hcju/isabelleps huggingface dataset:
eq_uptoI: fixes g :: "nat \<Rightarrow> 'b"   and B :: "(nat \<Rightarrow> 'b) set"   and i :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "g \<in> B"   and "\<And>j. j < i \<Longrightarrow> f j = g j" shows "g \<in> eq_upto B f i"
can_lock_d_eq_can_lock: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id"   and xcp :: "'addr option"   and h :: "'heap"   and frs :: "'addr frame list"   and L :: "('addr + 'thread_id + 'thread_id) set" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "\<Phi> \<turnstile> t:(xcp, h, frs) \<surd>" shows "execd_mthr.can_sync P t (xcp, frs) h L = exec_mthr.can_sync P t (xcp, frs) h L"
simps_4: fixes x :: "'a"   and xs :: "[:'a:]"   and y :: "'a"   and ys :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>"   and "y \<noteq> \<bottom>"   and "ys \<noteq> \<bottom>" shows "prefix\<cdot>(x :# xs)\<cdot>(y :# ys) = (eq\<cdot>x\<cdot>y andalso prefix\<cdot>xs\<cdot>ys)"
while_unfold: fixes b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "while b c s = (if b s then while b c (c s) else s)"
intro: fixes n :: "V"   and f :: "V"   and A :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"   and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A" shows "nop_v11_axioms A n f"
less_eq_bot: fixes x :: "'a" shows "(x \<le> bot) = (x = bot)"
fw_invariant: fixes k' :: "nat"   and k :: "nat"   and i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "k' \<le> k"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "fw m n k i j \<le> fw m n k' i j"
assms_4: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "subterms f \<subseteq> Mapping.keys M"
determinant_one_element: fixes A :: "'b mat" assumes "A \<in> carrier_mat 1 1" shows "Determinant.det A = A $$ (0, 0)"
weakSimE: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and \<alpha> :: "act"   and Q' :: "ccs" assumes "P \<leadsto><Rel> Q"   and "Q \<longmapsto>\<alpha> \<prec> Q'" obtains P' :: "ccs" where "P \<Longrightarrow>\<alpha> \<prec> P'" and "(P', Q') \<in> Rel"
msetext_huet_imp_dersh: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "msetext_huet gt ys xs" shows "msetext_dersh gt ys xs"
inter_restrict: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and B :: "'c set" assumes "finite A" shows "prod g (A \<inter> B) = (\<Prod>x\<in>A. if x \<in> B then g x else (1::'a))"
chain: shows "\<forall>i. (F i, F (Suc i)) \<in> S O S\<^sup>* O NS\<^sup>*"
pcompose_def: fixes p :: "'a poly"   and q :: "'a poly" shows "p \<circ>\<^sub>p q = fold_coeffs (\<lambda>a c. [:a:] + q * c) p 0"
lset_nth_member_inf: fixes xs :: "'c llist"   and A :: "'c set"   and n :: "nat" assumes "\<not> lfinite xs"   and "lset xs \<subseteq> A" shows "xs $ n \<in> A"
mr_is_C: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators" assumes "applied_rule_rev C x p = \<lfloor>a\<rfloor>" shows "C (list2FWpolicy p) x = C a x"
sub_num_simps_1: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one num.One num.One = zero"
