hcju/isabelleps huggingface dataset:
zero_least2: fixes x :: "'a" shows "o \<cdot> !x = o"
bind_cong_simp: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> 'b measure"   and g :: "'a \<Rightarrow> 'b measure" assumes "M = N"   and "\<And>x. x \<in> space M =simp=> f x = g x" shows "M \<bind> f = N \<bind> g"
rec_penc_def: shows "rec_penc = CN rec_add [CN rec_triangle [CN rec_add [recf.Id 2 0, recf.Id 2 1]], recf.Id 2 0]"
enabled_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and s :: "nat \<Rightarrow> nat option" shows "ast_problem.enabled problem name s \<equiv> case ast_problem.lookup_operator problem name of None \<Rightarrow> False | Some (x, pres, effs, xa) \<Rightarrow> s \<in> ast_problem.subsuming_states problem (map_of pres) \<and> s \<in> ast_problem.subsuming_states problem (map_of (implicit_pres effs))"
"0_3": assumes "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> in_trancl TI (the_Abs f) (the_Abs g)"
ZIf: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>2 :: "com" assumes "\<turnstile>\<^sub>Z {\<lambda>s. P s + \<up> (bval b s)} c\<^sub>1 {Q}"   and "\<turnstile>\<^sub>Z {\<lambda>s. P s + \<up> (\<not> bval b s)} c\<^sub>2 {Q}" shows "\<turnstile>\<^sub>Z {\<lambda>s. eSuc (P s)} IF b THEN c\<^sub>1 ELSE c\<^sub>2 {Q}"
rk_A'B'PR: fixes P :: "Points"   and Q :: "Points"   and R :: "Points"   and A' :: "Points"   and B' :: "Points" assumes "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {A', B', P, Q} = 4" shows "rk {A', B', P, R} = 4"
set_relE: fixes s :: "'a set"   and s' :: "'b set"   and R :: "('a \<times> 'b) set"   and x :: "'a" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "x \<in> s" obtains x' :: "'b" where "x' \<in> s'" and "(x, x') \<in> R"
fixespointwise2_imp_eq_on: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'b" assumes "fixespointwise f A"   and "fixespointwise g A" shows "fun_eq_on f g A"
that: fixes s :: "nat \<Rightarrow> 'a" assumes "\<And>i. P i (s i)"   and "strict_mono s"   and "\<And>n. g n \<le> s n" shows "thesis"
disc_iff_1: fixes xs :: "('c, 'd) tllist" shows "lnull (llist_of_tllist xs) = is_TNil xs"
mult_sign_intros_1: fixes a :: "'e"   and b :: "'e" assumes "(0::'e) \<le> a"   and "(0::'e) \<le> b" shows "(0::'e) \<le> a * b"
case_11: 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"   and x111 :: "'b ltln"   and x112 :: "'b ltln" shows "(case x111 M\<^sub>n x112 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) = f11 x111 x112"
i_1: shows "i + 1 < length (map ((\<inter>) K) \<GG>)"
dis_log_break: fixes d' :: "nat"   and y :: "nat"   and m' :: "nat"   and d :: "nat"   and m :: "nat" assumes "\<^bold>g [^] d' \<otimes> (\<^bold>g [^] y) [^] m' = \<^bold>g [^] d \<otimes> (\<^bold>g [^] y) [^] m"   and "y < order \<G>"   and "m' < m"   and "m < order \<G>" shows "y = nat ((int d' - int d) * Number_Theory_Aux.inverse (m - m') (order \<G>) mod int (order \<G>))"
inf_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B = rel_ext (inf (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
assms: shows "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
fn_lf_rep_def: shows "fn_lf_rep \<equiv> \<lambda>(mR, pR). {\<lambda>i. ValF\<cdot>(fs i) |fs. \<forall>xs\<in>unlr (undual mR). (\<lambda>j. fs j\<cdot>(xs j)) \<in> unlr pR}"
dom: shows "\<D>\<^sub>\<circ> vrat_uminus = \<rat>\<^sub>\<circ>"
