hcju/isabelleps huggingface dataset:
ext_hd_or_tl_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_hd_or_tl ext \<equiv> ext ext \<and> ext_hd_or_tl_axioms ext"
order_iff: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a = a \<sqinter> b)"
distinct_rev: fixes xs :: "'g list" shows "distinct (rev xs) = distinct xs"
finsum_infinite: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a vec" assumes "infinite A" shows "finsum V f A = 0\<^sub>v n"
abs_benv_def: fixes \<beta> :: "nat \<Rightarrow> contour option" shows "|\<beta>| = map_option AbsCFCorrect.abs \<circ> \<beta>"
trms\<^sub>s\<^sub>s\<^sub>t_mono: fixes M :: "('a, 'b) stateful_strand_step list"   and N :: "('a, 'b) stateful_strand_step list" assumes "set M \<subseteq> set N" shows "trms\<^sub>s\<^sub>s\<^sub>t M \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t N"
of_complex_def: fixes c :: "complex" shows "of_complex c = c *\<^sub>C (1::'b)"
trivial_imp_no_distinct: fixes X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "trivial X"   and "x \<in> X"   and "y \<in> X" shows "x = y"
preorder_on_axioms: shows "Order_Predicates.preorder_on alts (Pareto R)"
well_shaped_lr: fixes a :: "'b tree" assumes "well_shaped a" shows "l a \<le> r a"
meet_idem: fixes x :: "('a \<times> 'a set) set" shows "x \<inter> x = x"
less_eq_introp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- (x \<squnion> - (y \<squnion> z)) = - (x \<squnion> y \<squnion> - z)" shows "y \<sqsubseteq> x"
insertion_sum_var: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b mpoly"   and n :: "nat" shows "insertion f (sum g {0..<n}) = (\<Sum>i = 0..<n. insertion f (g i))"
BackwardSlice_wf_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and sem :: "'f \<Rightarrow> 'c \<Rightarrow> 'f \<Rightarrow> 'c \<Rightarrow> bool"   and identifies :: "'b \<Rightarrow> 'f \<Rightarrow> bool" shows "BackwardSlice_wf sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice sem identifies \<equiv> BackwardSlice sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice \<and> CFG_semantics_wf sourcenode targetnode kind valid_edge Entry sem identifies"
eq1: shows "Q $$ (0, 0) * Q $$ (1, 1) * a + Q $$ (1, 0) * - Q $$ (0, 1) * a = u * a"
next_subset_dfs: fixes xs :: "node list"   and g :: "(node \<times> node) list"   and ys :: "node list" shows "set xs \<subseteq> set (dfs g xs ys)"
star_n_one_num: shows "1 = star_n (\<lambda>n. 1::'b)"
sp_from_old_verts_imp_sp_in_old: fixes e :: "('a \<times> nat) edge"   and g' :: "('a \<times> nat, 'b) rgraph_scheme"   and v1 :: "'a \<times> nat"   and v2 :: "'a \<times> nat"   and es :: "('a \<times> nat) edge list" assumes "extends g e g'"   and "v1 \<in> Graph.vertices g"   and "v2 \<in> Graph.vertices g"   and "SubRel.subpath g' v1 es v2 subs" shows "SubRel.subpath g v1 es v2 subs"
d_def: shows "d = \<Sqinter> {\<parallel>x\<parallel>\<^sup>2 |x. x \<in> M}"
insert_iff: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
Nadd_normN: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "isnormNum (x +\<^sub>N y)"
homotopic_with_linear: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and t :: "'b set" assumes "continuous_on S f"   and "continuous_on S g"   and "\<And>x. x \<in> S \<Longrightarrow> closed_segment (f x) (g x) \<subseteq> t" shows "homotopic_with_canon (\<lambda>z. True) S t f g"
fds_converges_deriv_aux: fixes f :: "'a fds"   and s0 :: "'a"   and s :: "'a" assumes "fds_converges f s0"   and "s0 \<bullet> (1::'a) < s \<bullet> (1::'a)" shows "fds_converges (fds_deriv f) s"
