hcju/isabelleps huggingface dataset:
measurable_embed_measure1: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e"   and M :: "'c measure"   and N :: "'d measure" assumes "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M N" shows "g \<in> embed_measure M f \<rightarrow>\<^sub>M N"
less_irrefl: fixes x :: "'b list" shows "\<not> strict_suffix x x"
simps_13: fixes v :: "'a local.term"   and va :: "'a local.term" shows "Nml (v \<^bold>\<cdot> va) = False"
Nil: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn" shows "G,A\<turnstile>{Normal (P\<leftarrow>In3 [])} []\<doteq>\<succ> {P}"
assms_1: shows "sound_system \<Sigma> M\<^sub>0 s\<^sub>0 f"
rGamma_zeros: fixes n :: "nat" shows "(\<lambda>z. rGamma z / (z + of_nat n)) \<midarrow>- of_nat n\<rightarrow> (- (1::'a)) ^ n * fact n"
Abs_Ainteg_inverse: fixes y :: "int \<times> int" assumes "y \<in> Ainteg" shows "Rep_Ainteg (Abs_Ainteg y) = y"
that: fixes c :: "'c" assumes "homotopic_with_canon Q t U f (\<lambda>x. c)" shows "thesis"
Abs_x_plus_x_pos: fixes x :: "ant" shows "0 \<le> Abs x + x"
assms_2: shows "m \<equiv> p ## trace (w ||| r) p"
measurable_If: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and M' :: "'b measure"   and g :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "g \<in> M \<rightarrow>\<^sub>M M'"   and "{x \<in> space M. P x} \<in> sets M" shows "(\<lambda>x. if P x then f x else g x) \<in> M \<rightarrow>\<^sub>M M'"
T_on_embed: fixes s0 :: "'state"   and qs :: "'request list"   and A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)" shows "real (T_on A s0 qs) = T_on_rand' (local.embed A) (fst (local.embed A) s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs"
simps: fixes x :: "'d"   and xs :: "'d list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
f\<^sub>1_def: shows "f\<^sub>1 = \<tau> i(gap i (LEAST t. \<forall>T. gap i T \<le> gap i t) \<mapsto> 1)"
UNIV_not_singleton: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.perfect_space open" shows "UNIV \<noteq> {x}"
span_add: fixes U :: "rat vec set"   and v :: "rat vec"   and w :: "rat vec" assumes "U \<subseteq> Rn"   and "v \<in> gs.span U"   and "w \<in> Rn" shows "(w \<in> gs.span U) = (v + w \<in> gs.span U)"
keyh: fixes r :: "real" shows "\<not> (\<exists>(a', b', c')\<in>set c. ((a' \<noteq> 0 \<or> b' \<noteq> 0) \<and> a' * r\<^sup>2 + b' * r + c' = 0) \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
pairwise_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set" shows "Set.pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> R x y)"
evaln_abrupt_lemma: fixes G :: "prog"   and s :: "state"   and e :: "Term.term"   and n :: "nat"   and v :: "vals"   and s' :: "state"   and xc :: "abrupt" assumes "G\<turnstile>s \<midarrow>e\<succ>\<midarrow>n\<rightarrow> (v, s')" shows "fst s = Some xc \<longrightarrow> s' = s \<and> v = undefined3 e"
cc_Suc_le: fixes j :: "nat" assumes "j < card X - 3" shows "int (aa (Suc j) + bb (Suc j)) \<le> 2 + (int (aa (j + 2) + bb (j + 2)) gchoose 2) + (\<Sum>i = j + 4..card X - 1. int (aa i + bb i) gchoose (i - j))"
D_det: fixes P :: "'c process"   and Q :: "'c process" shows "D (P [+] Q) = D P \<union> D Q"
prems_1: shows "list_all (\<lambda>x. x = dir.Left) (a # p)"
AE_T_enabled: fixes s :: "'a cfg" shows "almost_everywhere (MDP.MC.T s) (MDP.MC.enabled s)"
closed_segment_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "linear f" shows "closed_segment (f a) (f b) = f ` closed_segment a b"
mk_scheme_Fun: fixes t :: "typ"   and sch1 :: "type_scheme"   and sch2 :: "type_scheme" assumes "mk_scheme t = sch1 =-> sch2" shows "\<exists>t1 t2. sch1 = mk_scheme t1 \<and> sch2 = mk_scheme t2"
