hcju/isabelleps huggingface dataset:
equiv_lemma: fixes eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list" assumes "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0)) \<or> (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)) \<or> (\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)" shows "(\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> ((\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<or> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0))) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
simps_2: fixes f :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'a"   and xs :: "('b \<times> 'a) list" shows "lookup f ((x, y) # xs) = (if f x then Some y else lookup f xs)"
that: fixes t :: "'a"   and v :: "'t" assumes "t \<in> Keys (Poly_Mapping.range r)"   and "v \<in> Keys (keys r)"   and "u = t \<oplus> v" shows "thesis"
r_into_trancl': fixes p :: "'c \<times> 'c"   and r :: "('c \<times> 'c) set" assumes "p \<in> r" shows "p \<in> r\<^sup>+"
intermediary_def: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'d"   and u :: "'d" assumes "Separation_Kernel kstep output_f current cswitch kprecondition control kinvolved ifp vpeq" shows "Separation_Kernel.intermediary ifp d u \<equiv> \<exists>v. ifp\<^sup>*\<^sup>* v d \<and> ifp d u \<and> \<not> ifp v u \<and> d \<noteq> u"
prj_chine_assoc_2: shows "\<mu>\<nu>\<pi>.Prj\<^sub>1\<^sub>0 \<cdot> \<mu>\<nu>\<pi>.chine_assoc = \<mu>\<nu>\<pi>.Prj\<^sub>0\<^sub>1"
no_prime_divisors_imp_unit: fixes a :: "'a" assumes "a \<noteq> (0::'a)"   and "\<And>b. \<lbrakk>b dvd a; normalize b = b\<rbrakk> \<Longrightarrow> \<not> prime_elem b" shows "is_unit a"
not_eq_extremum: fixes a :: "'b multiset" shows "(a \<noteq> {#}) = ({#} \<subset># a)"
finite_labels: fixes prog :: "cmd" shows "finite {l. \<exists>c. labels prog l c}"
ins_list_Cons: fixes x :: "'b"   and xs :: "'b list" assumes "Sorted_Less.sorted (x # xs)" shows "ins_list x xs = x # xs"
sub: shows "(\<lambda>i. cis (real i * 2 * pi / real n)) ` {0..<n} \<subseteq> {x. x ^ n = 1}"
gcd_ge_0_int: fixes x :: "int"   and y :: "int" shows "0 \<le> gcd x y"
last_map_One: shows "last (map (h2 i) xs) = One"
simps_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" assumes "flow_attainability \<Delta>" shows "flow_attainability.lift \<Delta> f (Inner x, Inner y) = f (x, y)"
INF_greatest: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'d set"   and u :: "'b"   and f :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq u (f i)" shows "less_eq u (Inf (f ` A))"
2: shows "int (card X - j) = int (card X) - int j"
invariant_1_pos_3: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" assumes "invariant_1 (p, l, r)" shows "(real_of_1 (p, l, r) \<le> 0) = (r \<le> 0)"
CHECK_monadic_def: fixes c :: "('b + bool) nres"   and e :: "'b" shows "CHECK_monadic c e \<equiv> ebind c (\<lambda>b. CHECK b e)"
Says_Server_KeyWithNonce: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and NB' :: "nat"   and X :: "msg"   and evs :: "event list"   and NB :: "nat" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB'\<rbrace>, X\<rbrace> \<in> set evs"   and "NB \<noteq> NB'"   and "evs \<in> yahalom" shows "\<not> KeyWithNonce K NB evs"
distinct_11: fixes x7 :: "'b ptrm" shows "PUnit \<noteq> PSnd x7"
p_4: fixes K :: "'a set" assumes "K \<in> p" shows "\<exists>a b. K = cbox a b"
a_3: shows "irreducible (Ring_Divisibility.mult_of R) a"
wf_Fields_Ex: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list" assumes "wf_prog wf_md P"   and "is_class P C" shows "\<exists>FDTs. P \<turnstile> C has_fields FDTs"
n: fixes n :: "nat" assumes "residues_nat n" shows "1 < n"
dom_def: fixes m :: "'f \<Rightarrow> 'g option" shows "dom m = {a. m a \<noteq> None}"
