># image_mset f B"
lsum_list_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lfinite xs"   and "lfinite ys" shows "lsum_list (lappend xs ys) = lsum_list xs + lsum_list ys"
flipped_reflections_odd_lconjseq: fixes S :: "'b set"   and ss :: "'b list" assumes "CoxeterSystem S"   and "ss \<in> lists S" shows "CoxeterSystem.flipped_reflections S (sum_list ss) = {t \<in> \<Union>w\<in>\<langle>S\<rangle>. lconjby w ` S. \<not> gcd_nat.greater_eq (count_list (lconjseq ss) t) 2}"
assms_2: shows "A = P\<^sup>-\<^sup>1 ** diag_mat f ** P"
interleave_append: fixes s :: "'a event list"   and t1 :: "'a event list"   and t2 :: "'a event list"   and C :: "'a event set"   and u :: "'a event list" assumes "s setinterleaves ((t1 @ t2, u), C)" shows "\<exists>u1 u2 s1 s2. u = u1 @ u2 \<and> s = s1 @ s2 \<and> s1 setinterleaves ((t1, u1), C) \<and> s2 setinterleaves ((t2, u2), C)"
less_1: fixes y :: "'s" assumes "(y, s) \<in> R"   and "P y"   and "y \<in> S1" shows "0 < n y"
c': shows "\<forall>x y. T (i (i y x) (i (n x) (n y)))"
proj_scaleR: fixes a :: "real"   and x :: "'a" assumes "0 < a" shows "proj (a *\<^sub>R x) = proj x"
fsafeD: shows "fsafe (foldr (||) \<Delta> FF)"
rel_sel: fixes R :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and a :: "'l fundef"   and b :: "'m fundef" shows "rel_fundef R a b = (list_all2 R (body a) (body b) \<and> arity a = arity b)"
complete_iff_size: fixes kdt :: "'a kdt" shows "KD_Tree.complete kdt = (KD_Tree.size_kdt kdt = 2 ^ height kdt)"
le_onorm: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "bounded_linear f" shows "norm (f x) / norm x \<le> onorm f"
vmc_path_axioms: shows "vmc_path G P' (P $ n) p \<sigma>'"
f_def: shows "f = (\<lambda>d. mangoldt d / real d)"
d_2: fixes x' :: "'d" assumes "x' \<in> T"   and "dist x' t < d" shows "dist (g x') (g t) < u"
cont: fixes I :: "nat set"   and i :: "nat"   and j :: "nat" assumes "I \<in> P"   and "i \<in> I"   and "j \<in> F I" shows "local.cont (fst cf1) (snd cf1) i \<approx>s local.cont (fst cf2) (snd cf2) j"
real_int_of_real: fixes x :: "real" assumes "real_is_int x" shows "real_of_int (int_of_real x) = x"
constr_rep_1: fixes x3 :: "subject"   and x1 :: "name"   and x2 :: "pi" shows "Rep_residual (x3\<guillemotleft>x1\<guillemotright> \<prec> x2) = BoundR_Rep x3 ([x1].x2)"
longest_Lyndon_suf_Lyndon_id: fixes w :: "'a list" assumes "Lyndon w" shows "LynSuf w = w"
UnOp: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and A :: "assigned"   and unop :: "unop" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>UnOp unop e\<rangle>\<guillemotright> A"
g_def: fixes mrk :: "'node set"   and lbl :: "'a \<Rightarrow> Index"   and ptr :: "'a \<Rightarrow> 'node"   and x :: "'a" shows "g mrk lbl ptr x \<equiv> ptr x \<noteq> nil \<and> ptr x \<notin> mrk \<and> lbl x = none"
mat_of_cblinfun_def: fixes f :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "mat_of_cblinfun f = mat (length canonical_basis) (length canonical_basis) (\<lambda>(i, j). crepresentation (set canonical_basis) (f *\<^sub>V canonical_basis ! j) (canonical_basis ! i))"
minGraphProps8a': fixes g :: "graph"   and v :: "nat" assumes "minGraphProps g"   and "v < countVertices g" shows "distinct (normFaces (faceListAt g ! v))"
convex_halfspace_intersection: fixes S :: "'b set" assumes "closed S"   and "convex S" shows "S = \<Inter> {h. S \<subseteq> h \<and> (\<exists>a b. h = {x. a \<bullet> x \<le> b})}"
