hcju/isabelleps huggingface dataset:
Units_closed: fixes x :: "'a" assumes "x \<in> Units (units_group K)" shows "x \<in> carrier (units_group K)"
ltl_prop_equiv_true_implies_true: fixes x :: "'b ltln"   and y :: "'b ltln" assumes "x \<sim>\<^sub>P true\<^sub>n"   and "x \<longrightarrow>\<^sub>P y" shows "y \<sim>\<^sub>P true\<^sub>n"
cases: fixes a1 :: "('d \<times> 'e) set"   and a2 :: "('d \<times> 'e) list"   and a3 :: "('d \<times> 'e, 'f) ta_rule set"   and Q10 :: "'d set"   and Q20 :: "'e set" assumes "(a1, a2, a3) \<in> frp_initial Q10 Q20" obtains W :: "('d \<times> 'e) list" where "a1 = Q10 \<times> Q20" and "a2 = W" and "a3 = {}" and "distinct W" and "set W = Q10 \<times> Q20"
set_encode_vimage_Suc: fixes A :: "nat set" shows "set_encode (Suc -` A) = set_encode A div 2"
encode_problem_has_model_then_also_partial_encodings_3: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "\<A> \<Turnstile> encode_operators \<Pi> t"
simps_1: shows "matcheq_matchNone MatchAny = False"
mem_scale: fixes x :: "'a"   and a :: "real" assumes "x \<in> S" shows "a *\<^sub>R x \<in> S"
is_subset_2: fixes c :: "'Q" assumes "c \<in> \<delta>_states \<delta>" shows "c \<in> Q"
lformula_lderiv: fixes \<phi> :: "('a, 'k) aformula"   and x :: "'x" assumes "lformula \<phi>" shows "lformula (lderiv x \<phi>)"
meetI: fixes x :: "'a"   and y :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "\<And>i. is_glb L i {x, y} \<Longrightarrow> P i"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "P (x \<sqinter> y)"
dim_vec_first: fixes v :: "'b vec"   and n :: "nat" shows "dim_vec (vec_first v n) = n"
edom_evalHeap_subset: fixes h :: "('d \<times> 'f) list"   and eval :: "'f \<Rightarrow> 'e" shows "edom (evalHeap h eval) \<subseteq> domA h"
splay_elemsD: fixes x :: "'b"   and t :: "'b tree"   and l :: "'b tree"   and a :: "'b"   and r :: "'b tree" assumes "splay x t = \<langle>l, a, r\<rangle>"   and "Sorted_Less.sorted (inorder t)" shows "(x \<in> set (inorder t)) = (x = a)"
wt_step_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and \<tau>s :: "'c list" shows "wt_step r T step \<tau>s = (\<forall>p<length \<tau>s. \<tau>s ! p \<noteq> T \<and> stable r step \<tau>s p)"
that: fixes l :: "'a list" assumes "mset l = m" shows "thesis"
cs_order: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and \<pi>' :: "nat \<Rightarrow> 'a"   and i :: "nat"   and i' :: "nat"   and j :: "nat"   and j' :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.is_path prob \<pi>'"   and "IFC_def.cs prob \<pi> i = IFC_def.cs prob \<pi>' i'"   and "IFC_def.cs prob \<pi> j = IFC_def.cs prob \<pi>' j'"   and "\<pi> i \<noteq> IFC_def.return prob"   and "i < j" shows "i' < j'"
indicator_def: fixes S :: "'b set"   and x :: "'b" shows "chi S x = (if x \<in> S then 1::'a else (0::'a))"
restrict_restrict: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" shows "restrict (restrict f A) B = restrict f (A \<inter> B)"
exhaust: fixes y :: "'a tree" obtains x11 :: "nat" and x12 :: "'a" where "y = Leaf x11 x12"   | x21 :: "nat" and x22 :: "'a tree" and x23 :: "'a tree" where "y = Node x21 x22 x23"
is_parser_bind: fixes p :: "char list \<Rightarrow> char list + 'a \<times> char list"   and q :: "'a \<Rightarrow> char list \<Rightarrow> char list + 'b \<times> char list" assumes "is_parser p"   and "\<And>x. is_parser (q x)" shows "is_parser (p \<bind> q)"
tl_def: fixes list :: "'c list" shows "tl list = (case list of [] \<Rightarrow> [] | x21 # x22 \<Rightarrow> x22)"
