hcju/isabelleps huggingface dataset:
healthy_intros_15: fixes f :: "'a \<Rightarrow> 'a" shows "healthy (wp (Apply f))"
exhaust: fixes y :: "tvl" obtains "y = Yes"   | "y = No"   | "y = Maybe"
sublistI: fixes A :: "'a list"   and B :: "'a list" assumes "\<And>x. List.member A x \<Longrightarrow> List.member B x" shows "sublist A B"
assertion_axioms: shows "assertion (\<otimes>) SImp (\<bottom>) SChanEq"
valid_history: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "history_consistent \<theta> (hd_prog p sb) sb"
cos_differentiable_at: fixes f :: "'b \<Rightarrow> real"   and x :: "'b"   and X :: "'b set" assumes "f differentiable at x within X" shows "(\<lambda>x. cos (f x)) differentiable at x within X"
that: fixes q :: "'a mpoly"   and r :: "'a mpoly" assumes "p = MPoly_Type.monom (monomial (Suc 0) x') (1::'a) * q + r"   and "x' \<notin> vars r" shows "thesis"
that: fixes l :: "nat" assumes "i < l"   and "l \<le> j"   and "tts (s l) t = HOLD tk" shows "thesis"
simps_10: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and \<nu> :: "'a set" shows "af_letter (\<phi> U \<psi>) \<nu> = \<phi> U \<psi> and af_letter \<phi> \<nu> or af_letter \<psi> \<nu>"
Bs_1: shows "B1 \<in> carrier_mat n1 n1"
norm_fstzero_zerosnd_3: shows "norm (no_real_inner.fstzero - no_real_inner.zerosnd) = 1"
xy_in_set_cases: fixes x :: "'b"   and l :: "'b list"   and y :: "'b" assumes "x \<in> set l"   and "y \<in> set l" obtains l1 :: "'b list" and l2 :: "'b list" where "x = y" and "l = l1__ @ y # l2__"   | l1 :: "'b list" and l2 :: "'b list" and l3 :: "'b list" where "x \<noteq> y" and "l = l1a__ @ x # l2a__ @ y # l3__"   | l1 :: "'b list" and l2 :: "'b list" and l3 :: "'b list" where "x \<noteq> y" and "l = l1 @ y # l2 @ x # l3"
frontier_closures: fixes S :: "'b set" shows "frontier S = closure S \<inter> closure (- S)"
all_ex_finite_set: fixes m :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> bool" assumes "\<forall>j\<in>{0..<m}. \<exists>n. P j n"   and "\<forall>n1 n2. \<forall>j\<in>{0..<m}. P j n1 \<longrightarrow> P j n2 \<longrightarrow> n1 = n2" shows "finite {n. \<exists>j\<in>{0..<m}. P j n}"
replace_comp_trading_strat: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'c" assumes "trading_strategy pf1"   and "trading_strategy pf2" shows "trading_strategy (qty_replace_comp pf1 x pf2)"
mpoly_induct: fixes P :: "'a mpoly \<Rightarrow> bool"   and p :: "'a mpoly" assumes "\<And>m a. P (monom m a)"   and "\<And>p1 p2 m a. \<lbrakk>P p1; P p2; p2 = monom m a; m \<notin> monomials p1; a \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> P (p1 + p2)" shows "P p"
trunc_closed: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "Cring_Poly.truncate R f \<in> carrier P"
Star_def: fixes r :: "'a language" shows "Star r = StarLR One r"
inj_on_span_independent_image: fixes B :: "'b set" assumes "m2.independent (f ` B)"   and "inj_on f B" shows "inj_on f (m1.span B)"
globalsimp: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and s :: "(nat \<Rightarrow> 'a) \<times> 'b" shows "global P s = P (fst s)"
simps_11: fixes var :: "nat"   and uw :: "atom fm" shows "grab_atoms var (AllQ uw) = None"
and_dfa_def: shows "and_dfa = binop_dfa (\<and>)"
has_integral_component_ubound: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a"   and k :: "'b"   and B :: "real" assumes "(f has_integral i) (cbox a b)"   and "\<forall>x\<in>cbox a b. f x \<bullet> k \<le> B"   and "k \<in> Basis" shows "i \<bullet> k \<le> B * content (cbox a b)"