pinduct: fixes a0 :: "nat"   and a1 :: "'a list"   and P :: "nat \<Rightarrow> 'a list \<Rightarrow> bool" assumes "mom_select_dom (a0, a1)"   and "\<And>k xs. \<lbrakk>mom_select_dom (k, xs); \<not> length xs \<le> 20 \<Longrightarrow> P (((length xs + 4) div 5 - 1) div 2) (map slow_median (chop 5 xs)); \<And>x xa xb y xaa ya. \<lbrakk>\<not> length xs \<le> 20; x = mom_select (((length xs + 4) div 5 - 1) div 2) (map slow_median (chop 5 xs)); xa = partition3 x xs; (xb, y) = xa; (xaa, ya) = y; k < length xb\<rbrakk> \<Longrightarrow> P k xb; \<And>x xa xb y xaa ya. \<lbrakk>\<not> length xs \<le> 20; x = mom_select (((length xs + 4) div 5 - 1) div 2) (map slow_median (chop 5 xs)); xa = partition3 x xs; (xb, y) = xa; (xaa, ya) = y; \<not> k < length xb; \<not> k < length xb + length xaa\<rbrakk> \<Longrightarrow> P (k - length xb - length xaa) ya\<rbrakk> \<Longrightarrow> P k xs" shows "P a0 a1"
assms_9: shows "mod_arr = arr'(m - 1 := Min_mset (arr_mset arr l\<^sub>0 m))"
overlap_eq: fixes x :: "i"   and y :: "i" shows "O x y = (\<exists>z. P z x \<and> P z y)"
1: shows "\<mu>.leg0 \<cdot> Prj\<^sub>1 = \<nu>.leg1 \<cdot> Prj\<^sub>1\<^sub>0"
subset_of_opens: fixes U :: "'c set" assumes "U \<in> target.neighborhoods (f x)" shows "is_open\<^sub>Y U"
AbstForm_imp_abst_dbfm: fixes v :: "hf"   and i :: "hf"   and x :: "hf"   and x' :: "hf"   and e :: "(name, hf) finfun" assumes "AbstForm v i x x'" shows "\<exists>A. x = \<lbrakk>quot_dbfm A\<rbrakk>e \<and> x' = \<lbrakk>quot_dbfm (abst_dbfm (decode_Var v) (nat_of_ord i) A)\<rbrakk>e"
affine_affine_hull: fixes s :: "'a set" shows "affine (affine hull s)"
simps_1: fixes A :: "'a"   and uu :: "'a"   and uv :: "'a set" shows "local.Dom (MkArr A uu uv) = A"
set_func_lhs: shows "set_func (F \<^bsub>\<a>\<^esub> f \<odot> \<sigma>(A,a) B) = (\<lambda>g\<in>Hom A B. set_func (F \<^bsub>\<a>\<^esub> (f \<bullet> g)) a)"
Streams_Connected_defs_2: fixes outS :: "nat \<Rightarrow> 'd message_af"   and inS :: "nat \<Rightarrow> 'd message_af" shows "i_Streams_Connected outS inS \<equiv> inS = [NoMsg] \<frown> outS"
compact_cbox: fixes a :: "'a"   and b :: "'a" shows "compact (cbox a b)"
conj_subset_def: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "(A \<subseteq> {x. P x \<and> Q x}) = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
rtrancl_impl: fixes set_of :: "'b \<Rightarrow> 'c set"   and memb :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and empty :: "'b"   and un :: "'c list \<Rightarrow> 'b \<Rightarrow> 'b"   and gen_succ :: "('c \<times> 'c) list \<Rightarrow> 'c list \<Rightarrow> 'c list"   and rel :: "('c \<times> 'c) list"   and as :: "'c list" assumes "set_access_gen set_of memb empty un gen_succ" shows "set_of (rtrancl_impl gen_succ un memb empty rel as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set rel)\<^sup>*}"
fls_integral_def: fixes a :: "'a fls" shows "fls_integral a = Abs_fls (\<lambda>n. if n = 0 then 0::'a else inverse (of_int n) * fls_nth a (n - 1))"
weakSimAct_def: fixes P :: "pi"   and Rs :: "residual"   and C :: "'a"   and Rel :: "(pi \<times> pi) set" shows "weakSimAct P Rs C Rel \<equiv> (\<forall>Q' a x. Rs = a<\<nu>x> \<prec> Q' \<longrightarrow> x \<sharp> C \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>Q' a x. Rs = a<x> \<prec> Q' \<longrightarrow> x \<sharp> C \<longrightarrow> (\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel)) \<and> (\<forall>Q' \<alpha>. Rs = \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
top_def: fixes x :: "'a" shows "top = - x \<squnion> - - x"
Smith_normal_form_upt_k_intro: fixes k :: "nat"   and A :: "(('d, 'c) vec, 'b) vec" assumes "\<And>a b. mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a + 1 < k \<and> mod_type_class.to_nat b + 1 < k \<Longrightarrow> A $ a $ b dvd A $ (a + (1::'b)) $ (b + (1::'c))"   and "\<And>a b. mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b \<and> (mod_type_class.to_nat a < k \<or> mod_type_class.to_nat b < k) \<Longrightarrow> A $ a $ b = (0::'d)" shows "Smith_normal_form_upt_k A k"