rank_invar_not_empty_hd: fixes t :: "('c, 'd) BinomialTree"   and bq :: "('c, 'd) BinomialTree list" assumes "rank_invar (t # bq)"   and "bq \<noteq> []" shows "rank t < rank (hd bq)"
step: fixes A :: "('a \<times> ('b, real) cconstraint \<times> 'c \<times> 'b list \<times> 'a) set \<times> ('a \<Rightarrow> ('b, real) cconstraint)"   and \<R> :: "('b \<Rightarrow> real) set set"   and l :: "'a"   and R :: "('b \<Rightarrow> real) set"   and l' :: "'a"   and R' :: "('b \<Rightarrow> real) set"   and l'' :: "'a"   and R'' :: "('b \<Rightarrow> real) set" assumes "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle>"   and "A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto> \<langle>l'', R''\<rangle>" shows "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l'', R''\<rangle>"
invar3_def: shows "invar3 \<equiv> \<lambda>(c, n, ts). n = c + num_ni ts"
simps_121: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a" shows "(case false\<^sub>n of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f2"
streamValue8: fixes d :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "'a" assumes "\<forall>j\<le>d. str (t + i + k + d + Suc j) = x"   and "str (t + i + k + d) = x" shows "\<forall>j\<le>Suc d. str (t + i + k + d + j) = x"
not_eq_order_implies_strict: fixes a :: "'a list"   and b :: "'a list" assumes "a \<noteq> b"   and "prefix b a" shows "strict_prefix b a"
assms: shows "heterogeneous\<^sub>l\<^sub>s\<^sub>t t A S"
fps_compose_add_distrib: fixes a :: "'a fps"   and b :: "'a fps"   and c :: "'a fps" shows "a + b oo c = (a oo c) + (b oo c)"
cinner_gt_zero_iff: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(0 < cinner x x) = (x \<noteq> zero)"
case_6: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a"   and x61 :: "Arraytype"   and x62 :: "nat" shows "(case arrV x61 x62 of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = f6 x61 x62"
reduced_row_echelon_form_upt_k_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" shows "reduced_row_echelon_form_upt_k A k = ((\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)) \<and> (\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'d)) = (1::'d)) \<and> (\<forall>i. i < i + (1::'f) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'f)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ (i + (1::'f)) $ n \<noteq> (0::'d))) \<and> (\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'d)) = (0::'d))))"
rPar2: fixes \<Psi>\<^sub>P :: "'b"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* C" shows "Prop (P \<parallel> Q')"
assms_2: fixes x :: "'n" assumes "x \<notin> S" shows "f x = (0::'a)"
bind_spmf_const_return_None: fixes p :: "'c spmf" shows "p \<bind> (\<lambda>_. return_pmf None) = return_pmf None"
is_arg_min_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "ord.is_arg_min less f P x = (P x \<and> (\<nexists>y. P y \<and> less (f y) (f x)))"
abs_eq: fixes xa :: "'b ltl"   and x :: "'b set" shows "\<up>step (Abs xa) x = Abs (step xa x)"
lossless_R1_14_OT: fixes m1 :: "bool"   and m2 :: "bool"   and m3 :: "bool"   and m4 :: "bool"   and c0 :: "bool"   and c1 :: "bool" shows "lossless_spmf (R1_14_OT (m1, m2, m3, m4) (c0, c1))"
rev_is_Nil_conv: fixes xs :: "'g list" shows "(rev xs = []) = (xs = [])"
simps_3: fixes disc :: "'b \<Rightarrow> bool"   and neg :: "bool"   and m :: "'b match_expr" shows "has_disc_negated disc neg (MatchNot m) = has_disc_negated disc (\<not> neg) m"
alist_and_append: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and l1 :: "'b negation_type list"   and l2 :: "'b negation_type list"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (alist_and (l1 @ l2)) a p = matches \<gamma> (MatchAnd (alist_and l1) (alist_and l2)) a p"