simps_9: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and x :: "String.literal"   and p :: "fol fm" shows "semantics e f g (Exists x p) = (\<exists>v. semantics (e(x := v)) f g p)"
l_comb_mem: fixes A :: "'b set"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "m \<in> {j. j \<le> n} \<rightarrow> H" shows "l_comb R M n s m \<in> carrier M"
undo_normalize: fixes a :: "'a poly" shows "a = Polynomial.smult (unit_factor (lead_coeff a)) (normalize a)"
init_config_def: fixes c :: "('c, 'd) configuration" shows "init_config c = ((\<forall>p. c_temp c p = {#}\<^sub>z) \<and> (\<forall>p1 p2. c_msg c p1 p2 = []) \<and> (\<forall>p. c_glob c p = c_records c) \<and> (\<forall>t. 0 \<le> zcount (c_records c) t))"
thetaWhileW0_RetrT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaWhileW0 tval aval indis \<subseteq> PL_Indis.RetrT tval aval indis (PL_Indis.thetaWhileW0 tval aval indis \<union> PL_Indis.BisT tval aval indis)"
is_Comp_def: fixes "term" :: "'b local.term" shows "is_Comp term = (\<exists>x41 x42. term = x41 \<^bold>\<cdot> x42)"
intros_3: fixes s :: "'a"   and b :: "'a set"   and c1 :: "'a com"   and s' :: "'a"   and c2 :: "'a com" assumes "s \<in> b"   and "Sem c1 s s'" shows "Sem (Cond b c1 c2) s s'"
det_def: fixes A :: "(('b, 'c) vec, 'c) vec" shows "det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. A $ i $ p i))"
a: shows "\<forall>i\<in>dom \<gamma>. \<forall>j\<in>dom \<gamma>. i \<noteq> j \<longrightarrow> W (the (\<gamma> i)) \<inter> W (the (\<gamma> j)) = {}"
not_in_interior_convex_hull_3_2: fixes b :: "complex"   and a :: "complex"   and c :: "complex" shows "b \<notin> interior (convex hull {a, b, c})"
pkt_def: shows "pkt = \<lparr>p_iiface = ''+'', p_oiface = ''+'', p_src = 0, p_dst = 0, p_proto = TCP, p_sport = 0, p_dport = 0, p_tcp_flags = {TCP_SYN}, p_payload = [], p_tag_ctstate = CT_New\<rparr>"
either_or: fixes r :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "\<exists>y'>r. \<forall>x\<in>{r<..y'}. aEvalUni (EqUni (a, b, c)) x \<or> aEvalUni (LessUni (a, b, c)) x" shows "(\<exists>y'>r. \<forall>x\<in>{r<..y'}. aEvalUni (EqUni (a, b, c)) x) \<or> (\<exists>y'>r. \<forall>x\<in>{r<..y'}. aEvalUni (LessUni (a, b, c)) x)"
applyLearn_def: fixes state :: "State" shows "applyLearn state = (if getC state = [opposite (getCl state)] then state else let state' = state\<lparr>getF := getF state @ [getC state]\<rparr>; l = getCl state; ll = getLastAssertedLiteral (removeAll l (oppositeLiteralList (getC state))) (elements (getM state)); clauseIndex = length (getF state); state'' = setWatch1 clauseIndex (opposite l) state' in Let (setWatch2 clauseIndex (opposite ll) state'') (getCll_update (\<lambda>_. ll)))"
coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> c" shows "min a b \<le> c"
function_ring_is_comm_monoid: shows "Group.comm_monoid F"
compositionality: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and multiset :: "'e multiset" shows "image_mset f (image_mset g multiset) = image_mset (f \<circ> g) multiset"
VV_equivalence: fixes v :: "'a"   and p :: "Player" assumes "v \<in> V\<^bsub>G'\<^esub>" shows "(v \<notin> G'.VV p) = (v \<in> G'.VV p**)"
lessThan_def: fixes u :: "'a word" shows "signed.lessThan u \<equiv> {x. x <s u}"