finsert_not_fempty: fixes a :: "'b"   and A :: "'b fset" shows "finsert a A \<noteq> {||}"
mrbst_inter_diff_reduce: fixes b :: "bool"   and l1 :: "'a tree"   and x :: "'a"   and r1 :: "'a tree" shows "mrbst_inter_diff b \<langle>l1, x, r1\<rangle> = (\<lambda>t2. case split_bst' x t2 of (sep, l2, r2) \<Rightarrow> mrbst_inter_diff b l1 l2 \<bind> (\<lambda>l. mrbst_inter_diff b r1 r2 \<bind> (\<lambda>r. if sep = b then return_pmf \<langle>l, x, r\<rangle> else mrbst_join l r)))"
to_nat_def: fixes a :: "'b" shows "to_nat a = (SOME i. univ_list ! i = a \<and> i < length univ_list)"
hom_deg_set_closed_minus: fixes A :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and z :: "nat"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>a1 a2. \<lbrakk>a1 \<in> A; a2 \<in> A\<rbrakk> \<Longrightarrow> a1 - a2 \<in> A"   and "p \<in> hom_deg_set z A"   and "q \<in> hom_deg_set z A" shows "p - q \<in> hom_deg_set z A"
rel_refl_strong: fixes x :: "'b message_af"   and Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z. z \<in> set_message_af x \<Longrightarrow> Ra z z" shows "rel_message_af Ra x x"
orient1: shows "\<forall>x y. (x, y) \<in> set xysp \<longrightarrow> delta_gt \<delta> x y"
lm052: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> Pow (N \<times> (Pow G - {{}}))"
Match: fixes P :: "pi"   and Rs :: "residual"   and b :: "name" assumes "P \<longmapsto> Rs" shows "[b\<frown>b]P \<longmapsto> Rs"
vR_2: shows "X = v +o \<langle>R\<rangle>"
lift: fixes d :: "nat"   and dm :: "nat"   and p :: "(nat \<times> int) list"   and lm :: "nat"   and F :: "nat \<Rightarrow> nat \<Rightarrow> (nat \<times> int) list \<Rightarrow> ((nat \<times> int) list \<Rightarrow> real) \<Rightarrow> (nat \<times> int) list \<Rightarrow> real"   and S :: "real \<Rightarrow> (nat \<times> int) list \<Rightarrow> (nat \<times> int) list \<Rightarrow> real"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "d < dm"   and "p \<in> sparsegrid dm lm"   and "\<And>l b p \<alpha>. \<lbrakk>b \<in> lgrid (start dm) ({0..<dm} - {d}) lm; l + level b = lm; p \<in> sparsegrid dm lm\<rbrakk> \<Longrightarrow> F d l b \<alpha> p = (if b = base {d} p then \<Sum>p'\<in>lgrid b {d} lm. S (\<alpha> p') p p' else \<alpha> p)" shows "lift F dm lm d \<alpha> p = (\<Sum>p'\<in>lgrid (base {d} p) {d} lm. S (\<alpha> p') p p')"
assms_3: shows "Q_step = make_mat n (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)))))"
preserves_iso: fixes f :: "'a" assumes "local.iso f" shows "local.iso (R f)"
vimage2p_def: fixes f :: "'n \<Rightarrow> 'q"   and g :: "'o \<Rightarrow> 'r"   and R :: "'q \<Rightarrow> 'r \<Rightarrow> 'p" shows "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
dynamically_complete_calculus_axioms: shows "dynamically_complete_calculus Bot Inf (\<Turnstile>) Red_I Red_F"
member_Id_on: fixes x :: "'d"   and y :: "'d"   and A :: "'d set" shows "((x, y) \<in> Id_on A) = (x = y \<and> y \<in> A)"
prodag_sameTr3: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and B :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)"   and "\<forall>k\<in>I. A k = B k" shows "prod_mOp I A = prod_mOp I B"
lm048: fixes P :: "('b \<times> 'a) set"   and Q :: "('b \<times> 'a) set" assumes "Range P \<inter> Range Q = {}"   and "runiq (P\<inverse>)"   and "runiq (Q\<inverse>)" shows "runiq ((P +< Q)\<inverse>)"
vcomp_vimage: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<circ>\<^sub>\<circ> s) `\<^sub>\<circ> A = r `\<^sub>\<circ> s `\<^sub>\<circ> A"
esize_mono: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "esize x \<le> esize y"