p_def: shows "p \<equiv> cf_pos_poly (poly_add_rat r1 p2)"
prv_prv_imp_trans: fixes \<phi> :: "fm"   and \<chi> :: "fm"   and \<psi> :: "fm" assumes "\<phi> \<in> UNIV"   and "\<chi> \<in> UNIV"   and "\<psi> \<in> UNIV"   and "{} \<turnstile> \<phi> IMP \<chi>"   and "{} \<turnstile> \<chi> IMP \<psi>" shows "{} \<turnstile> \<phi> IMP \<psi>"
y_def: shows "y \<equiv> (x - 1) / (x + 1)"
substRes3: fixes b :: "name"   and P :: "pi"   and a :: "name" assumes "b \<sharp> P" shows "<\<nu>a>P[a::=b] = <\<nu>b>P[a::=b]"
bisimilar_def: fixes a :: "('c, 'd) Stream"   and b :: "('c, 'e) Stream" shows "(a \<approx> b) = (unstream\<cdot>a = unstream\<cdot>b \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
smap2_szip: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and s1 :: "'f stream"   and s2 :: "'g stream" shows "smap2 f s1 s2 = smap (\<lambda>(x, y). f x y) (szip s1 s2)"
acyclicI: fixes r :: "('a \<times> 'a) set" assumes "\<forall>x. (x, x) \<notin> r\<^sup>+" shows "acyclic r"
GFP_closed: fixes f :: "'a \<Rightarrow> 'a" shows "GFP\<^bsub>L\<^esub> f \<in> carrier L"
inf_sup_aci_6: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and exp :: "exp" assumes "\<not> nonrec \<Gamma>"   and "\<And>x. x \<in> domA \<Gamma> \<Longrightarrow> P (the (map_of \<Gamma> x))"   and "P exp" shows "P (Let \<Gamma> exp)"
conv_abscissa_leI_weak: fixes d :: "ereal"   and f :: "'a fds" assumes "\<And>x. d < ereal x \<Longrightarrow> fds_converges f (of_real x)" shows "conv_abscissa f \<le> d"
rhss_union_distrib: fixes A :: "'b trm set"   and B :: "'b trm set" shows "rhss (A \<union> B) = rhss A \<union> rhss B"
S\<^sub>i: fixes x :: "i"   and y :: "i" assumes "AxiomsSetMcL" shows "E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)"
TryReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>try e catch(C V) e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>try e' catch(C V) e\<^sub>2,s',b'\<rangle>"
case_option_apply: fixes none :: "'c \<Rightarrow> 'b"   and some :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'d option"   and y :: "'c" shows "(case x of None \<Rightarrow> none | Some x \<Rightarrow> some x) y = (case x of None \<Rightarrow> none y | Some a \<Rightarrow> some a y)"
nonneg_power_le: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and n :: "nat" assumes "class.semiring_1_real_line one times plus zero minus less_eq less"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq (power.power one times a n) (power.power one times b n) = (n = 0 \<or> less_eq a b)"
ennreal_inverse_positive: fixes x :: "ennreal" shows "(0 < inverse x) = (x \<noteq> \<top>)"
Compl_eq: fixes A :: "'a set" shows "- A = {x. x \<notin> A}"
division_weak_partial_order: shows "weak_partial_order (division_rel G)"
fsmult': shows "(\<sharp>\<star>) \<equiv> aezfun_scalar_mult.fsmult (\<star>)"
Unbal: shows "ins (size t + dl) 0 a t = Unbal x3_"
has_path_refl: fixes n :: "nat"   and S :: "(nat \<times> nat) set"   and a :: "nat" assumes "is_valid_graph n S"   and "a < n" shows "has_path n S a a"
polytope_interval: fixes a :: "'b"   and b :: "'b" shows "polytope (cbox a b)"
MStep_Exec1_Aux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and kb :: "nat"   and l1 :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and hh :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" assumes "(M, l, s, kb, l1, t) \<in> MStep"   and "\<exists>k. (M, l, s, k, hh, v) \<in> Exec" shows "\<exists>n. (M, l1, t, n, hh, v) \<in> Exec"