wpo_ns_mono: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term"   and f :: "'f"   and bef :: "('f, 'v) Term.term list"   and aft :: "('f, 'v) Term.term list" assumes "s \<succeq> t" shows "Fun f (bef @ s # aft) \<succeq> Fun f (bef @ t # aft)"
swap_commute: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and zs' :: "'varSort"   and x :: "'var"   and y :: "'var"   and x' :: "'var"   and y' :: "'var" assumes "good X"   and "zs \<noteq> zs' \<or> {x, y} \<inter> {x', y'} = {}" shows "X #[x \<and> y]_zs #[x' \<and> y']_zs' = X #[x' \<and> y']_zs' #[x \<and> y]_zs"
emeasure_erlang_density: fixes l :: "real"   and k :: "nat"   and a :: "real" assumes "0 < l" shows "emeasure (density lborel (\<lambda>x. ennreal (erlang_density k l x))) {..a} = ennreal (erlang_CDF k l a)"
image_empty: fixes f :: "'d \<Rightarrow> 'c" shows "f ` {} = {}"
add_ac_2: fixes a :: "'g"   and b :: "'g" shows "a + b = b + a"
has_derivative_def: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "D f \<mapsto> f' F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (netlimit F) - f' (y - netlimit F)) /\<^sub>R (\<parallel>y - netlimit F\<parallel>)) \<longlongrightarrow> (0::'d)) F)"
list_to_nat_def: shows "list_to_nat = (\<lambda>ls. if ls = [] then 0 else c_pair (length ls - 1) (c_fold ls) + 1)"
lm015: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G"   and "finite G" shows "card G = sum card (Range a)"
matrix_to_iarray_Gauss_Jordan_in_ij: fixes i :: "'c"   and j :: "'d"   and A :: "(('b, 'd) vec, 'c) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))" shows "matrix_to_iarray (Gauss_Jordan_in_ij A i j) = Gauss_Jordan_in_ij_iarrays (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j)"
pm_pth_def: fixes m :: "('a, 'b) pointermap_scheme"   and p :: "nat" shows "pm_pth m p \<equiv> entries m ! p"
less_eq_multiset\<^sub>H\<^sub>O: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(M \<le> N) = (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x))"
reduced_run_invisible: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and init :: "'b \<Rightarrow> bool"   and int :: "'b \<Rightarrow> 'c"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and src :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ren :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b"   and v\<^sub>1 :: "'a list"   and v\<^sub>2 :: "'a stream"   and l :: "'a list"   and w :: "'a list"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and u :: "'a stream" assumes "ample_abstract ex en init int ind src ren"   and "ample_abstract.reduced_run ex en int ind ren q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "set w\<^sub>2 \<subseteq> - transition_system_interpreted.visible ex en int"
InvariantImpliedLiteralsHoldsForUnitLiteral: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and uClause :: "Literal list"   and uLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantImpliedLiterals F M"   and "formulaEntailsClause F uClause"   and "isUnitClause uClause uLiteral (elements M)"   and "M' = M @ [(uLiteral, False)]" shows "formulaEntailsLiteral (F @ val2form (decisionsTo uLiteral M')) uLiteral"
eqv: shows "int (\<Sum>\<^sub># (image_mset card S)) = \<v>"
frac_gt_0_iff: fixes x :: "'b" shows "((0::'b) < frac x) = (x \<notin> \<int>)"
span_eq_iff: fixes s :: "('a \<Rightarrow> real) set" assumes "s \<subseteq> diff_fun_space" shows "(sub.diff_fun_space.m1.span s = s) = sub.diff_fun_space.m1.subspace s"