wf_design_iff: fixes point_set :: "'c set"   and block_collection :: "'c set multiset"   and bl :: "'c set" assumes "design point_set block_collection"   and "bl \<in># block_collection" shows "design point_set block_collection = (bl \<subseteq> point_set \<and> finite point_set \<and> bl \<noteq> {})"
map_member_minus: fixes x :: "'c"   and y :: "'d"   and f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "((x, y) \<in>\<^sub>m f -- g) = ((x, y) \<in>\<^sub>m f \<and> \<not> (x, y) \<in>\<^sub>m g)"
l2_5_1_B: fixes S :: "('b \<times> 'b \<Rightarrow> bool) set"   and P :: "'b \<times> 'b \<Rightarrow> bool" shows "(\<forall>X\<in>S. \<forall>A. (X \<longrightarrow> P) A) \<longrightarrow> (\<forall>A. ((\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> S}) \<longrightarrow> P) A)"
h'_def: shows "h' \<equiv> \<lambda>z. if fst z \<in> {0} then f (snd z) else if fst z \<in> {1} then g (snd z) else h z"
fresh_symvar_def: fixes sv :: "'c \<times> nat"   and c :: "('c, 'd) conf" shows "Conf.fresh_symvar sv c = (sv \<notin> Conf.symvars c)"
0: shows "rawpsubst (exi y \<phi>) (zip (map Var ws) (map snd txs)) = exi y (rawpsubst \<phi> (zip (map Var ws) (map snd txs)))"
power3_eq_cube: fixes a :: "'a" shows "a ^ 3 = a * a * a"
tm_append_steps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and A :: "(action \<times> nat) list"   and stpa :: "nat"   and la :: "cell list"   and ra :: "cell list"   and B :: "(action \<times> nat) list"   and stpb :: "nat"   and sb :: "nat"   and lb :: "cell list"   and rb :: "cell list"   and off :: "nat" assumes "steps0 (s, l, r) A stpa = (Suc (length A div 2), la, ra)"   and "steps0 (Suc 0, la, ra) B stpb = (sb, lb, rb)"   and "sb \<noteq> 0"   and "off = length A div 2"   and "is_even (length A)" shows "steps0 (s, l, r) (A @ shift B off) (stpa + stpb) = (sb + off, lb, rb)"
f_lesbegue_integrable: shows "integrable lborel (\<lambda>x. if x \<in> Dy_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x else 0)"
rel_pmf_eqI: fixes x :: "'c pmf" shows "rel_pmf (=) x x"
simps: fixes a :: "assertion \<times> command \<times> assertion" shows "prov_triple a = ((\<exists>p c q x. a = (Exists x p, c, Exists x q) \<and> prov_triple (p, c, q)) \<or> (\<exists>p c1 q c2. a = (p, Choose c1 c2, q) \<and> prov_triple (p, c1, q) \<and> prov_triple (p, c2, q)) \<or> (\<exists>p c. a = (p, Loop c, p) \<and> prov_triple (p, c, p)) \<or> (\<exists>p c q r. a = (p \<star> r, c, q \<star> r) \<and> prov_triple (p, c, q) \<and> wr_com c \<inter> rd_ass r = {}) \<or> (\<exists>p. a = (p, Skip, p)) \<or> (\<exists>p c1 q c2 r. a = (p, c1 ;; c2, r) \<and> prov_triple (p, c1, q) \<and> prov_triple (q, c2, r)))"
iteration_unfoldr': fixes z :: "'a"   and y :: "'a"   and x :: "'a" shows "z \<cdot> y + z \<cdot> x\<^sup>\<infinity> \<cdot> x \<cdot> y = z \<cdot> x\<^sup>\<infinity> \<cdot> y"
size_eq_Suc_imp_elem: fixes M :: "'c multiset"   and n :: "nat" assumes "size M = Suc n" shows "\<exists>a. a \<in># M"
assms_3: shows "rk {A, B, C, A', B', C', P} = 3"
no_deadends: fixes v :: "'a" assumes "v \<in> V" shows "\<not> deadend v"
X_1: shows "X \<in> M \<rightarrow>\<^sub>M N"
inf_chain_bad: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "inf_chain p f" shows "bad p (f i)"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b fset" shows "fset (f |`| v) = f ` fset v"
assms: shows "Negligible.negligible (\<lambda>n. local.dis_log.advantage n (local.dis_log_alt.adversary2 (okamoto_\<Sigma>_commit.adversary n (\<A> n))))"
openin_subset_trans: fixes U :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "openin (top_of_set U) S"   and "S \<subseteq> T"   and "T \<subseteq> U" shows "openin (top_of_set T) S"