P: shows "P \<subseteq> {x. a \<bullet> x = 0}"
index_1_imp_mult_1: fixes bl :: "'a set" assumes "\<Lambda> = 1"   and "bl \<in># \<B>"   and "2 \<le> card bl" shows "multiplicity bl = 1"
id_act: fixes x :: "Vertex" shows "apply_rotation \<one>\<^bsub>tetrahedral_group\<^esub> x = x"
lex_def: fixes r :: "('a \<times> 'a) set" shows "List.lex r = \<Union> (range (lexn r))"
generalized_fw_joinI: fixes f1 :: "('b simple_match \<times> 'c) list"   and p :: "('b, 'd) simple_packet_scheme"   and r1 :: "'b simple_match"   and d1 :: "'c"   and f2 :: "('b simple_match \<times> 'e) list"   and r2 :: "'b simple_match"   and d2 :: "'e" assumes "generalized_sfw f1 p = Some (r1, d1)"   and "generalized_sfw f2 p = Some (r2, d2)" shows "generalized_sfw (generalized_fw_join f1 f2) p = Some (the (simple_match_and r1 r2), d1, d2)"
bit_signed_take_bit_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and m :: "nat"   and a :: "'c"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m a) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> bit a (min m n))"
sp_prefix: shows "sp_fundef (Fubx_get F) fd (take pc (body fd)) = Ok (None # map typeof \<Sigma>2)"
abs_eq: fixes x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "monomials (MPoly x) = keys x"
assms_2: shows "f \<in> UNIV \<rightarrow> carrier (UP R)"
assms_3: shows "w ! 0 \<noteq> z ! 0"
bv_to_int_Nil: shows "bv_to_int [] = 0"
cmod_wn_diff: fixes ea :: "real"   and w :: "complex" assumes "0 < ea"   and "cmod (w - z) < pe / 4"   and "cmod (w - z) < ea * pe\<^sup>2 / (8 * L)" shows "cmod (winding_number p w - winding_number p z) < ea"
path_from_toE: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "Digraph G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "(Digraph.walk G xs \<and> distinct xs) \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
conv_radius_leI: fixes f :: "nat \<Rightarrow> 'a"   and z :: "'a" assumes "\<not> summable (\<lambda>n. norm (f n * z ^ n))" shows "conv_radius f \<le> ereal (norm z)"
mssnth_sin_series_stream_aux: fixes b :: "bool"   and m :: "nat"   and n :: "nat" shows "mssnth (sin_series_stream_aux b (fact m) (real m)) n = (if b then - 1 else 1) * (- 1) ^ n / fact (2 * n + m)"
mult_sign_intros_2: fixes a :: "'e"   and b :: "'e" assumes "(0::'e) \<le> a"   and "b \<le> (0::'e)" shows "a * b \<le> (0::'e)"
matchC_MC_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and theta :: "(('b, 'c) com \<times> ('b, 'c) com) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.matchC_MC tval aval indis theta c d \<equiv> \<forall>s t c' s'. (s, t) \<in> indis \<and> PL.transC tval aval (c, s) (c', s') \<longrightarrow> (\<exists>d' t'. PL.MtransC tval aval (d, t) (d', t') \<and> (s', t') \<in> indis \<and> (c', d') \<in> theta)"
mset_subset_eq_mono_add: fixes A :: "'c multiset"   and B :: "'c multiset"   and C :: "'c multiset"   and D :: "'c multiset" assumes "A \<subseteq># B"   and "C \<subseteq># D" shows "A + C \<subseteq># B + D"
mor_comp: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "('f, 'g, 'h) F1 \<Rightarrow> 'g"   and s2' :: "('f, 'g, 'h) F2 \<Rightarrow> 'h"   and f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and B1'' :: "'i set"   and B2'' :: "'j set"   and s1'' :: "('f, 'i, 'j) F1 \<Rightarrow> 'i"   and s2'' :: "('f, 'i, 'j) F2 \<Rightarrow> 'j"   and f' :: "'g \<Rightarrow> 'i"   and g' :: "'h \<Rightarrow> 'j" assumes "mor B1 B2 s1 s2 B1' B2' s1' s2' f g"   and "mor B1' B2' s1' s2' B1'' B2'' s1'' s2'' f' g'" shows "mor B1 B2 s1 s2 B1'' B2'' s1'' s2'' (f' \<circ> f) (g' \<circ> g)"