continuous_complex_iff: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> complex" shows "continuous F f = (continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x)))"
P_new_decomp: shows "P_new = new_pre @ [new_last]"
span_eq: fixes S :: "'a set"   and T :: "'a set" shows "(ideal S = ideal T) = (S \<subseteq> ideal T \<and> T \<subseteq> ideal S)"
vsubstEnv_substEnv_compose1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodEnv rho"   and "good Y" shows "rho &[Y / y]_ys &[y1 // y]_ys = rho &[Y #[y1 // y]_ys / y]_ys"
assms_1: shows "SubRel.subpath g v1 es v2 subs"
empty_in_Fpow: fixes A :: "'a set" shows "{} \<in> Fpow A"
iso_trans: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and I :: "('h, 'i) monoid_scheme" assumes "G \<cong> H"   and "H \<cong> I" shows "G \<cong> I"
subcategory_axioms: shows "subcategory (\<cdot>) (right f)"
j_2: shows "hi - x \<in> G + {H_rcoset_reps ! j}"
rotate_rel_iff: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>" shows "r.Lyndon w = (rotate_rel w \<subseteq> {(x, y). (x, y) \<in> r})"
star_zero: shows "(0::'a)\<^sup>\<star> = (1::'a)"
r_sigma_neq: fixes x\<^sub>1 :: "nat"   and y\<^sub>1 :: "nat"   and x\<^sub>2 :: "nat"   and y\<^sub>2 :: "nat" shows "(\<sigma> x\<^sub>1 y\<^sub>1 \<noteq> \<sigma> x\<^sub>2 y\<^sub>2) = (the (\<sigma> x\<^sub>1 y\<^sub>1) \<noteq> the (\<sigma> x\<^sub>2 y\<^sub>2))"
list_mset_rel_def: shows "list_mset_rel \<equiv> br mset (\<lambda>_. True)"
size_random_bst: fixes t :: "'a tree"   and A :: "'a set" assumes "t \<in> set_pmf (random_bst A)"   and "finite A" shows "size t = card A"
cGDERIV_DERIV_compose: fixes f :: "'a \<Rightarrow> complex"   and x :: "'a"   and df :: "'a"   and g :: "complex \<Rightarrow> complex"   and dg :: "complex" assumes "cGDERIV f x :> df"   and "(g has_field_derivative cnj dg) (at (f x))" shows "cGDERIV (\<lambda>x. g (f x)) x :> dg *\<^sub>C df"
bot_apply: fixes x :: "'b" shows "\<bottom> x = \<bottom>"
inv_on_left_moving_in_middle_B_Bk: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" assumes "inv_on_left_moving_norm (as, lm) (s, l, Oc # r) ires"   and "hd l = Bk"   and "l \<noteq> []" shows "inv_on_left_moving_in_middle_B (as, lm) (s, tl l, Bk # Oc # r) ires"
find_cong: fixes xs :: "'a list"   and ys :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "find P xs = find Q ys"
fixpt_dist_nodes': fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list"   and v :: "nat"   and m :: "nat"   and q :: "nat"   and p :: "nat" assumes "wf_tr M T"   and "wf_dfa M v"   and "\<forall>p q. dfa_is_node M q \<and> p < q \<longrightarrow> tr_lookup T q p = (\<exists>n<m. dist_nodes M n v p q)"   and "0 < m"   and "dfa_is_node M q"   and "p < q" shows "tr_lookup (fixpt M T) q p = (\<exists>n. dist_nodes M n v p q)"
retraction_prop: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "f \<circ> f = f" shows "(f x = x) = (x \<in> range f)"
progress_set_mono: fixes B :: "'b set"   and B' :: "'b set"   and C :: "'b set set"   and F :: "'b program"   and T :: "'b set" assumes "B \<subseteq> B'"   and "B' \<in> C"   and "C \<in> progress_set F T B" shows "C \<in> progress_set F T B'"
totalize_invariant_iff: fixes F :: "'b program"   and A :: "'b set" shows "(totalize F \<in> invariant A) = (F \<in> invariant A)"