exponential_distributedD_gt: fixes X :: "'a \<Rightarrow> real"   and l :: "real"   and a :: "real" assumes "distributed M lborel X (\<lambda>x. ennreal (exponential_density l x))"   and "0 \<le> a"   and "0 < l" shows "prob {x \<in> space M. a < X x} = exp (- a * l)"
is_word: shows "is_word (take (Min {j. j < length a \<and> is_nonterminal (a ! j)}) a)"
rcos_mult_step1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "H #> x <#> (H #> y) = H <#> (x <# H) #> y"
Sup: fixes M :: "'a set"   and a :: "'a" assumes "M \<in> Chains leq"   and "M \<noteq> {x. bot x}"   and "\<And>x. x \<in> M \<Longrightarrow> iterates_abovep a x" shows "iterates_abovep a (lub M)"
decomp_rm\<^sub>e\<^sub>s\<^sub>t_append: fixes S :: "('fun, 'var) extstrand_step list"   and S' :: "('fun, 'var) extstrand_step list" shows "decomp_rm\<^sub>e\<^sub>s\<^sub>t (S @ S') = decomp_rm\<^sub>e\<^sub>s\<^sub>t S @ decomp_rm\<^sub>e\<^sub>s\<^sub>t S'"
rBang: fixes \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> \<alpha> \<prec> P'"   and "guarded P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) \<alpha> P'"   and "guarded P"   and "distinct (bn \<alpha>)" shows "Prop C \<Psi> (!P) \<alpha> P'"
GE_wtFsym_def: shows "GE_wtFsym \<equiv> ProblemIkTpartG.GE_wtFsym wtFsym resOf tpCD"
tcons: fixes x :: "'d"   and xs :: "('d, 'a) tllist" shows "monotone orda ordb (\<lambda>f. tcons f x xs)"
g_2: shows "\<forall>x\<in>s'. finite (g -` {x})"
WTrtCast: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and T :: "ty"   and U :: "ty" assumes "P,E,h \<turnstile> e : T"   and "is_type P U" shows "P,E,h \<turnstile> Cast U e : U"
carrier_FBinds: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and x :: "Vars.var" shows "carrier ((FBinds \<Gamma>\<cdot>ae) x) \<subseteq> fv \<Gamma>"
add_edge_def: fixes v :: "'b"   and v' :: "'b"   and G :: "('b, 'c) graph_scheme" shows "FiniteGraph.add_edge v v' G = \<lparr>nodes = nodes G \<union> {v, v'}, edges = {(v, v')} \<union> edges G\<rparr>"
measurable_embed_measure1: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd"   and M :: "'b measure"   and N :: "'c measure" assumes "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M N" shows "g \<in> embed_measure M f \<rightarrow>\<^sub>M N"
gebub_inBound: fixes n :: "nat"   and m :: "nat" assumes "n < length paid_A"   and "m < length (paid_A ! n)" shows "gebub n m < length init"
exhaust: fixes y :: "pexpr" obtains x1 :: "pexpr1" where "y = PExpr1 x1"   | x2 :: "pexpr2" where "y = PExpr2 x2"
apply_chart_restrict_chart: fixes S :: "'b set"   and c :: "('b, 'c) chart" shows "apply_chart (restrict_chart S c) = apply_chart c"
literalsWithSameVariableAreEqualOrOpposite: fixes literal1 :: "Literal"   and literal2 :: "Literal" shows "(var literal1 = var literal2) = (literal1 = literal2 \<or> opposite literal1 = literal2)"
deriv: shows "chain (\<leadsto>\<^sub>w) Sts"
fimage_constant_conv: fixes c :: "'d"   and A :: "'e fset" shows "(\<lambda>x. c) |`| A = (if A = {||} then {||} else {|c|})"
ide_exp: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "ide (hfsetcat.exp b c)"
arev_permutes_arcs: shows "edge_rev M permutes arcs (del_vert u)"
dy_fake_msg_monotone_chan: fixes c :: "msg"   and bad :: "agent set"   and I :: "msg set"   and G :: "chan set"   and H :: "chan set" assumes "c \<in> dy_fake_msg bad I G"   and "G \<subseteq> H" shows "c \<in> dy_fake_msg bad I H"
BotL: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<bottom> \<in># \<Gamma>" shows "\<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