elims: fixes x :: "'a"   and xa :: "'a tree \<times> nat"   and y :: "real" assumes "t\<^sub>s x xa = y" obtains x :: "'a" and tn :: "'a tree \<times> nat" where "xa__ = x" and "xa = tn" and "y = real (T_insert3 x tn)"
p_residue_mod: fixes k :: "nat"   and m :: "nat"   and n :: "int" assumes "k < m" shows "p_residue k (p_residue m n) = p_residue k n"
Gr_def: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "BNF_Def.Gr A f = {(a, f a) |a. a \<in> A}"
one_is_normal: shows "{\<one>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub>} \<lhd> G\<lparr>carrier := generate G {a}\<rparr>"
simps_2: fixes uv :: "String.literal" shows "is_Num (value.Str uv) = False"
SCF_completeD: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and Pcond :: "'d set \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> ('d \<times> 'd) set" assumes "SCF scf A Is Pcond"   and "Pcond A Is P" shows "complete A (scf P)"
invar_deq: fixes empty :: "'b"   and enq :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and first :: "'b \<Rightarrow> 'c"   and deq :: "'b \<Rightarrow> 'b"   and is_empty :: "'b \<Rightarrow> bool"   and list :: "'b \<Rightarrow> 'c list"   and invar :: "'b \<Rightarrow> bool"   and q :: "'b" assumes "Queue empty enq first deq is_empty list invar"   and "invar q" shows "invar (deq q)"
fwi_outermost_mono: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "fwi m n k i j i j \<le> m i j"
residual_network_sel_1: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" shows "edge (res.residual_network f) x y = (edge (residual_network f) x y \<or> edge (residual_network f) y x \<and> y \<noteq> source (residual_network f))"
primerec_constn_1: fixes n :: "nat" shows "primerec (constn n) (Suc 0)"
finite_fv\<^sub>s\<^sub>t: fixes S :: "('b, 'a) strand_step list" shows "finite (fv\<^sub>s\<^sub>t S)"
bounds_compare_Some_5: fixes x :: "'a"   and c :: "'a" shows "x \<ge>\<^sub>l\<^sub>b Some c = (c \<le> x)"
minsky_reaching_s_def: fixes s :: "nat" shows "minsky_reaching_s s = {uu_. \<exists>M m vs vs'. uu_ = m \<and> (M, vs) = decode_minsky_state m \<and> deterministic M \<and> 0 \<notin> fst ` M \<and> 1 \<notin> fst ` M \<and> ((2, vs), s, vs') \<in> (step M)\<^sup>*}"
vs2: shows "valid_sop (eval_expr (t + used_tmps e\<^sub>1) e\<^sub>2)"
l2_inv7D: fixes x :: "l2_state" assumes "x \<in> l2_inv7" shows "secret x \<subseteq> {Exp (Exp Gen (NonceF (R' $ N'))) (NonceF (test $ N)) |N N' R'. R' \<in> partners \<and> (N = nx \<or> N = ny) \<and> (N' = nx \<or> N' = ny)}"
basis_in_ex: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> insert x X" shows "\<exists>B. X_insert.basis_in \<E> B"
cases: fixes G :: "prog"   and a1 :: "qtname"   and a2 :: "qtname" assumes "G\<turnstile>a1\<leadsto>a2" obtains C :: "qtname" and J :: "qtname" where "a1 = C__" and "a2 = J__" and "G\<turnstile>C__\<leadsto>1J__"   | C :: "qtname" and I :: "qtname" and J :: "qtname" where "a1 = Ca__" and "a2 = Ja__" and "G\<turnstile>Ca__\<leadsto>1I" and "G\<turnstile>I\<preceq>I Ja__"   | C :: "qtname" and D :: "qtname" and J :: "qtname" where "a1 = C" and "a2 = J" and "G\<turnstile>C\<prec>\<^sub>C1D" and "G\<turnstile>D\<leadsto>J"
next_action_from_execs: shows "\<forall>s execs. next_action s execs \<rightharpoonup> (\<lambda>a. a \<in> actions_in_execution (execs (current s)))"
Rep_\<I>: fixes x :: "\<I>" shows "Rep_\<I> x \<in> {(x, j). enat x \<le> j}"
Cauchy_def: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
def_lfp_unfold: fixes h :: "'d"   and f :: "'d \<Rightarrow> 'd" assumes "h \<equiv> lfp f"   and "mono f" shows "h = f h"