set_pmf_poisson: fixes rate :: "real" assumes "0 < rate" shows "set_pmf (poisson_pmf rate) = UNIV"
A': shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g' \<eta>' \<epsilon>'"
pell_square_solution_int_aux: fixes x :: "int"   and y :: "int" assumes "0 < D"   and "x\<^sup>2 = 1 + int D * y\<^sup>2" shows "x \<in> {- 1, 1} \<and> y = 0"
map_option_eq_Some: fixes f :: "'c \<Rightarrow> 'b"   and xo :: "'c option"   and y :: "'b" shows "(map_option f xo = Some y) = (\<exists>z. xo = Some z \<and> f z = y)"
\<phi>J_def: fixes \<psi> :: "'fmla" shows "\<phi>J \<psi> \<equiv> subst \<psi> (tJ \<psi>) xx"
oghoare_Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P Throw Q, A" shows "\<exists>P'. P = AnnExpr P' \<and> P' \<subseteq> A"
runState_def: fixes state :: "('a, 'b) state" shows "runState state = (case state of State x \<Rightarrow> x)"
AbstForm_abst_dbfm: fixes i :: "name"   and n :: "nat"   and e :: "(name, hf) finfun"   and fm :: "dbfm" shows "AbstForm (q_Var i) (ord_of n) \<lbrakk>quot_dbfm fm\<rbrakk>e \<lbrakk>quot_dbfm (abst_dbfm i n fm)\<rbrakk>e"
lD_def: shows "lD = list_of_mset (C - A)"
s: shows "sub2_wit [] fs = (fst (sub2_wit [] fs), snd (sub2_wit [] fs))"
fd_map_split'': fixes list :: "fd list" shows "map ((\<lambda>(x, y). fd_def x y) \<circ> case_fd Pair) list = list"
M: fixes x :: "'a"   and n :: "nat" assumes "x \<in> S" shows "norm (f n x) \<le> M"
encR: fixes S :: "'procS" shows "SourceTerm S \<R>\<lbrakk>\<cdot>\<rbrakk>R TargetTerm (\<lbrakk>S\<rbrakk>)"
eta_F: fixes f :: "ValD \<rightarrow> ValD"   and f' :: "(('o ValK \<rightarrow> 'o) \<rightarrow> 'o) \<rightarrow> ('o ValK \<rightarrow> 'o) \<rightarrow> 'o" assumes "\<theta>: f \<mapsto> f'" shows "\<eta>: ValF\<cdot>f \<mapsto> unitK\<cdot>(ValKF\<cdot>f')"
case_1: fixes f1 :: "'h"   and f2 :: "'i \<Rightarrow> 'h" shows "(case Fault of Fault \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = f1"
run_owner_iii_unused: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "\<exists>D. PriKey D \<notin> used (fst (run_owner_iii n s))"
HPT_ETI: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<in> HPT ST" shows "TS \<subseteq> ET ST"
approx_mem_monad: fixes x :: "'b star"   and u :: "'b star" assumes "x \<approx> u" shows "u \<in> monad x"
corec_sel_3: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and q22 :: "'a \<Rightarrow> bool"   and g221 :: "'a \<Rightarrow> ('b, 'c) tllist"   and g222 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "ttl (corec_tllist p g1 g21 q22 g221 g222 a) = (if q22 a then g221 a else corec_tllist p g1 g21 q22 g221 g222 (g222 a))"
do_uncurry: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "f a b = uncurry f (a, b)"
a_Inj1_inv_I: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> Inj1 e, eP, eV : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2" obtains eP :: "Syntax.term" and eV :: "Syntax.term" where "ePa__ = Inj1 eP" and "eVa__ = Inj1 eV" and "\<Gamma> \<turnstile> e, eP, eV : \<tau>\<^sub>1"
rel_refl: fixes R1a :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "('d, 'e, 'f) gpv" assumes "\<And>x. R1a x x"   and "\<And>x. R2a x x" shows "rel_gpv R1a R2a x x"
\<J>_0_0_item_end: fixes x :: "item" assumes "x \<in> \<J> 0 0" shows "item_end x = 0"
zero_index: fixes i :: "'b" shows "0 $ i = (0::'a)"
one_less_numeral_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less one (numeral.numeral one plus n) = (num.One < n)"