relpow_Suc_I2: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set"   and z :: "'b"   and n :: "nat" assumes "(x, y) \<in> R"   and "(y, z) \<in> R ^^ n" shows "(x, z) \<in> R ^^ Suc n"
l2_down2: fixes d :: "nat"   and pd :: "(nat \<times> int) list"   and pc :: "(nat \<times> int) list"   and dir :: "dir"   and p :: "(nat \<times> int) list" assumes "d < length pd"   and "pc \<in> grid (child pd dir d) {d}"   and "pd = child p dir d" shows "l2_\<phi> (pc ! d) (pd ! d) / 2 = l2_\<phi> (pc ! d) (p ! d)"
balance_non_Leaf: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "balance l k v r \<noteq> Leaf"
of_nat_power_le_of_nat_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "nat"   and b :: "nat"   and w :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (semiring_1.of_nat one plus zero x) (power.power one times (semiring_1.of_nat one plus zero b) w) = (x \<le> b ^ w)"
simps_3: fixes ODE1 :: "('e, 'd) ODE"   and ODE2 :: "('e, 'd) ODE" shows "FVO (OProd ODE1 ODE2) = FVO ODE1 \<union> FVO ODE2"
sup_lower_upper_bound_b: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> c"   and "b \<le> c" shows "a \<squnion> b \<le> c"
levelsCounter_aux_startIrellevant: fixes a :: "('b \<times> bool) list"   and x :: "nat list" shows "\<forall>y. y \<noteq> [] \<longrightarrow> levelsCounter_aux a (x @ y) = x @ levelsCounter_aux a y"
hull_Un_right: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" shows "P hull (S \<union> T) = P hull (S \<union> P hull T)"
assms_2: shows "\<phi> \<in> ring_hom R S"
norm1_0_iff: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(norm1 v = 0) = (v = 0)"
gstep_succ_def: fixes gc :: "(cmd, local_state, global_state) Gen_Scheduler.global_config" shows "li.gstep_succ gc \<equiv> let lcs = global_config.processes gc in {(lc, lcs'). lcs = {#lc#} + lcs'} \<bind> (\<lambda>(lc, lcs'). let gs = global_config.state gc; ls = local_config.state lc; cmd = command lc in {(a, cmd'). cfg cmd a cmd'} \<bind> (\<lambda>(a, cmd'). case a of Inl a \<Rightarrow> case la_en (ls, gs) a of None \<Rightarrow> {None} | Some True \<Rightarrow> {la_ex (ls, gs) a \<bind> (\<lambda>(ls, gs). Some \<lparr>global_config.processes = {#\<lparr>command = cmd', state = ls\<rparr>#} + lcs', state = gs\<rparr>)} | Some False \<Rightarrow> {} | Inr x \<Rightarrow> {None}))"
log_less_zero_cancel_iff: fixes a :: "real"   and x :: "real" assumes "1 < a"   and "0 < x" shows "(log a x < 0) = (x < 1)"
ChamberSubcomplexD_sub: fixes Y :: "'a set set" assumes "ChamberSubcomplex Y" shows "order.greater_eq X Y"
span_is_monotone: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T" shows "local.module.span S \<subseteq> local.module.span T"
in_hom_def: fixes f :: "'d"   and a :: "'d"   and b :: "'d" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright> \<equiv> a \<in> D.domains f \<and> b \<in> D.codomains f"
eq1: shows "(\<Union>b\<in>set bs. map_component (\<lambda>k. k + length bs) ` keys b) = (\<Union>i\<in>{0..<length bs}. map_component (\<lambda>k. k + length bs) ` keys (bs ! i))"
simps_14: fixes x14 :: "'a exp"   and y14 :: "'a exp" shows "(throw x14 = throw y14) = (x14 = y14)"
lnth_inf_llist: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "lnth (inf_llist f) n = f n"
dropWhile_eq_Nil_conv: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "(dropWhile P xs = []) = (\<forall>x\<in>set xs. P x)"