ctr_2: fixes sys :: "'a \<Rightarrow> ('b, 'a) process_term"   and T :: "('b, 'a) process_term" assumes "\<not> isACT sys T" shows "solution sys T = Choice (solution sys (CH1 sys T)) (solution sys (CH2 sys T))"
cases: fixes a1 :: "nat"   and a2 :: "'a set"   and a3 :: "'a set" assumes "dimension a1 a2 a3" obtains K :: "'a set" where "a1 = 0" and "a2 = Ka__" and "a3 = {\<zero>}"   | v :: "'a" and E :: "'a set" and n :: "nat" and K :: "'a set" where "a1 = Suc n" and "a2 = K" and "a3 = line_extension K v E" and "v \<in> carrier R" and "v \<notin> E" and "dimension n K E"
refine: fixes n :: "nat"   and Xi :: "real list \<times> real list"   and X :: "'b set"   and sctnsi :: "real list sctn list"   and sctns :: "'b sctn set" assumes "DIM_precond TYPE('b) n"   and "(Xi, X) \<in> lvivl_rel"   and "(sctnsi, sctns) \<in> sctns_rel" shows "(nres_of (restrict_to_halfspaces_ivl n sctnsi Xi), restrict_to_halfspaces sctns X) \<in> \<langle>lvivl_rel\<rangle>nres_rel"
Q_def: shows "Q = (\<lambda>n a k c1. \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> f n a i j k \<le> c1 + real k ^ (N - 1))"
binary_power_code_unfold: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.monoid_mult one times" shows "power.power one times = monoid_mult.binary_power one times"
simps_1: fixes uu :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "insort_wrt uu x [] = [x]"
in_br_conv: fixes c :: "'e"   and a :: "'f"   and \<alpha> :: "'e \<Rightarrow> 'f"   and I :: "'e \<Rightarrow> bool" shows "((c, a) \<in> br \<alpha> I) = (a = \<alpha> c \<and> I c)"
l_create_character_data_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_character_data_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr DocumentClass.type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_val set_val_locs set_disconnected_nodes set_disconnected_nodes_locs create_character_data DocumentClass.known_ptrs"
absorb1: fixes a :: "'e"   and b :: "'e" assumes "a \<le> b" shows "inf a b = a"
Im_Arcsin: fixes z :: "complex" shows "Im (Arcsin z) = - ln (cmod (\<i> * z + csqrt (1 - z\<^sup>2)))"
collapse: fixes inference :: "'a inference" shows "Infer (prems_of inference) (concl_of inference) = inference"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_goto_start_pos m lm (l, r) = (wprepare_goto_start_pos_B m lm (l, r) \<or> wprepare_goto_start_pos_O m lm (l, r))"
index_append: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a" shows "index (xs @ ys) x = (if x \<in> set xs then index xs x else length xs + index ys x)"
Inter_subset: fixes A :: "'f set set"   and B :: "'f set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B"   and "A \<noteq> {}" shows "\<Inter> A \<subseteq> B"
epiI: fixes f :: "'j" assumes "J.arr f"   and "\<And>g g'. J.seq g f \<and> J.seq g' f \<and> g \<cdot>\<^sub>J f = g' \<cdot>\<^sub>J f \<Longrightarrow> g = g'" shows "J.epi f"
inf_le2: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<inter># y \<subseteq># y"
axioms_5: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<tau>" shows "natural_transformation_axioms A B F G \<tau>"
lexordp_simps_2: fixes less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" shows "ord.lexordp less xs [] = False"
Prim_Impl_Defs_axioms: shows "Prim_Impl_Defs G.invar G_adj G_empty G_add_edge RBT_Set.empty RBT_Map.update RBT_Map.delete lookup M.invar PST_RBT.empty PST_RBT.update PST_RBT.delete PM.invar lookup rbt_is_empty pst_getmin G.\<alpha>w G.\<alpha>g"