cases: fixes x :: "'b formula" obtains k :: "'b" where "(x::'b formula) = Atom (k::'b)"   | "(x::'b formula) = \<bottom>"   | F :: "'b formula" where "(x::'b formula) = \<^bold>\<not> (Fa__::'b formula)"   | F :: "'b formula" and G :: "'b formula" where "(x::'b formula) = (Fb__::'b formula) \<^bold>\<and> (G__::'b formula)"   | F :: "'b formula" and G :: "'b formula" where "(x::'b formula) = (Fc__::'b formula) \<^bold>\<or> (Ga__::'b formula)"   | F :: "'b formula" and G :: "'b formula" where "(x::'b formula) = (F::'b formula) \<^bold>\<rightarrow> (G::'b formula)"
is_type_int: fixes P :: "'a prog" shows "is_type P Integer = True"
map_gpv_Fail: fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e" shows "map_gpv f g Fail = Fail"
lunit_char: fixes a :: "'a" assumes "ide a" shows "\<I> \<otimes> \<l>[a] = (\<iota> \<otimes> a) \<cdot> MC.assoc' MC.unity MC.unity a"
vcard_vlepoll: fixes A :: "V"   and B :: "V" shows "(vcard A \<subseteq>\<^sub>\<circ> vcard B) = (A \<lesssim>\<^sub>\<circ> B)"
Id_on_iff: fixes x :: "'d"   and y :: "'d"   and A :: "'d set" shows "((x, y) \<in> Id_on A) = (x = y \<and> x \<in> A)"
ccpo_Sup_upper: fixes A :: "'a set"   and x :: "'a" assumes "Complete_Partial_Order.chain (\<le>) A"   and "x \<in> A" shows "x \<le> \<Squnion> A"
set_eq_subset: fixes A :: "'g set"   and B :: "'g set" shows "(A = B) = (A \<subseteq> B \<and> B \<subseteq> A)"
read_tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'a"   and \<O> :: "'b" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>read_tmps sb. i < t"
dmu_array_row_def: fixes fs :: "int vec iarray"   and dmus :: "int iarray iarray"   and i :: "nat" shows "dmu_array_row fs dmus i = (let fi = fs !! i in dmu_array_row_main fs fi i (iarray_append dmus (IArray [fi \<bullet> fs !! 0])) 0)"
one_zero_def: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "local.one_zero A i j \<equiv> (Suc i < j \<longrightarrow> A $$ (i, Suc i) = (1::'a) \<longrightarrow> A $$ (i, j) = (0::'a)) \<and> (j < i \<longrightarrow> A $$ (i, j) = (0::'a)) \<and> (i < j \<longrightarrow> A $$ (i, i) \<noteq> A $$ (j, j) \<longrightarrow> A $$ (i, j) = (0::'a))"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_tag_name type_wf set_tag_name set_tag_name_locs"   and "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_set_tag_name_get_child_nodes_axioms set_tag_name_locs get_child_nodes_locs" shows "l_set_tag_name_get_child_nodes type_wf set_tag_name set_tag_name_locs known_ptr get_child_nodes get_child_nodes_locs"
eq: shows "(\<lambda>x. (X x, X x, Y x)) ` space M = (\<lambda>(x, y). (x, x, y)) ` (\<lambda>x. (X x, Y x)) ` space M"
quad_form_scale_m: fixes v :: "complex \<times> complex"   and k :: "complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "quad_form v (k *\<^sub>s\<^sub>m H) = k * quad_form v H"
coinduct: fixes X :: "pi \<Rightarrow> pi \<Rightarrow> bool"   and x1 :: "pi"   and x2 :: "pi" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>P Q. x1 = P \<and> x2 = Q \<and> P \<leadsto><{(xa, x). X xa x \<or> xa \<approx> x}> Q \<and> (X Q P \<or> Q \<approx> P)" shows "x1 \<approx> x2"
mor_image1': fixes A :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B :: "'g set"   and B2' :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and x :: "'d" assumes "mor A B2 s1 s2 B B2' s1' s2' f g"   and "x \<in> A" shows "f x \<in> B"