simps_2: fixes x :: "name"   and xs :: "name list"   and F :: "'d frame" shows "\<lparr>\<nu>*(x # xs)\<rparr>F = \<lparr>\<nu>x\<rparr>\<lparr>\<nu>*xs\<rparr>F"
zeroring_def: fixes R :: "('c, 'd) Ring_scheme" shows "zeroring R = (Ring R \<and> carrier R = {\<zero>\<^bsub>R\<^esub>})"
left_neutral: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and a :: "'d" assumes "monoid f z" shows "f z a = a"
neg_divideC_le_eq: fixes c :: "complex"   and b :: "'b"   and a :: "'b" assumes "c < 0" shows "(b /\<^sub>C c \<le> a) = (c *\<^sub>C a \<le> b)"
rel_def: fixes x :: "'c set \<times> 'd"   and y :: "'c set \<times> 'd" shows "stfx.rel x y \<equiv> (fst x \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U} \<and> fst y \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}) \<and> (snd x \<in> \<O>\<^sub>Y (fst x) \<and> snd y \<in> \<O>\<^sub>Y (fst y)) \<and> (\<exists>W. W \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U} \<and> W \<subseteq> fst x \<inter> fst y \<and> \<rho>\<^sub>Y (fst x) W (snd x) = \<rho>\<^sub>Y (fst y) W (snd y))"
union_ac_2: fixes M :: "'a multiset"   and N :: "'a multiset" shows "M + N = N + M"
red: shows "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<^sub>1\<rangle>"
eventually_at_filter: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and s :: "'b set" shows "eventually P (at a within s) = (\<forall>\<^sub>F x in nhds a. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x)"
has: shows "class_add P (C, cdec) \<turnstile> C\<^sub>0 has F,b:T in D"
simps: fixes a :: "('s, 'v) tm" shows "wary_fo a = (\<exists>s. a = s \<and> (is_Hd s \<or> is_Sym (head s)) \<and> enat (num_args s) = arity_hd (head s) \<and> Ball (set (args s)) wary_fo)"
intro: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and S' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<phi> :: "'b \<Rightarrow> 'd" assumes "set_category S \<AA>"   and "set_category S' \<AA>"   and "two_set_categories_bij_betw_Univ_axioms S S' \<phi>" shows "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"
is_bezout_ext_def: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "is_bezout_ext bezout = (\<forall>a b. let (p, q, u, v, gcd_a_b) = bezout a b in p * a + q * b = gcd_a_b \<and> gcd_a_b dvd a \<and> gcd_a_b dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd gcd_a_b) \<and> gcd_a_b * u = - b \<and> gcd_a_b * v = a)"
pseudo_divmod_list_def: fixes p :: "'a list"   and q :: "'a list" shows "pseudo_divmod_list p q = (if q = [] then ([], p) else let rq = rev q; (qu, re) = pseudo_divmod_main_list (hd rq) [] (rev p) rq (1 + length p - length q) in (qu, rev re))"
Dom_f: shows "\<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = b"
expr_elist_inj_term: fixes t :: "expr"   and w :: "expr list" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
pure_def: fixes R :: "('c \<times> 'b) set" shows "pure R \<equiv> \<lambda>a c. \<up> ((c, a) \<in> R)"
elims: fixes x :: "'b \<Rightarrow> 'a option set"   and xa :: "'b option"   and y :: "'a option set" assumes "map_option_set x xa = y" obtains f :: "'b \<Rightarrow> 'a option set" where "x = fa__" and "xa = None" and "y = {None}"   | f :: "'b \<Rightarrow> 'a option set" and xb :: "'b" where "x = f" and "xa = Some xb" and "y = f xb"
strategyproof_scf_axioms: shows "strategyproof_scf {A1, A2, A3} {a, b, c} lowered Fishb"
Seq_def: fixes f :: "unit \<Rightarrow> 'c seq" shows "Seq f = pred_of_seq (f ())"
HEndPhase0_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4b s q"   and "HInv4d s p"   and "Inv1 s"   and "Inv2a s"   and "Inv2c_inner s p" shows "HInv4b s' q"
a1: shows "j < dim_col (Matrix.mat (2 ^ Suc n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ Suc n)))"