of_hypreal_mult: fixes x :: "real star"   and y :: "real star" shows "of_hypreal (x * y) = of_hypreal x * of_hypreal y"
real_exp_bound_lemma: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1 / 2" shows "exp x \<le> 1 + 2 * x"
read': shows "a' \<in> read_actions (justifying_exec (J n))"
nz: shows "(deriv ^^ nat n) f z \<noteq> 0"
b_rel_subtyping_merge_tR: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and Am :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool" assumes "hn_ctxt A x y \<or>\<^sub>A hn_ctxt A' x y \<Longrightarrow>\<^sub>t hn_ctxt Am x y" shows "hn_ctxt (b_assn A P) x y \<or>\<^sub>A hn_ctxt A' x y \<Longrightarrow>\<^sub>t hn_ctxt Am x y"
lsc_at_MInfty: fixes f :: "'b \<Rightarrow> ereal"   and x0 :: "'b" assumes "f x0 = - \<infinity>" shows "lsc_at x0 f"
assms_2: shows "x\<^sup>2 + y\<^sup>2 = p"
safe: shows "\<O>s,i\<turnstile> (is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
aform_isFDERIV_def: fixes prec :: "nat"   and n :: "nat"   and ns :: "nat list"   and fas :: "floatarith list"   and xs :: "(real \<times> real pdevs) list" shows "aform_isFDERIV prec n ns fas xs = dRETURN (isFDERIV_approx prec n ns fas (ivls_of_aforms prec xs))"
hom_def: fixes G :: "('d, 'f) monoid_scheme"   and H :: "('e, 'g) monoid_scheme" shows "hom G H = {h \<in> carrier G \<rightarrow> carrier H. \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y}"
Ssegment_inj: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "(a = b) = (Ssegment D a = Ssegment D b)"
length_mapWithIndex: fixes f :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and xs :: "'e list" shows "length (mapWithIndex f xs) = length xs"
eq_\<I>_converter_mono: fixes \<I>1 :: "('a, 'b) \<I>"   and \<I>2 :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter"   and conv' :: "('a, 'b, 'c, 'd) converter"   and \<I>1' :: "('a, 'b) \<I>"   and \<I>2' :: "('c, 'd) \<I>" assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv \<sim> conv'"   and "\<I>1' \<le> \<I>1"   and "\<I>2 \<le> \<I>2'" shows "\<I>1',\<I>2' \<turnstile>\<^sub>C conv \<sim> conv'"
self_codomain_iff_ide: fixes a :: "(bool, 'arr list) concrete_category.arr" shows "(a \<in> codomains a) = ide a"
of_int_dirichlet_inverse: fixes f :: "nat \<Rightarrow> int"   and i :: "int"   and n :: "nat" shows "of_int (dirichlet_inverse f i n) = dirichlet_inverse (\<lambda>n. of_int (f n)) (of_int i) n"
carr_mat_3: shows "mat_of_rows n G \<in> carrier_mat n n"
unsigned_of_nat: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (word_of_nat n) = semiring_1.of_nat one plus zero (take_bit LENGTH('l) n)"
Invariants_quasi_Invariants_functions: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M" shows "(\<exists>g\<in>borel_measurable Invariants. AE x in M. f x = g x) = (AE x in M. f (T x) = f x)"
leq: shows "rel_language letter_eq (Language1 (init1 r)) (Language2 (init2 s)) = (\<forall>(r', s')\<in>{((r, s), delta1 a r, delta2 b s) |a b r s. a \<in> set alphabet1 \<and> b \<in> set alphabet2 \<and> letter_eq a b}\<^sup>* `` {(init1 r, init2 s)}. accept1 r' = accept2 s')"
the_elem_simp: fixes x :: "'a" assumes "x \<in> carrier R" shows "the_elem (h ` (a_kernel R S h +> x)) = h x"
lc_eq_zero_iff: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "(lcf p = (0::'b)) = (p = 0)"
mult_imp_div_pos_less: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c"   and z :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less x (times z y)" shows "less (divide x y) z"