par_def: fixes ioa1 :: "('d set \<times> 'd set \<times> 'd set) \<times> 'e set \<times> ('e \<times> 'd \<times> 'e) set"   and ioa2 :: "('d set \<times> 'd set \<times> 'd set) \<times> 'f set \<times> ('f \<times> 'd \<times> 'f) set" shows "ioa1 || ioa2 \<equiv> (asig_comp (asig_of ioa1) (asig_of ioa2), {pr. fst pr \<in> starts_of ioa1 \<and> snd pr \<in> starts_of ioa2}, {tr. let s = fst tr; a = fst (snd tr); t = snd (snd tr) in (a \<in> actions (asig_of ioa1) \<or> a \<in> actions (asig_of ioa2)) \<and> (if a \<in> actions (asig_of ioa1) then (fst s, a, fst t) \<in> trans_of ioa1 else fst t = fst s) \<and> (if a \<in> actions (asig_of ioa2) then (snd s, a, snd t) \<in> trans_of ioa2 else snd t = snd s)})"
cong: fixes p :: "'b poly" shows "monic_factorization.A p \<equiv> monic_factorization.A p"
Objectmheads_def: fixes G :: "prog"   and S :: "qtname" shows "Objectmheads G S = (\<lambda>sig. (\<lambda>(Cls, mthd). (ClassT Cls, mhead mthd)) ` set_option (filter_tab (\<lambda>sig m. accmodi m \<noteq> Private) (accmethd G S Object) sig))"
\<tau>instr_stk_append_check: fixes i :: "'addr instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'addr frame list"   and vs :: "'addr val list" assumes "check_instr' i P h stk loc C M pc frs" shows "\<tau>instr P h (stk @ vs) i = \<tau>instr P h stk i"
simps_8: fixes c :: "int"   and e :: "Cooper.num" shows "\<beta> (Ge (CN 0 c e)) = [Sub (C (- 1)) e]"
is_red_implies_0_red_dgrad_p_set': fixes d :: "'a \<Rightarrow> nat"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "B \<subseteq> dgrad_p_set d m"   and "pmdl B \<subseteq> pmdl A"   and "\<And>q. \<lbrakk>q \<in> pmdl A; q \<noteq> 0\<rbrakk> \<Longrightarrow> is_red B q"   and "p \<in> pmdl A" shows "(red B)\<^sup>*\<^sup>* p 0"
functor_axioms: shows "functor (\<cdot>) CC.comp Diag"
simps: fixes a :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) \<times> (nat \<times> ('sf, 'sc, 'sz) step) list" shows "proof_ok a = (\<exists>D S. a = (D, S) \<and> deriv_ok (start_proof D) S)"
distinct_28: fixes ccs' :: "ccs"   and ccs1 :: "ccs"   and ccs2 :: "ccs" shows "!ccs' \<noteq> ccs1 \<parallel> ccs2"
while_option_commute_invariant: fixes P :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and b' :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and c' :: "'b \<Rightarrow> 'b"   and s :: "'a" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "\<And>s. P s \<Longrightarrow> b s = b' (f s)"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> f (c s) = c' (f s)"   and "P s" shows "map_option f (while_option b c s) = while_option b' c' (f s)"
cases_scheme: fixes r :: "('c, 'd) monoid_scheme" obtains carrier :: "'c set" and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" and one :: "'c" and more :: "'d" where "r = \<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>"
le_utrans_antisym: fixes x :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and y :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "le_utrans x y"   and "le_utrans y x" shows "equiv_utrans x y"
Bossy''_CH_BCh': fixes X :: "'a set" shows "Bossy''.CH X = BCh' (Xh d1h1) X \<union> BCh' (Xh d1h2) X \<union> BCh' (Xh d1h3) X"
lu_props_5: fixes x :: "real" assumes "x \<le> l" shows "sgn (poly p x) = sgn (poly p l)"
lex_mirror_point: fixes x0 :: "real \<times> real"   and a :: "real \<times> real"   and b :: "real \<times> real" assumes "Counterclockwise_2D_Arbitrary.lex (mirror_point x0 a) (mirror_point x0 b)" shows "Counterclockwise_2D_Arbitrary.lex b a"