pred_preds_update: fixes mem' :: "'Var \<Rightarrow> 'Val"   and mem :: "'Var \<Rightarrow> 'Val"   and x :: "'Var"   and e :: "'AExp"   and P' :: "'BExp set"   and P :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set" assumes "mem' = mem(x := ev\<^sub>A mem e)"   and "P' = assign_post P x e |` {v. stable \<S> v}"   and "pred P mem" shows "pred P' mem'"
assms: shows "\<exists>p r i. j = make_full_observation e None r p i"
encode_problem_parallel_complete_iii: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "is_valid_problem_strips \<Pi>"   and "\<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi>"   and "\<forall>k op. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<longrightarrow> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"   and "\<forall>l op. length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<le> l \<and> l < length \<pi> \<longrightarrow> \<not> \<A> (Operator l (index (\<Pi>\<^sub>\<O>) op))"   and "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)" shows "\<A> \<Turnstile> encode_operators \<Pi> (length \<pi>)"
simps_1: fixes i :: "int" shows "dvdnumcoeff (C i) = (\<lambda>g. g dvd i)"
summable_zero: shows "summable (\<lambda>n. 0::'a)"
inj_eq_image: fixes f :: "'a \<Rightarrow> 'b"   and X :: "'a set"   and Y :: "'a set" assumes "inj f"   and "f ` X = f ` Y" shows "X = Y"
is_derivation_def: fixes u :: "symbol list" shows "is_derivation u = derives [\<SS>] u"
trss_return_cases: fixes u :: "'a"   and r :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and r' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((u # r, c), w, r', c') \<in> trcl (trss fg)" obtains s' :: "'a list" and u' :: "'a" where "r' = s' @ u' # r" and "(([u], c), w, s' @ [u'], c') \<in> trcl (trss fg)"   | waa :: "('b, 'c) label list" and wb :: "('b, 'c) label list" and ch :: "'a list multiset" where "w = waa @ wb" and "(([u], c), waa, [], ch) \<in> trcl (trss fg)" and "((r, ch), wb, r', c') \<in> trcl (trss fg)"
inputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a<b>.P \<longmapsto> Rs" obtains x :: "name" and aa :: "name" and Pa :: "pi" where "a = aa \<and> [b].P = [x].Pa" and "Rs = aa<x> \<prec> Pa" and "x \<noteq> aa"
space_measure_pmf: fixes p :: "'d pmf" shows "space (measure_pmf p) = UNIV"
eval_simps'_2: fixes f :: "'c local.term" assumes "Arr f" shows "src \<lbrace>f\<rbrace> = \<lbrace>Src f\<rbrace>"
nprv_exi_commute: fixes x :: "'var"   and y :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "y \<in> var"   and "\<phi> \<in> fmla" shows "nprv {exi x (exi y \<phi>)} (exi y (exi x \<phi>))"
event_can_go_back_if_no_sender_trace: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event" assumes "trace c t c'"   and "list_all (\<lambda>ev. occurs_on ev \<noteq> occurs_on ev') t"   and "can_occur ev' c'"   and "\<not> isRecvMarker ev'"   and "list_all (\<lambda>ev. \<not> isSend ev) t" shows "can_occur ev' c"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "enat \<Rightarrow> 'a"   and enat :: "enat" shows "P (case enat of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = (\<not> (enat = 0 \<and> \<not> P f1 \<or> (\<exists>x2. enat = eSuc x2 \<and> \<not> P (f2 x2))))"
Assign': fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and a :: "aexp" assumes "\<forall>s. P s \<longrightarrow> Q (s[a/x])" shows "\<turnstile>\<^sub>t {P} x ::= a {Q}"
\<psi>_conv_\<theta>_sum: fixes x :: "real" assumes "0 < x" shows "\<psi> x = sum_upto (\<lambda>m. \<theta> (root m x)) (log 2 x)"
eqvt: fixes p :: "perm"   and a :: "exp" shows "p \<bullet> isVal a = isVal (p \<bullet> a)"