uniform_limit_add: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and X :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> 'd" assumes "uniform_limit X f l F"   and "uniform_limit X g m F" shows "uniform_limit X (\<lambda>a b. f a b + g a b) (\<lambda>a. l a + m a) F"
monom_one: shows "MPoly_Type.monom 0 (1::'b) = 1"
simps_2: fixes l :: "'b tree"   and uu :: "'b"   and r :: "'b tree" shows "min_height \<langle>l, uu, r\<rangle> = min (min_height l) (min_height r) + 1"
simps: fixes f :: "nat discr Box \<rightarrow> 'a" shows "unwrapS\<cdot>f = f\<cdot>0 && unwrapS\<cdot>(f oo (\<Lambda> x. 1 + x))"
vmatch_induct: fixes P :: "pat \<Rightarrow> value \<Rightarrow> bool"   and a0 :: "pat"   and a1 :: "value" assumes "\<And>name v. P (Patvar name) v"   and "\<And>name ps name' vs. (\<And>x y. \<lbrakk>name = name' \<and> length ps = length vs; x \<in> set ps; y \<in> set vs\<rbrakk> \<Longrightarrow> P x y) \<Longrightarrow> P (Patconstr name ps) (Vconstr name' vs)"   and "\<And>v va vb vc. P (Patconstr v va) (Vabs vb vc)"   and "\<And>v va vb vc vd. P (Patconstr v va) (Vrecabs vb vc vd)" shows "P a0 a1"
finsum_vec_insert: fixes F :: "'a set"   and a :: "'a"   and f :: "'a \<Rightarrow> 'b vec"   and n :: "nat" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier_vec n"   and "f a \<in> carrier_vec n" shows "finsum_vec TYPE('b) n f (insert a F) = f a + finsum_vec TYPE('b) n f F"
P2: shows "Col P Q A' \<and> P Q Perp A A'"
Csafe_def: fixes ode_ops :: "ode_ops" shows "ode_operations.Csafe ode_ops = Collect (ode_operations.safe ode_ops)"
map_of_domAD: fixes \<Gamma> :: "('e \<times> 'd) list"   and x :: "'e"   and e :: "'d" assumes "map_of \<Gamma> x = Some e" shows "x \<in> domA \<Gamma>"
m_def_F: shows "m = length (prems_of (to_F \<iota>))"
Di_take_subset: fixes i :: "nat"   and l :: "nat"   and y :: "nat list" shows "Di i (take l y) \<subseteq> Di i y"
parse_attribute_value_def: shows "parse_attribute_value = exactly [CHR 0x22] \<bind> (\<lambda>_. many ((\<noteq>) CHR 0x22) \<bind> (\<lambda>v. exactly [CHR 0x22] \<bind> (\<lambda>_. Parser_Monad.return v)))"
that: fixes X :: "nat list set" assumes "X \<subseteq> WW"   and "ordertype X (lenlex less_than) = \<omega> \<up> \<omega>"   and "\<And>u. u \<in> [X]\<^bsup>2\<^esup> \<Longrightarrow> \<exists>l. Form l u \<and> (0 < l \<longrightarrow> [enum N l] < inter_scheme l u \<and> list.set (inter_scheme l u) \<subseteq> N)" shows "thesis"
finfun_comp_const: fixes g :: "'c \<Rightarrow> 'b"   and c :: "'c" shows "finfun_comp g (finfun_const c) = finfun_const (g c)"
elims: fixes x :: "bool"   and xa :: "'a \<times> 'a"   and y :: "'a" assumes "sel2 x xa = y" obtains b :: "bool" and a1 :: "'a" and a2 :: "'a" where "x = b" and "xa = (a1, a2)" and "y = (if b then a2 else a1)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. nat = Suc x2 \<longrightarrow> P (f2 x2)))"
Dg_def: fixes n :: "nat"   and t :: "real" shows "Dg n t = Df (line t) n H"
sup_state_opt_None_any: fixes P :: "'a prog"   and s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> None \<le>' s"
simps: fixes a1 :: "'b sequent"   and a2 :: "nat"   and R :: "('b sequent list \<times> 'b sequent) set" shows "((a1, a2) \<in> derivable R) = ((\<exists>C. a1 = C \<and> a2 = 0 \<and> ([], C) \<in> R) \<or> (\<exists>r m. a1 = snd r \<and> a2 = m + 1 \<and> r \<in> R \<and> fst r \<noteq> [] \<and> (\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R)))"