DiaS: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<diamond>\<phi> in v] = (\<exists>v. [\<phi> in v])"
cong: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" shows "comm_monoid_mult.prod times one \<equiv> comm_monoid_mult.prod times one"
NOT_def: fixes x1 :: "ty" shows "NOT x1 \<equiv> Abs_ty (NOT_Rep (Rep_ty x1))"
max2_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "((local.max2 a b, c) \<in> r) = ((a, c) \<in> r \<and> (b, c) \<in> r)"
Tau: fixes P :: "pi" shows "\<tau>.P \<Longrightarrow>\<^sub>l \<tau> \<prec> P"
greaterThanAtMost_empty: fixes l :: "'a"   and k :: "'a" assumes "l \<le> k" shows "{k<..l} = {}"
assms_1: shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p S"
acquired_all_acquired_in: fixes x :: "nat"   and pending_write :: "bool"   and xs :: "'b memref list"   and A :: "nat set" assumes "x \<in> acquired pending_write xs A" shows "x \<in> A \<union> all_acquired xs"
b: shows " \<Gamma>' \<subseteq> \<Gamma>'' "
subst_terms_NF: fixes ts :: "dB list"   and i :: "nat"   and j :: "nat" assumes "listall NF ts"   and "listall (\<lambda>t. \<forall>i j. NF (t[Var i/j])) ts" shows "listall NF (map (\<lambda>t. t[Var i/j]) ts)"
e2ls_ls: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2ls (encode_config ((f, xs, ls) # ss, rv)) = Partial_Recursive.list_encode ls"
subset_Compl_self_eq: fixes A :: "'a set" shows "(A \<subseteq> - A) = (A = {})"
gcd_0_left_int: fixes x :: "int" shows "gcd 0 x = \<bar>x\<bar>"
finite_infinite: fixes P :: "'a \<Rightarrow> bool" assumes "finite {b. P b}"   and "infinite UNIV" shows "infinite {b. \<not> P b}"
head_def: fixes tm :: "('c, 'd) tm" shows "head tm = (case tm of Hd x1 \<Rightarrow> x1 | App s _dummy_ \<Rightarrow> head0 s)"
returns_heap_def: fixes h :: "'l"   and p :: "('l, 'm, 'n) prog"   and h' :: "'l" shows "h \<turnstile> p \<rightarrow>\<^sub>h h' = (case h \<turnstile> p of Inl x \<Rightarrow> False | Inr (x, h'') \<Rightarrow> h' = h'')"
factors: fixes fs :: "'a list"   and y :: "'a" shows "comm_semiring_1_class.factors fs y = (prod_list fs = y \<and> Ball (set fs) irred)"
aec_cec_subset: fixes t :: "'s Trace"   and aec :: "'s odlist"   and a :: "'a"   and cec :: "'s odlist"   and x :: "'s" assumes "t \<in> Clock.jkbpC"   and "toSet aec = agent_abs a t"   and "toSet cec = common_abs t"   and "x \<in> toSet aec" shows "x \<in> toSet cec"
flatten_monomial: fixes t :: "'b"   and c :: "'b \<Rightarrow>\<^sub>0 'c" shows "flatten (monomial c t) = MPoly_Type_Class.punit.monom_mult (1::'c) t c"
addr2thread_id_inverse: fixes h :: "'heap"   and a :: "'addr"   and C :: "String.literal" assumes "typeof_addr h a = \<lfloor>Class_type C\<rfloor>"   and "P \<turnstile> C \<preceq>\<^sup>* Thread" shows "thread_id2addr (addr2thread_id a) = a"
it_paths_def: fixes t :: "'a rose_tree" shows "it_paths t = Collect (it_pathsP t)"
leadsto_WF: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> (Enabled (<A>_v) \<leadsto> <A>_v) \<longrightarrow> WF(A)_v"
singleton_null_set: fixes x :: "'a" assumes "x \<in> space M" shows "{x} \<in> null_sets M"
denormalize_shift: fixes f :: "Float.float"   and m :: "int"   and e :: "int" assumes "f = Float m e"   and "f \<noteq> 0" obtains i :: "nat" where "m = mantissa f * 2 ^ i" and "e = Float.exponent f - int i"
rcos_assoc_lcos: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>add_monoid G\<^esub> x <#>\<^bsub>add_monoid G\<^esub> K = H <#>\<^bsub>add_monoid G\<^esub> (x <#\<^bsub>add_monoid G\<^esub> K)"
subst_compose_left_idemp: fixes \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and x :: "'c"   and t :: "('a, 'c) Term.term" assumes "\<sigma> x = t \<cdot> \<sigma>" shows "subst x t \<circ>\<^sub>s \<sigma> = \<sigma>"