fin_finite: fixes r :: "'b llist"   and A :: "'b set" assumes "r \<in> A\<^sup>\<star>"   and "r \<notin> UNIV\<^sup>\<star>" shows "False"
simps_1: fixes I :: "nat set" shows "I \<leftarrow> 0 = Max I"
c3_4: shows "lst_conv (conv_mirror \<gamma>3) = fst \<gamma>1'''"
transpose_mat_def: fixes A :: "'a Matrix.mat" shows "A\<^sup>t \<equiv> Matrix.mat (dim_col A) (dim_row A) (\<lambda>(i, j). A $$ (j, i))"
B: shows "\<Theta>,\<Gamma> \<turnstile> B"
iso_simple: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e" assumes "simple_group G"   and "Group.group H"   and "\<phi> \<in> Group.iso G H" shows "simple_group H"
real_vector_space_pair_on_def: fixes S :: "'c set"   and T :: "'d set" shows "real_vector_space_pair_on S T \<equiv> real_vector_space_on S \<and> real_vector_space_on T"
stutter_equiv_sym: fixes \<sigma> :: "nat \<Rightarrow> 'f"   and \<tau> :: "nat \<Rightarrow> 'f" assumes "\<sigma> \<approx> \<tau>" shows "\<tau> \<approx> \<sigma>"
ipresIGSubstAll_def: fixes h :: "'u \<Rightarrow> 'v"   and hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'varSort, 'sort, 'aa, 'ab, 'u, 'w, 'ac) model_scheme"   and MOD' :: "('ad, 'ae, 'varSort, 'af, 'ag, 'ab, 'v, 'x, 'ah) model_scheme" shows "ipresIGSubstAll h hA MOD MOD' \<equiv> ipresIGSubst h MOD MOD' \<and> ipresIGSubstAbs h hA MOD MOD'"
keys_lift_poly_syz_aux: fixes i :: "nat"   and n :: "nat"   and b :: "'t \<Rightarrow>\<^sub>0 'c" shows "{x. (if pair_of_term x = (0::'a, i) then 1::'c else if n \<le> component_of_term x then lookup b (map_component (\<lambda>k. k - n) x) else (0::'c)) \<noteq> (0::'c)} \<subseteq> insert (term_of_pair (0::'a, i)) (map_component (\<lambda>k. k + n) ` keys b)"
inj_on_image_set_diff: fixes f :: "'c \<Rightarrow> 'd"   and C :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f C"   and "A - B \<subseteq> C"   and "B \<subseteq> C" shows "f ` (A - B) = f ` A - f ` B"
aprimedivisor_prime_power: fixes p :: "'a"   and k :: "nat" assumes "prime p"   and "0 < k" shows "aprimedivisor (p ^ k) = p"
inv_chart_in_domain: fixes y :: "'c"   and c :: "('d, 'c) chart" assumes "y \<in> codomain c" shows "inv_chart c y \<in> domain c"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "in_hom f a b"   and "in_hom g b c" shows "seq g f"
simps_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and c :: "color"   and a :: "('a, 'b) rbt"   and y :: "'a"   and s :: "'b"   and b :: "('a, 'b) rbt" shows "ord.rbt_del less x (Branch c a y s b) = (if less x y then ord.rbt_del_from_left less x a y s b else if less y x then ord.rbt_del_from_right less x a y s b else combine a b)"
WhileTrue: fixes b :: "state \<Rightarrow> bool"   and s :: "state"   and c :: "com"   and n :: "nat"   and t :: "state"   and u :: "state" assumes "b s"   and "s -c-n\<rightarrow> t"   and "t -WHILE b DO c-n\<rightarrow> u" shows "s -WHILE b DO c-n\<rightarrow> u"
t_cvt: fixes t :: "t"   and sx :: "sx option"   and v :: "v"   and v' :: "v" assumes "cvt t sx v = Some v'" shows "t = typeof v'"
subadditive: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E" shows "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>"
content_subset: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "cbox a b \<subseteq> cbox c d" shows "content (cbox a b) \<le> content (cbox c d)"
gauss_cnj_dvd: fixes a :: "gauss_int"   and b :: "gauss_int" assumes "a dvd b" shows "gauss_cnj a dvd gauss_cnj b"
star_n_eq_starfun_whn: fixes X :: "nat \<Rightarrow> 'a" shows "star_n X = (*f* X) whn"
normal_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.group_add minus plus zero uminus" shows "group_add.normal plus = {A. \<forall>a. group_add.add_set plus A {a} = group_add.add_set plus {a} A}"