orthogonal_matrix_exists_basis: fixes a :: "(real, 'a) vec"   and k :: "'a" assumes "norm a = 1" obtains A :: "((real, 'a) vec, 'a) vec" where "orthogonal_matrix A" and "A *v axis k 1 = a"
ih_9: shows "InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
map_scale_eq_times: fixes c :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "c \<cdot> p = monomial c (0::'b) * p"
GSMP_Union: fixes A :: "('a strand_label \<times> ('fun, 'var) strand_step) list" shows "GSMP (trms\<^sub>l\<^sub>s\<^sub>t A) = (\<Union>l. GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l A)))"
assms_3: shows "is_Groebner_basis (fst ` set gs)"
hom_0_iff: fixes x :: "int" shows "(of_int x = (0::'a)) = (x = 0)"
a: shows " N \<longrightarrow>\<^sub>1 N'"
content_eq_L: fixes A :: "'a set" assumes "A \<in> sets borel"   and "A \<subseteq> cbox x y" shows "content A = Sigma_Algebra.measure (lebesgue_on (cbox x y)) A"
TR_submodule: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "submodule R P (TR\<^bsub>R\<^esub> M N P)"
r_def: shows "r = (w - z) / (\<gamma> 0 - z)"
bounded_arity_def: fixes b :: "nat"   and F :: "('c \<times> nat) set" shows "bounded_arity b F = (\<forall>(f, n)\<in>F. n \<le> b)"
add_le_same_cancel2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus a b) b = less_eq a zero"
Explanation_expl: shows "\<lfloor>\<lambda>w. universal (\<lambda>x xa. (xa explains x) w = (x dependsOn xa) w)\<rfloor>"
assms_3: shows "X \<in> obj\<^bsub>C\<^esub>"
less_bit0_def: fixes a :: "'b bit0"   and b :: "'b bit0" shows "(a < b) = (Rep_bit0 a < Rep_bit0 b)"
assms_4: shows "frac (a + d) \<le> frac (b + d)"
next_sibling_pure: fixes new_child :: "(_) node_ptr"   and h :: "(_) heap" shows "pure (local.a_next_sibling new_child) h"
a_rcos_const: fixes h :: "('c multiset \<Rightarrow> 'a) list" assumes "h \<in> PIdl\<^bsub>poly_ring L\<^esub> q" shows "L.rupture_surj (carrier L) q h = PIdl\<^bsub>poly_ring L\<^esub> q"
delay_bisimulation: shows "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"
calculus_axioms: shows "calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) empty_ord.Red_I Red_F_\<G>_empty"
church_rosser_var: fixes y :: "'a"   and x :: "'a" assumes "y ; x\<^sup>\<star> \<le> x\<^sup>\<star> ; y\<^sup>\<star>" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> ; y\<^sup>\<star>"
ntrunc_Scons: fixes k :: "nat"   and M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" shows "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"
in_set_insert: fixes x :: "'g"   and xs :: "'g list" assumes "x \<in> set xs" shows "List.insert x xs = xs"
diff_eq: fixes x :: "'a"   and y :: "'a" shows "x - y = inf_class.inf x (- y)"
impl_of_RBT_HM_invar: fixes hm :: "('a, 'b) hashmap" shows "HashMap_Impl.invar (impl_of_RBT_HM hm)"
set_mode_locs_def: shows "set_mode_locs \<equiv> l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_mode_locs"
closed_Int: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T" shows "closed (S \<inter> T)"
mkeps_flat: fixes r :: "'b rexp" assumes "nullable r" shows "flat (mkeps r) = []"
length_tensor_vec_from_lookup: fixes ds :: "nat list"   and e :: "nat list \<Rightarrow> 'a" shows "length (tensor_vec_from_lookup ds e) = prod_list ds"
poly_roots_degree: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "card {x. poly p x = (0::'a)} \<le> degree p"
b: shows "(x, T\<^sub>1) \<in> set \<Gamma>"
tau_ret: fixes x :: "'a" shows "\<tau> (\<tau> x) = \<tau> x"