axioms_3: fixes R :: "('a, 'b) ring_scheme" assumes "semiring R" shows "semiring_axioms R"
arg_min_nat_lemma: fixes P :: "'d \<Rightarrow> bool"   and k :: "'d"   and m :: "'d \<Rightarrow> nat" assumes "P k" shows "P (arg_min m P) \<and> (\<forall>y. P y \<longrightarrow> m (arg_min m P) \<le> m y)"
plus_one_in_nonpos_Ints_imp: fixes z :: "'a" assumes "z + (1::'a) \<in> \<int>\<^sub>\<le>\<^sub>0" shows "z \<in> \<int>\<^sub>\<le>\<^sub>0"
assms: shows "FI_RESULT [(B, B), (A, A)] C D F"
simps_4: fixes x :: "('f, 'g, 'h) Resumption.resumption"   and f :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption" assumes "\<not> Resumption.resumption.is_Done x \<or> \<not> (Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x))))" shows "Resumption.resumption.output (x \<bind> f) = (if Resumption.resumption.is_Done x then Resumption.resumption.output (f (the (Resumption.resumption.result x))) else Resumption.resumption.output x)"
isotone_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" shows "ord.isotone less_eq f \<equiv> \<forall>x y. less_eq x y \<longrightarrow> less_eq (f x) (f y)"
not_in_iff: fixes x :: "'a"   and M :: "'a multiset" shows "(x \<notin># M) = (count M x = 0)"
set_dfs_Cons: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and x :: "('d, 'e) bintree option"   and xs :: "('d, 'e) bintree option list" shows "set (dfs f (x # xs)) = set (dfs f xs) \<union> set (dfs f [x])"
enc_atom_bool_take_drop_True: fixes r :: "nat"   and I :: "('c + 'c set) list"   and p :: "'c" assumes "r < length I"   and "case I ! r of Inl p' \<Rightarrow> p = p' | Inr P \<Rightarrow> p \<in> P" shows "enc_atom_bool I p = take r (enc_atom_bool I p) @ True # drop (Suc r) (enc_atom_bool I p)"
finite_fourier_poly'_def: fixes ws :: "nat \<Rightarrow> complex"   and k :: "nat" shows "finite_fourier_poly' ws k = poly_of_list (map (\<lambda>n. 1 / of_nat k * (\<Sum>m<k. ws m * unity_root k (- int n * int m))) [0..<k])"
in_homE: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "C.arr f" and "C.dom f = a" and "C.cod f = b"
simps_1: fixes t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.lconsNlist [] t = t"
simps_2: fixes var :: "nat" shows "grab_atoms var FalseF = Some []"
list_of_def: fixes xs :: "'b llist" shows "list_of xs = (if lfinite xs then inv llist_of xs else undefined)"
one_circ_split: shows "(1::'a)\<^sup>\<circ> = L \<squnion> (1::'a)"
wf_mdecl_wwf_mdecl: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Md :: "char list \<times> ty list \<times> ty \<times> char list list \<times> expr" assumes "wf_C_mdecl P C Md" shows "wwf_mdecl P C Md"
maximal_independent_subset: fixes V :: "'b set" obtains B :: "'b set" where "B \<subseteq> V" and "independent B" and "V \<subseteq> span B"
sum_list_const_mult: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.semiring_0 plus zero times" shows "monoid_add.sum_list plus zero (map (\<lambda>x. times c (f x)) xs) = times c (monoid_add.sum_list plus zero (map f xs))"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "size ta + size sa < size t + size s"   and "ta >\<^sub>t sa" shows "vars sa \<subseteq> vars ta"
cupcake_single_correct: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and e :: "exp0"   and res :: "(v, v) result"   and s :: "'a state" assumes "cakeml_static_env static_cenv"   and "cupcake_evaluate_single env e res"   and "cakeml_static_env.is_cupcake_all_env static_cenv env"   and "is_cupcake_exp e" shows "\<exists>s'. Big_Step_Unclocked_Single.evaluate env s e (s', res)"
IH: fixes vs :: "v list" assumes "n_zeros ts = vs" shows "ts = map typeof vs"
mem_box_1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x \<in> box a b) = (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)"