cSUP_subset_mono: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd multiset"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_above (g ` B)"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># g x" shows "Sup (f ` A) \<subseteq># Sup (g ` B)"
"1_4": shows "(WHILE b_ DO c_, s2_) \<Rightarrow> y_ \<Down> s3_"
existence_ivl_initial_time_iff: fixes x0 :: "'a" shows "(t0 \<in> existence_ivl t0 x0) = (t0 \<in> T \<and> x0 \<in> X)"
elims_3: fixes x :: "'a val" assumes "\<not> is_Bool x" obtains "x = Unit"   | "x = Null"   | v :: "32 word" where "x = Intg v__"   | v :: "'a" where "x = Addr v"
inject: fixes x1 :: "nat"   and y1 :: "nat" shows "(pnat x1 = pnat y1) = (x1 = y1)"
szx: fixes x :: "'a" assumes "x \<in> S" shows "a \<bullet> z \<le> a \<bullet> x"
Open: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>a[b] \<prec> P'"   and "a \<noteq> b" shows "<\<nu>b>P \<Longrightarrow>a<\<nu>b> \<prec> P'"
ereal_diff_add_inverse: fixes x :: "ereal"   and y :: "ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "x + y - x = y"
assms_1: shows "\<bar>m\<bar> \<le> 2 ^ (2 * n)"
complex_homo_coords_eq_reflp: shows "reflp (\<approx>)"
cindependent_cfinite_dim_finite: fixes S :: "'a set" assumes "cindependent S" shows "finite S"
lcm_gcd_prod: fixes a :: "'a"   and b :: "'a" shows "lcm a b * gcd a b = normalize (a * b)"
homogeneous_idealE: fixes F :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "p \<in> ideal F" obtains F' :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" and q :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" where "finite F'" and "F' \<subseteq> F" and "p = (\<Sum>f\<in>F'. q f * f)" and "\<And>f. poly_deg (q f * f) \<le> poly_deg p" and "\<And>f. f \<notin> F' \<Longrightarrow> q f = 0"
simps: fixes st :: "nat"   and ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_halt_case_inv st ires rs (l, r) = (if st = 0 then wcode_stop ires rs (l, r) else if st = Suc 0 then wcode_on_left_moving_3 ires rs (l, r) else if st = Suc (Suc 0) then wcode_on_checking_3 ires rs (l, r) else if st = 7 then wcode_goon_checking_3 ires rs (l, r) else False)"
IH: shows "gate (m * n) (A \<otimes>\<^bsup>n\<^esup>)"
inject_1: fixes x1 :: "('a, 'b) hd"   and y1 :: "('a, 'b) hd" shows "(Hd x1 = Hd y1) = (x1 = y1)"
Kdelta_in_Zinf1: fixes j :: "nat"   and n :: "nat"   and k :: "nat" assumes "j \<le> n"   and "k \<le> n" shows "\<delta>\<^bsub>j k\<^esub> \<in> Z\<^sub>\<infinity>"
ih: shows "\<exists>xs. chain R (llist_of (x # xs @ [y]))"
unity_def: shows "\<I>\<^sub>C \<equiv> C.cod \<iota>\<^sub>C"
IH: fixes s :: "('a, 'b) fmap"   and p :: "('a, 'b) fmap list" assumes "statelist' s as = p" shows "\<forall>n\<le>length as. exec_plan s (take n as) = p ! n"
complex_roots_complex_prod: fixes f :: "complex poly"   and g :: "complex poly" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "mset (complex_roots_complex (f * g)) = mset (complex_roots_complex f) + mset (complex_roots_complex g)"
MWLsSteps_det_cases_1: fixes \<iota> :: "nat"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "\<langle>skip\<^bsub>\<iota>\<^esub>,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m'\<rangle>" obtains "\<alpha> = []" and "p = None" and "m' = m"
lunit_def: fixes f :: "'a" shows "\<l>[f] \<equiv> left_hom_with_unit.lunit (\<cdot>) (\<star>) \<a> \<i>[trg f] (trg f) f"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier L; y \<in> carrier L\<rbrakk> \<Longrightarrow> \<exists>s. is_glb L s {x, y}" shows "weak_lower_semilattice_axioms L"