dvd_lcm_I1_nat: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k dvd m" shows "k dvd lcm m n"
restrict_one_left_unit: fixes ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and ls :: "'c list" shows "ks \<langle>mone\<rangle> ks \<odot> ks \<langle>f\<rangle> ls = ks \<langle>f\<rangle> ls"
tranclE: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> r\<^sup>+" obtains "(a, b) \<in> r"   | c :: "'a" where "(a, c) \<in> r\<^sup>+" and "(c, b) \<in> r"
thmD8: shows "\<J> 0 0 = Gen (\<P> 0 0)"
MPair: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<in> synth H"   and "Y \<in> synth H" shows "\<lbrace>X, Y\<rbrace> \<in> synth H"
is_run_def: fixes r :: "nat \<Rightarrow> 'Q" shows "is_run r \<equiv> r 0 \<in> V0 \<and> ipath E r"
remdups_adj_singleton_iff: fixes xs :: "'b list" shows "(\<^bold>|remdups_adj xs\<^bold>| = Suc 0) = (xs \<noteq> \<epsilon> \<and> xs = replicate \<^bold>|xs\<^bold>| (hd xs))"
assms_2: shows "\<not> (\<exists>i>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. \<parallel>c\<parallel>\<^bsub>t i\<^esub>)"
inline2_fixp_induct: fixes P :: "(('d, 'e, 'f) gpv \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> ('d \<times> 'g \<times> 'h + 'i \<times> ('k \<times> 'h, 'i, 'j) rpv \<times> ('f \<times> 'g, 'l, 'k) rpv \<times> ('d, 'e, 'f) rpv) spmf) \<Rightarrow> bool"   and callee1 :: "'h \<Rightarrow> 'l \<Rightarrow> ('k \<times> 'h, 'i, 'j) gpv"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g, 'l, 'k) gpv" assumes "spmf.admissible (\<lambda>inline2. P (\<lambda>gpv s2 s1. inline2 ((gpv, s2), s1)))"   and "P (\<lambda>_ _ _. return_pmf None)"   and "\<And>inline2'. P inline2' \<Longrightarrow> P (\<lambda>gpv s2 s1. the_gpv gpv \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf (Inl (x, s2, s1)) | IO out rpv \<Rightarrow> inline1 callee1 (callee2 s2 out) s1 \<bind> (\<lambda>lr. case lr of Inl ((r2, s2), c) \<Rightarrow> inline2' (rpv r2) s2 c | Inr (x, rpv'', rpv') \<Rightarrow> return_spmf (Inr (x, rpv'', rpv', rpv)))))" shows "P (inline2 callee1 callee2)"
cases: fixes a1 :: "'d DBMEntry"   and a2 :: "'d DBMEntry" assumes "a1 \<prec> a2" obtains uu_ :: "'d" where "a1 = Lt uu_" and "a2 = \<infinity>"   | uv_ :: "'d" where "a1 = Le uv_" and "a2 = \<infinity>"   | a :: "'d" and b :: "'d" where "a1 = Le a__" and "a2 = Le b__" and "a__ < b__"   | a :: "'d" and b :: "'d" where "a1 = Le aa__" and "a2 = Lt ba__" and "aa__ < ba__"   | a :: "'d" and b :: "'d" where "a1 = Lt ab__" and "a2 = Le bb__" and "ab__ \<le> bb__"   | a :: "'d" and b :: "'d" where "a1 = Lt a" and "a2 = Lt b" and "a < b"
Delta_null_of_null_is_null: fixes B :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "B \<in> sets M"   and "A \<Delta> B \<in> null_sets M"   and "A \<in> null_sets M" shows "B \<in> null_sets M"
piecewise_C1_differentiable_on_def: fixes f :: "real \<Rightarrow> 'c"   and i :: "real set" shows "f piecewise_C1_differentiable_on i \<equiv> continuous_on i f \<and> (\<exists>S. finite S \<and> f C1_differentiable_on i - S)"
pol_deg_le_n1: fixes p :: "'a"   and c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "p \<in> carrier R"   and "pol_coeff S c"   and "k \<le> fst c"   and "p = polyn_expr R X k c" shows "deg_n R S X p \<le> k"
take_rev: fixes n :: "nat"   and xs :: "'c list" shows "take n (rev xs) = rev (drop (length xs - n) xs)"