intro: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "Matrix_Kernel.kernel nr nc A"
filt_equiv_triv_init: fixes F :: "'b \<Rightarrow> bool stream measure"   and N :: "bool stream measure"   and G :: "'c \<Rightarrow> bool stream measure" assumes "filt_equiv F M N"   and "init_triv_filt M G" shows "init_triv_filt N G"
emeasure_PiM: fixes I :: "'i set"   and A :: "'i \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)" shows "emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
borel_measurable_times_ennreal: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x * g x) \<in> borel_measurable M"
distinct_sorted_list_of_set: fixes A :: "'a word set" shows "distinct (signed.sorted_list_of_set A)"
element_ptr_kinds_eq_h: shows "element_ptr_kinds h' = element_ptr_kinds h"
ennreal_enn2real: fixes r :: "ennreal" assumes "r < \<top>" shows "ennreal (enn2real r) = r"
splitAtRec_ram: fixes ram :: "'a"   and vs :: "'a list"   and a :: "'a list"   and b :: "'a list"   and us :: "'a list" assumes "ram \<in> set vs"   and "(a, b) = splitAtRec ram us vs" shows "us @ vs = a @ [ram] @ b"
iso_trans: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and I :: "('e, 'f) monoid_scheme" assumes "G \<cong> H"   and "H \<cong> I" shows "G \<cong> I"
bot_Assertion_def: shows "\<bottom> = abs_wpt \<bottom>"
idom_def: fixes g :: "'a"   and n :: "'node" shows "gE2.idom g n \<equiv> THE m. gE2.isIdom g n m"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_before_State_Idle_imp: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a" assumes "Suc 0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c"   and "\<not> State_Idle localState output_fun trans_fun (localState (s t1)). t1 \<U> t2 [0\<dots>] \<oplus> t0. output_fun (s t2) = m \<and> (\<circle> t3 t2 [0\<dots>]. output_fun (s t4) = NoMsg. t4 \<U> t5 [0\<dots>] \<oplus> t3. output_fun (s t5) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t5)))" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m"
oconf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and obj :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)" shows "P,h \<turnstile> obj \<surd> \<equiv> let (C, fs) = obj in \<forall>F D T. P \<turnstile> C has F:T in D \<longrightarrow> (\<exists>v. fs (F, D) = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
Suc: fixes \<sigma> :: "'state" shows "csmall_nstep P \<sigma> (Suc n) = {a. case a of (\<sigma>2, coll) \<Rightarrow> \<exists>\<sigma>1 coll1 coll2. (\<sigma>1, coll1) \<in> csmall P \<sigma> \<and> (\<sigma>2, coll2) \<in> csmall_nstep P \<sigma>1 n \<and> combine coll1 coll2 = coll}"
that: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" assumes "\<And>x y. 0 < h x y \<Longrightarrow> (x, y) \<in> R"   and "\<And>x y. h x y \<noteq> \<top>"   and "\<And>x. x \<in> A \<Longrightarrow> integral\<^sup>N (count_space B) (h x) = f x"   and "\<And>y. y \<in> B \<Longrightarrow> (\<Sum>\<^sup>+ x\<in>A. h x y) = g y" shows "thesis"
that_2: fixes x :: "'a" assumes "x \<in> X" shows "f x = g x"
ALength1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and a' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list" assumes "uf,P,t \<turnstile>1 \<langle>a,s\<rangle> -ta\<rightarrow> \<langle>a',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>a\<bullet>length,s\<rangle> -ta\<rightarrow> \<langle>a'\<bullet>length,s'\<rangle>"
simps_1: fixes b :: "'a"   and u :: "'a local.term" shows "\<^bold>\<langle>b\<^bold>\<rangle>\<^sub>0 \<^bold>\<Down> u = \<^bold>\<l>\<^bold>[u\<^bold>]"
store_size1: fixes m :: "mem"   and n :: "nat"   and off :: "nat"   and v :: "bytes"   and l :: "nat" shows "(store m n off v l = None) = (mem_size m < off + n + l)"