is_arg_min_linorder: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "is_arg_min f P x = (P x \<and> (\<forall>y. P y \<longrightarrow> order.greater_eq (f y) (f x)))"
ParallelRuleAnn: fixes as :: "(('d, 'e, 'f) ann \<times> 'd set \<times> 'd set) list"   and cs :: "('d, 'e, 'f) com list"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and Q :: "'d set"   and A :: "'d set" assumes "length as = length cs"   and "\<forall>i<length cs. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (fst (as ! i)) (cs ! i) (fst (snd (as ! i))), (snd (snd (as ! i)))"   and "interfree \<Gamma> \<Theta> F as cs"   and "\<Inter> (set (map (\<lambda>a. fst (snd a)) as)) \<subseteq> Q"   and "\<Union> (set (map (\<lambda>a. snd (snd a)) as)) \<subseteq> A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnPar as) (Parallel cs) Q, A"
bound: shows "interaction_bounded_by' (\<lambda>_. True) \<A> q"
vec_to_iarray_column': fixes k :: "nat"   and A :: "(('b, 'c) vec, 'd) vec" assumes "k < ncols A" shows "vec_to_iarray (column (mod_type_class.from_nat k) A) = column_iarray k (matrix_to_iarray A)"
condition_def: fixes P :: "'d \<Rightarrow> bool"   and L :: "'d \<Rightarrow> ('e \<times> 'd) \<times> bool"   and R :: "'d \<Rightarrow> ('e \<times> 'd) \<times> bool" shows "condition P   L   R \<equiv> \<lambda>s. if P s then L s else R s"
fps_XD'_power_Suc: fixes a :: "'a fps"   and b :: "'a fps"   and n :: "nat" shows "fps_XD' a (b ^ Suc n) = of_nat (Suc n) * b ^ n * fps_XD' a b"
polyexp_ratfps_def: fixes xs :: "('b \<times> nat \<times> 'b) list" shows "polyexp_ratfps xs = (\<Sum>(a, k, b)\<leftarrow>xs. ratfps_of_poly (smult a (fps_monom_poly b k)) / ratfps_of_poly ([:1::'b, - b:] ^ (k + 1)))"
set_relD2: fixes s :: "'d set"   and s' :: "'e set"   and R :: "('d \<times> 'e) set"   and x' :: "'e" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "x' \<in> s'" shows "\<exists>x\<in>s. (x, x') \<in> R"
mn_unhalt_case: fixes n :: "nat"   and f :: "recf"   and ap :: "abc_inst list"   and ar :: "nat"   and ft :: "nat"   and args :: "nat list"   and anything :: "nat list" assumes "rec_ci (Mn n f) = (ap, ar, ft) \<and> length args = ar"   and "\<forall>i. terminate f (args @ [i]) \<and> 0 < rec_exec f (args @ [i])" shows "{\<lambda>nl. nl = args @ 0 \<up> (ft - ar) @ anything} ap \<up>"
blocks_3: fixes bb :: "nat"   and be :: "nat" assumes "(bb, be) \<in> set ((i_begin, i_end) # bs)" shows "be - bb \<le> l - lb"
subgroups_Inter_pair: fixes I :: "'a set"   and J :: "'a set" assumes "subgroup I G"   and "subgroup J G" shows "subgroup (I \<inter> J) G"
collapse_2: fixes generat :: "('g, 'h, 'i) generat" assumes "\<not> is_Pure generat" shows "IO (generat.output generat) (continuation generat) = generat"
j_of_gt: fixes k :: "nat" shows "k < j_of k"
ih_3: shows "InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
S1_def: shows "local.S1 = {0<..int p - 1}"
lower_circle_def: fixes l :: "real"   and r :: "real" shows "lower_circle l r = {x. cmod ((x - complex_of_real l) / complex_of_real (r - l) - (1 / 2 - complex_of_real (sqrt 3) / 6 * \<i>)) < sqrt 3 / 3}"
tmsi: fixes F :: "'a formula"   and S :: "'a formula multiset" shows "tms (F, S) = to_mini_formula F, tms S"
assms: shows "Deliver (i, Insert e n) \<in> set xs"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_uminus) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_uminus \<and> -\<^sub>\<real> a = b)"
timpls_transformable_to'_inv'_2: fixes TI :: "('c set \<times> 'c set) list"   and f :: "('d, 'e, 'c) prot_fun"   and T :: "(('d, 'e, 'c) prot_fun, 'f) Term.term list"   and t :: "(('d, 'e, 'c) prot_fun, 'f) Term.term" assumes "timpls_transformable_to' TI (Fun f T) t" shows "length T = length (args t)"