convex_finite: fixes S :: "'a set" assumes "finite S" shows "convex S = (\<forall>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<longrightarrow> (\<Sum>x\<in>S. u x *\<^sub>R x) \<in> S)"
det_mul: fixes A :: "(('c, 'd) vec, 'd) vec"   and B :: "(('c, 'd) vec, 'd) vec" shows "det (A ** B) = det A * det B"
renaming_set_minus: fixes Sel :: "'at set"   and C :: "'at Literal set"   and L :: "'at Literal" shows "rename_clause Sel (C - {L}) = rename_clause Sel C - {rename_literal Sel L}"
that_2: fixes a :: "'a"   and dl :: "'a dlist" assumes "y = DList.insert a dl"   and "\<not> member dl a" shows "thesis"
CompDefined_def: fixes CC :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" shows "f \<approx>>\<^bsub>CC\<^esub> g \<equiv> f \<in> mor\<^bsub>CC\<^esub> \<and> g \<in> mor\<^bsub>CC\<^esub> \<and> cod\<^bsub>CC\<^esub> f = dom\<^bsub>CC\<^esub> g"
exp_double: fixes z :: "'a" shows "exp ((2::'a) * z) = (exp z)\<^sup>2"
simps_3: fixes x1 :: "'a" shows "tick \<noteq> ev x1"
orthogonal_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner x y = (inner x y = 0)"
meet_related: fixes x :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> y" shows "x \<sqinter> y = x"
set2_FGf_def: fixes x :: "('q, 'p, 'r, 's, 't, 'u, 'v, 'w, 'x, ('y, 'r, 'z, 'u, 'aa, 'x, 'ab) G) F" shows "set2_FGf x = set2_F x"
Abs_inverse': fixes r :: "bool list"   and a :: "'b word" assumes "r \<in> {bl. length bl = LENGTH('b)}"   and "of_bl r = a" shows "to_bl a = r"
surjective_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.surjective f = (\<forall>j. \<exists>i. f (i, j) = top)"
Vset_succ_rank_le: fixes k :: "hf"   and a :: "hf" assumes "Ord k"   and "a \<^bold>\<in> Vset (succ k)" shows "rank a \<le> k"
SemGet: fixes s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and y :: "Var"   and l :: "Location"   and C :: "Class"   and Flds :: "(Field \<times> Val) list"   and F :: "Field"   and v :: "Val"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var" assumes "fst s y = RVal (Loc l)"   and "lookup (snd s) l = Some (C, Flds)"   and "lookup Flds F = Some v"   and "t = (update (fst s) x v, snd s)" shows " s , Get x y F \<rightarrow>\<^sub>1  t "
ex_is_arg_min_on_if_finite: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>x. is_arg_min_on f S x"
set_cases: fixes e :: "'b"   and a :: "'b tree" assumes "e \<in> set_tree a" obtains z1 :: "'b tree" and z2 :: "'b" and z3 :: "'b tree" where "a = \<langle>z1__, z2__, z3__\<rangle>" and "e \<in> set_tree z1__"   | z1 :: "'b tree" and z3 :: "'b tree" where "a = \<langle>z1a__, e, z3a__\<rangle>"   | z1 :: "'b tree" and z2 :: "'b" and z3 :: "'b tree" where "a = \<langle>z1, z2, z3\<rangle>" and "e \<in> set_tree z3"
conv_abscissa_cmult_left: fixes c :: "'a"   and f :: "'a fds" assumes "c \<noteq> (0::'a)" shows "conv_abscissa (fds_const c * f) = conv_abscissa f"
simps: fixes uu :: "('form, 'rule) graph_node"   and a :: "('form, 'var) antecedent" shows "local_vars uu a = a_fresh a"
prod_map_Pair: fixes f :: "'g \<rightarrow> 'e"   and g :: "'h \<rightarrow> 'f"   and x :: "'g"   and y :: "'h" shows "prod_map\<cdot>f\<cdot>g\<cdot>(x, y) = (f\<cdot>x, g\<cdot>y)"
set_keys_Mapping_RBT: fixes t :: "('b, unit) rbt" shows "set (RBT_Set2.keys (Mapping_RBT t)) = set (RBT_Impl.keys t)"
hom_mult: fixes x :: "'a"   and y :: "'a" shows "f (x * y) = f x * f y"
sublinear_intros_6: fixes x :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sublinear (wp x)"   and "sublinear (wp y)" shows "sublinear (wp (x \<Sqinter> y))"