append_Cons: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "append (Cons x xs) ys = Cons x (append xs ys)"
mergeSL_def: fixes S :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and L :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'c"   and i :: "'a" shows "mergeSL S L f i \<equiv> if isSucc i then S (pred i) (f (pred i)) else L f i"
assms_1: shows "m1 \<in> carrier_mat (2 ^ n) (2 ^ n)"
rel_star_def: fixes X :: "('a \<times> 'a) set" shows "X\<^sup>* = \<Union> (range (rel_d.power X))"
a: shows "R[x\<turnstile>n>y] = AndL2 v.N u"
select_convs_3: fixes sprFst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprLst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprCRel :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set"   and more :: "'f" shows "sprCRel \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprCRel"
le_sup_iff: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "less_eq_prod (sup_prod x y) z = (less_eq_prod x z \<and> less_eq_prod y z)"
Succ_refl: fixes \<R> :: "('a \<Rightarrow> real) set set"   and X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and R :: "('a \<Rightarrow> real) set" assumes "\<R> = {Regions.region X I r |I r. Regions.valid_region X k I r}"   and "finite X"   and "R \<in> \<R>" shows "R \<in> Succ \<R> R"
rank_invertible_subset_cols: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a vec list" assumes "invertible_mat A"   and "A \<in> carrier_mat n n"   and "set B \<subseteq> set (cols A)"   and "distinct B" shows "vec_space.rank n (mat_of_cols n B) = length B"
merge_finite': fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat"   and j :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "i < j"   and "finite (mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F j)" shows "finite (mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F i)"
C0_def: fixes \<psi> :: "'a \<Rightarrow> complex" shows "C0 \<psi> = (\<lambda>b. case inv_map \<pi> b of None \<Rightarrow> 0 | Some x \<Rightarrow> \<psi> x)"
lookup_pair_update_by_pair: fixes xs :: "('a \<times> 'd) list"   and k1 :: "'a"   and v :: "'d"   and k2 :: "'a" assumes "oalist_inv_raw xs" shows "lookup_pair (update_by_pair (k1, v) xs) k2 = (if k1 = k2 then v else lookup_pair xs k2)"
Converter_def: fixes z :: "nat \<Rightarrow> bit list"   and x :: "nat \<Rightarrow> bit list" shows "Converter z x \<equiv> ts x \<and> (\<forall>t. hd (x t) = (if fin_make_untimed (inf_truncate z t) = [] then Zero else fin_make_untimed (inf_truncate z t) ! (length (fin_make_untimed (inf_truncate z t)) - 1)))"
A\<^sub>2_def: shows "A\<^sub>2 = {y \<in> A. x < y}"
UNIV_eq: shows "UNIV = Abs_bit0 ` {0..<int CARD('a bit0)}"
round_robin_update_thread_correct: fixes \<sigma> :: "'queue \<times> nat"   and T :: "'t set"   and t' :: "'t"   and n0 :: "nat"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" assumes "local.round_robin_invar \<sigma> T"   and "t' \<in> T" shows "round_robin_\<alpha> (local.round_robin_update_state n0 \<sigma> t ta) = Round_Robin.round_robin_update_state n0 (round_robin_\<alpha> \<sigma>) t ta"
ys: shows "(xs, ys) \<in> \<langle>R\<rangle>list_rel"
fmap_eq: shows "fmap\<cdot>d = PRJ('a\<cdot>'f) oo cast\<cdot>(TC('f)\<cdot>t) oo EMB('a\<cdot>'f)"
simps_1: fixes ys :: "'c list" shows "[] @ ys = ys"