le_def: fixes r\<^sub>A :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and r\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Product.le r\<^sub>A r\<^sub>B = (\<lambda>(a\<^sub>1, b\<^sub>1) (a\<^sub>2, b\<^sub>2). a\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>A\<^esub> a\<^sub>2 \<and> b\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>B\<^esub> b\<^sub>2)"
dominates_unsnoc: fixes g :: "'g"   and n :: "'node"   and m :: "'node"   and m' :: "'node" assumes "invar g"   and "old.dominates g n m"   and "m' \<in> set (old.predecessors g m)"   and "n \<noteq> m" shows "old.dominates g n m'"
span_span: fixes A :: "'a set" assumes "A \<subseteq> S" shows "span (span A) = span A"
H_star_inv: fixes p :: "'a"   and i :: "'a"   and x :: "'a"   and q :: "'a" assumes "\<tt>\<tt> p \<le> \<tt>\<tt> i"   and "H i x i"   and "\<tt>\<tt> i \<le> \<tt>\<tt> q" shows "H p (x\<^sup>\<star>) q"
map_key_compose: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and p :: "'c \<Rightarrow>\<^sub>0 'd" assumes "inj f"   and "inj g" shows "Poly_Mapping.map_key f (Poly_Mapping.map_key g p) = Poly_Mapping.map_key (g \<circ> f) p"
ideD_2: fixes a :: "'s" assumes "S.ide a" shows "S.dom a = a"
relcomp_Image: fixes X :: "('c \<times> 'd) set"   and Y :: "('d \<times> 'b) set"   and Z :: "'c set" shows "(X O Y) `` Z = Y `` X `` Z"
sublens'_implies_sublens: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "vwb_lens X"   and "vwb_lens Y"   and "X \<subseteq>\<^sub>L' Y" shows "X \<subseteq>\<^sub>L Y"
ord_of_nat_zero: shows "0\<^sub>\<nat> = 0"
ipv6_unparsed_compressed_to_preferred_def: fixes ls :: "16 word option list" shows "ipv6_unparsed_compressed_to_preferred ls = (if length (filter (\<lambda>p. p = None) ls) \<noteq> 1 \<or> 7 < length (filter (\<lambda>p. p \<noteq> None) ls) then None else let before_omission = map the (takeWhile (\<lambda>x. x \<noteq> None) ls); after_omission = map the (drop 1 (dropWhile (\<lambda>x. x \<noteq> None) ls)); num_omissions = 8 - (length before_omission + length after_omission); expanded = before_omission @ replicate num_omissions 0 @ after_omission in case expanded of [] \<Rightarrow> None | [a] \<Rightarrow> None | [a, b] \<Rightarrow> None | [a, b, c] \<Rightarrow> None | [a, b, c, d] \<Rightarrow> None | [a, b, c, d, e] \<Rightarrow> None | [a, b, c, d, e, f] \<Rightarrow> None | [a, b, c, d, e, f, g] \<Rightarrow> None | [a, b, c, d, e, f, g, h] \<Rightarrow> Some (IPv6AddrPreferred a b c d e f g h) | a # b # c # d # e # f # g # h # ai # x \<Rightarrow> Map.empty x)"
lesspoll_eq_trans: fixes X :: "'a set"   and Y :: "'b set"   and Z :: "'c set" assumes "X \<prec> Y"   and "Y \<approx> Z" shows "X \<prec> Z"
c1: shows "c $ 1 \<noteq> (0::'a)"
set_of_insert_mod: fixes x :: "'a"   and s :: "'s" assumes "I x"   and "S s \<and> (\<forall>x\<in>set_of s. I x)" shows "set_of (insert_mod x s) = insert x (set_of s) \<or> (\<exists>y\<in>set_of s. x \<preceq> y) \<and> set_of (insert_mod x s) = set_of s"
not_choice_def: fixes p :: "('e, 'f, 'g, 'h) seqp" shows "not_choice p \<equiv> \<forall>p1 p2. p \<noteq> p1 \<oplus> p2"
div_pCons_eq: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "pCons a p div q = (if q = 0 then 0 else pCons (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) (p div q))"
wellformed_\<J>_0_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.wellformed_items \<RR> Doc (LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc 0 0)"
adds_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a adds b"   and "b adds c" shows "a adds c"