real_cond_exp_cong: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "AE x in M. f x = g x"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. real_cond_exp M F f x = real_cond_exp M F g x"
remainder_recovery: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and K :: "'b set"   and A :: "'b set"   and \<psi> :: "'b"   and B :: "'b set"   and \<phi> :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "K = Cn A"   and "\<psi> \<in> Cn K"   and "B \<in> Tarskian_logic.remainder Cn K \<phi>" shows "imp_PL \<phi> \<psi> \<in> Cn B"
prod_dividef: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "class.field divide inverse times one plus zero minus uminus" shows "comm_monoid_mult.prod times one (\<lambda>x. divide (f x) (g x)) A = divide (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
M: fixes i :: "'a" assumes "i \<in> K" shows "prob_space (M i)"
Awhile_7: shows "\<forall>l s1 s2. s1 = s2 on LVQ \<longrightarrow> Q l s1 = Q l s2"
homotopic_loops_nearby_explicit: fixes g :: "real \<Rightarrow> 'b"   and h :: "real \<Rightarrow> 'b"   and S :: "'b set" assumes "path g"   and "path h"   and "pathfinish g = pathstart g"   and "pathfinish h = pathstart h"   and "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> S\<rbrakk> \<Longrightarrow> norm (h t - g t) < norm (g t - x)" shows "homotopic_loops S g h"
intro: fixes A :: "('h, 'i) graph_rec_scheme"   and B :: "('j, 'k) graph_rec_scheme"   and R :: "('h \<times> 'j) set" assumes "graph A"   and "graph B"   and "simulation_axioms R A B" shows "simulation R A B"
prefix_Cons: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "prefix xs (y # ys) = (xs = [] \<or> (\<exists>zs. xs = y # zs \<and> prefix zs ys))"
zero_fun_def: shows "0 \<equiv> \<lambda>s. none"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and f4 :: "'c"   and f5 :: "char list \<Rightarrow> 'c"   and f6 :: "'c"   and f7 :: "char list \<Rightarrow> 'c"   and f8 :: "'c"   and f9 :: "'c"   and action :: "action" shows "P (case action of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2 | Log \<Rightarrow> f3 | Reject \<Rightarrow> f4 | Call x \<Rightarrow> f5 x | Return \<Rightarrow> f6 | Goto x \<Rightarrow> f7 x | Empty \<Rightarrow> f8 | Unknown \<Rightarrow> f9) = (\<not> (action = Accept \<and> \<not> P f1 \<or> action = Drop \<and> \<not> P f2 \<or> action = Log \<and> \<not> P f3 \<or> action = Reject \<and> \<not> P f4 \<or> (\<exists>x5. action = Call x5 \<and> \<not> P (f5 x5)) \<or> action = Return \<and> \<not> P f6 \<or> (\<exists>x7. action = Goto x7 \<and> \<not> P (f7 x7)) \<or> action = Empty \<and> \<not> P f8 \<or> action = Unknown \<and> \<not> P f9))"
vbrelation_vconverse: shows "vbrelation (vrat_inverse\<inverse>\<^sub>\<circ>)"
rel_spmf_oracle_adv: fixes seed :: "'seed" assumes "seed \<in> set_spmf seed_gen" shows "rel_spmf (\<lambda>(x, s1) (y, s2). snd s1 \<noteq> inj_on (h seed) (dom s2) \<and> (inj_on (h seed) (dom s2) \<longrightarrow> x = y \<and> invar seed s1 s2)) (exec_gpv (semi_forgetful_RO seed) \<A> (Map.empty, False)) (exec_gpv prf.random_oracle \<A> Map.empty)"
B: shows "D x \<in> sinks I D u (xs @ [x'])"
separate_othogonal_commuted': fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and Q :: "'a \<times> 'b \<Rightarrow> bool"   and s :: "'a \<times> 'b" assumes "\<And>ps n. P (ps, n) \<Longrightarrow> ps = (0::'a)"   and "\<And>ps n. Q (ps, n) \<Longrightarrow> n = (0::'b)" shows "(P \<and>* Q) s = (P (0::'a, snd s) \<and> Q (fst s, 0::'b))"