convex_sum_bound_le: fixes I :: "'c set"   and x :: "'c \<Rightarrow> 'd"   and a :: "'c \<Rightarrow> 'd"   and b :: "'d"   and \<delta> :: "'d" assumes "\<And>i. i \<in> I \<Longrightarrow> (0::'d) \<le> x i"   and "sum x I = (1::'d)"   and "\<And>i. i \<in> I \<Longrightarrow> \<bar>a i - b\<bar> \<le> \<delta>" shows "\<bar>(\<Sum>i\<in>I. a i * x i) - b\<bar> \<le> \<delta>"
nAct_enat_the_nat: fixes n :: "enat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "cnf llist" assumes "n \<noteq> \<infinity>" shows "enat (the_enat (dynamic_component.nAct tCMP active c n t)) = dynamic_component.nAct tCMP active c n t"
ys: shows "ys = map p [0..<k] @ map p [k..<n]"
prefixE: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<le> ys" obtains zs :: "'a list" where "ys = xs @ zs"
unfold_congs_3: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "carrots r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "carrots_update f r = carrots_update f' r'"
healthy_unitary: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "healthy t"   and "unitary P" shows "unitary (t P)"
variables_of_def: shows "variables_of \<equiv> id \<circ> Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst sas_plus_problem_ext_Tuple_Iso"
t1_space_closed_map_image: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "closed_map X Y f"   and "f ` topspace X = topspace Y"   and "t1_space X" shows "t1_space Y"
take_map: fixes n :: "nat"   and f :: "'l \<Rightarrow> 'k"   and xs :: "'l list" shows "take n (map f xs) = map f (take n xs)"
d_neg: fixes x :: "'a" assumes "test x" shows "d (! x) = ! x"
generate_consistent: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H"   and "subgroup H G" shows "generate (G\<lparr>carrier := H\<rparr>) K = generate G K"
divide_zero_left: fixes z :: "complex_homo" assumes "z \<noteq> 0\<^sub>h" shows "0\<^sub>h :\<^sub>h z = 0\<^sub>h"
sum_list_ge_0_nth: fixes xs :: "'b list" assumes "\<And>i. i < length xs \<Longrightarrow> (0::'b) \<le> xs ! i" shows "(0::'b) \<le> sum_list xs"
emeasure_T_Int: fixes a :: "(int, 'n) vec"   and b :: "(int, 'n) vec" assumes "a \<noteq> b" shows "emeasure lebesgue (T a \<inter> T b) = 0"
match_prods: fixes X :: "'c"   and G :: "('c \<times> ('c, 'd) RHS) list"   and ls :: "'c list"   and rs :: "'c list" shows "(X \<in> set (match_prods G ls rs)) = (\<exists>A\<in>set ls. \<exists>B\<in>set rs. (X, Branch A B) \<in> set G)"
fresh_PairD_2: fixes a :: "atom"   and x :: "'b"   and y :: "'c" assumes "a \<sharp> (x, y)" shows "a \<sharp> y"
g_def: shows "g = (\<lambda>n. (ln (real n) + c * f n * ln (ln (real n))) / ln (f n))"
isometry_on_def: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "isometry_on X f = (\<forall>x\<in>X. \<forall>y\<in>X. dist (f x) (f y) = dist x y)"
mute_preserves_triangle: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "mute mf q k l (r, A) = (r', A')"   and "triangular_to l A"   and "l < k"   and "k < dim_row A"   and "l < dim_row A"   and "l < dim_col A" shows "triangular_to l A'"
is_lb_def: fixes s :: "val option"   and s1 :: "val option"   and s2 :: "val option" shows "is_lb s s1 s2 \<equiv> s \<preceq> s2 \<and> s \<preceq> s1"
choice: fixes k :: "nat" shows "fs' ! k = fs ! k \<or> fs' ! k = fs ! i \<or> fs' ! k = fs ! (i - 1)"
l12_18_d: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and P :: "'p" assumes "Cong A B C D"   and "Cong B C D A"   and "\<not> Col A B C"   and "B \<noteq> D"   and "Col A P C"   and "Col B P D" shows "A C TS B D"
node_in_scc_of_node: fixes v :: "'a"   and E :: "('a \<times> 'a) set" shows "v \<in> scc_of E v"