simps: fixes k :: "nat"   and m :: "nat" shows "prod_decode_aux k m = (if m \<le> k then (m, k - m) else prod_decode_aux (Suc k) (m - Suc k))"
IH: shows "list_all2 (\<lambda>x1 x2. veq_structure x1 x2 \<and> (vno_abs x1 \<longrightarrow> x1 = x2)) ts us"
b0: shows "bernoulli_pmf 0 = return_pmf False"
map_fst_zip_take: fixes xs :: "'c list"   and ys :: "'d list" shows "map fst (zip xs ys) = take (min (length xs) (length ys)) xs"
ad_def: shows "ad = f2 / (e * c) / (f2 / e - f2 / (e * c))"
intervalE: fixes a :: "'c"   and x :: "'c"   and b :: "'c" assumes "a \<le> x \<and> x \<le> b" obtains "x \<in> {a..b}"
llast_LCons: fixes x :: "'a"   and xs :: "'a llist" shows "llast (LCons x xs) = (if lnull xs then x else llast xs)"
map_add_dom_app_simps_2: fixes m :: "'b"   and l1 :: "'b \<Rightarrow> 'c option"   and l2 :: "'b \<Rightarrow> 'c option" assumes "m \<notin> dom l1" shows "(l2 \<Oplus> l1) m = l2 m"
opaque_dist_cong: fixes extra_rule :: "nat itrm \<Rightarrow> nat itrm \<Rightarrow> bool"   and itrm_bracket :: "nat \<Rightarrow> nat itrm \<Rightarrow> nat itrm option"   and x :: "nat itrm"   and y :: "nat itrm"   and vs :: "nat list" assumes "itrm_abstraction extra_rule itrm_bracket"   and "itrm_cong (special_idiom.idiom_ext_rule extra_rule) x y" shows "itrm_cong (special_idiom.idiom_ext_rule extra_rule) (opaque_dist vs x) (opaque_dist vs y)"
permutes_insert_lemma: fixes p :: "'a \<Rightarrow> 'a"   and a :: "'a"   and S :: "'a set" assumes "p permutes insert a S" shows "a \<rightleftharpoons>\<^sub>F p a \<circ> p permutes S"
IDirProds_incl: fixes G :: "('e, 'f) monoid_scheme"   and i :: "'g"   and I :: "'g set"   and S :: "'g \<Rightarrow> 'e set" assumes "Group.group G"   and "i \<in> I" shows "S i \<subseteq> IDirProds G S I"
cprod_infinite1': fixes r :: "('c \<times> 'c) set"   and p :: "('d \<times> 'd) set" assumes "Cinfinite r"   and "(p, czero) \<notin> ordIso \<and> Card_order p"   and "p \<le>o r" shows "r *c p =o r"
perm_empty: fixes xs :: "'c list" shows "([] <~~> xs) = (xs = [])"
simps_3: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<phi> :: "fml" shows "usubstappp \<sigma> U (? \<phi>) = (U, Testo (usubstappf \<sigma> U \<phi>))"
iarray_append_of_fun: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and x :: "'a" shows "iarray_append (IArray.of_fun f n) x = IArray.of_fun (f(n := x)) (Suc n)"
build_succ_def: fixes nds :: "('c, 'd) node_scheme set" shows "build_succ nds = FOREACH nds (\<lambda>q'. FOREACH (incoming q') (\<lambda>qn s. if qn = expand_init then RETURN s else RETURN (s(qn \<mapsto> insert (name q') (the_default {} (s qn)))))) Map.empty"
mat_eq_iff: fixes x :: "'a mat"   and y :: "'a mat" shows "(x = y) = (dim_row x = dim_row y \<and> dim_col x = dim_col y \<and> (\<forall>i j. i < dim_row y \<longrightarrow> j < dim_col y \<longrightarrow> x $$ (i, j) = y $$ (i, j)))"
less_ccSup_iff: fixes S :: "'b set"   and a :: "'b" assumes "countable S" shows "(a < Sup S) = (\<exists>x\<in>S. a < x)"
leftmost_unaffected_Derives1: fixes j :: "nat"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "leftmost j a"   and "j < i"   and "Derives1 a i r b" shows "leftmost j b"
dist_diff_1: fixes a :: "'c"   and b :: "'c" shows "dist a (a - b) = norm b"
finite_imp_bounded: fixes S :: "'c set" assumes "finite S" shows "bounded S"
empty_correct_2: shows "BinomialHeapStruc.queue_to_multiset [] = {#}"
orecvmsg_simps_3: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and ip :: "nat"   and m :: "'e" shows "orecvmsg P \<sigma> (unicast ip m) = True"