bounded_iff: fixes b :: "'e"   and c :: "'e"   and a :: "'e" shows "(sup b c \<le> a) = (b \<le> a \<and> c \<le> a)"
simps: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "kbo'.kbo s t = (if vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> weight t \<le> weight s then if weight t < weight s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict' (f, length ss) (g, length ts) then (True, True) else if pr_weak' (f, length ss) (g, length ts) then lex_ext_unbounded kbo'.kbo ss ts else (False, False) else (False, False))"
success_cond_effect: fixes p :: "bool Heap"   and h :: "heap" assumes "ro_cond p"   and "success p h"   and "cond p h" shows "effect p h h True"
zin_zexplode_eq: fixes x :: "ZF"   and A :: "ZF" shows "zin x (zexplode A) = x |\<in>| A"
Exit_path: fixes n :: "'b" assumes "valid_node n" shows "\<exists>as. n -as\<rightarrow>\<^sub>\<surd>* Exit"
new_asms_NoReadOrWrite_only_ignores_old_vars: shows "ignores_old_vars new_asms_NoReadOrWrite_only"
assms_3: shows "set (rs @ s # ss) \<subseteq> R"
Constrains_def: fixes A :: "'b set"   and B :: "'b set" shows "A Co B \<equiv> {F. F \<in> reachable F \<inter> A co B}"
generator_elimTr: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "f \<in> {j. j \<le> n} \<rightarrow> carrier M \<and> generator R M (f ` {j. j \<le> n}) \<and> (\<forall>i\<in>nset (Suc 0) n. f i \<in> linear_span R M (carrier R) (f ` {j. j \<le> i - Suc 0})) \<longrightarrow> linear_span R M (carrier R) {f 0} = carrier M"
simps_5: fixes edgesL' :: "('b \<times> 'b) list \<Rightarrow> ('b \<times> 'b) list"   and nodesL :: "'b list"   and edgesL :: "('b \<times> 'b) list"   and more :: "'c" shows "edgesL_update edgesL' \<lparr>nodesL = nodesL, edgesL = edgesL, \<dots> = more\<rparr> = \<lparr>nodesL = nodesL, edgesL = edgesL' edgesL, \<dots> = more\<rparr>"
minit_def: fixes \<phi> :: "'b MFOTL.formula" shows "minit \<phi> = (let n = MFOTL.nfv \<phi> in \<lparr>mstate_i = 0, mstate_m = minit0 n \<phi>, mstate_n = n\<rparr>)"
lemma2': fixes a :: "letter"   and vs :: "letter list list"   and ws :: "letter list list"   and as :: "letter list" assumes "R a vs ws"   and "L as vs" shows "L (a # as) ws"
parent_l_child: fixes l :: "int"   and n :: "int" shows "parent l (l_child l n) = n"
pair_fusion: fixes x :: "i"   and z :: "i"   and y :: "i" assumes "GEM1" shows "(P x z \<and> P y z) \<and> (\<forall>w. P w z \<longrightarrow> O w x \<or> O w y) \<longrightarrow> (\<sigma>z. z = x \<or> z = y) = z"
intro: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure" assumes "prob_space M"   and "filtrated_prob_space_axioms M F" shows "filtrated_prob_space M F"
lift: fixes x :: "'a"   and y :: "'a" assumes "x \<in> sset xs"   and "y \<in> sset xs"   and "P x y" shows "Q x y"
Always_includes_reachable: fixes F :: "'d program"   and A :: "'d set" assumes "F \<in> Always A" shows "reachable F \<subseteq> A"
assms_1: shows "f\<cdot>\<bottom> = \<bottom>"
case_2: fixes f1 :: "'b"   and f2 :: "'c set \<Rightarrow> 'b"   and x2 :: "'c set" shows "(case RES x2 of FAILi \<Rightarrow> f1 | RES x \<Rightarrow> f2 x) = f2 x2"
arith_mult_neq_nzero1: fixes i :: "nat"   and n :: "nat"   and t :: "nat"   and q :: "nat" assumes "i < n"   and "0 < i" shows "i + n * t \<noteq> n * q"
solvable_imp_solvable_img: fixes K :: "'a set" assumes "solvable_seq G K" shows "solvable_seq H (h ` K)"