simps_18: fixes n :: "nat"   and vs :: "real list" shows "interpret_floatarith (Var n) vs = vs ! n"
genset_order2_add: fixes S :: "'b set"   and s :: "'b" assumes "PreCoxeterSystem S"   and "s \<in> S" shows "s + s = (0::'b)"
strip_comb_size: fixes t' :: "'a"   and t :: "'a" assumes "t' \<in> set (snd (strip_comb t))" shows "size t' < size t"
rev_exhaust: fixes xs :: "'a list" obtains "xs = []"   | ys :: "'a list" and y :: "'a" where "xs = ys ## y"
inj_on_imageI: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd"   and A :: "'b set" assumes "inj_on (g \<circ> f) A" shows "inj_on g (f ` A)"
power_minus1_even: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus one) (2 * n) = one"
flow_cong: fixes Y :: "'a set"   and S :: "real set"   and g :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "real"   and x0 :: "'a" assumes "Y = Y"   and "UNIV = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> g x = g t x"   and "t \<in> Y.existence_ivl0 x0" shows "Y.flow0 x0 t = ll_on_open.flow S g Y 0 x0 t"
prod_mset_singleton: fixes x :: "'a" shows "prod_mset {#x#} = x"
assms: shows "low_equal s1 s2 \<and> (fst instr = load_store_type LDSB \<or> fst instr = load_store_type LDUB \<or> fst instr = load_store_type LDUH \<or> fst instr = load_store_type LD \<or> fst instr = load_store_type LDD) \<and> t1 = snd (fst (load_sub1 instr rd 0 s1)) \<and> t2 = snd (fst (load_sub1 instr rd 0 s2))"
restriction_to_subgroup_is_module: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and H :: "'a set" assumes "FGModule G smult V"   and "Group H \<and> H \<subseteq> G" shows "FGModule H smult V"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d pmf" shows "set_pmf (map_pmf f v) = f ` set_pmf v"
independent_explicit_module: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'d set" assumes "module scale" shows "(\<not> module.dependent scale s) = (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. scale (u v) v) = (0::'d) \<longrightarrow> v \<in> t \<longrightarrow> u v = (0::'c))"
run_one_step_basic_set_local_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x15 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Set_local x15) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
iso_class_def: fixes f :: "'a \<times> 'a" shows "VoV.iso_class f \<equiv> {f'. VoV.isomorphic f f'}"
tm_exclude: fixes t :: "real" assumes "t \<in> {tm<..<t2}" shows "flow0 x t \<notin> {a<--<b}"
MC_with_rewards_axioms: shows "MC_with_rewards \<iota> \<rho>"
finsert_is_funion: fixes a :: "'b"   and A :: "'b fset" shows "finsert a A = {|a|} |\<union>| A"
monotone_Pair: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'd" assumes "monotone ord orda f"   and "monotone ord ordb g" shows "monotone ord (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
w\<theta>'\<nu>: shows "ide w \<and> \<guillemotleft>\<theta>' : f \<star> w \<Rightarrow> src r\<guillemotright> \<and> \<guillemotleft>\<nu> : r \<Rightarrow> g \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> ((r \<star> \<theta>') \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w)) \<cdot> \<nu> = \<r>\<^sup>-\<^sup>1[r]"
dist_t: shows "(all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>t) \<union> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>t) \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>t)) \<inter> (all_shared sb' \<union> all_unshared sb' \<union> all_acquired sb') = {}"
simps: fixes x :: "'b extB"   and y :: "'b extB" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
eq_on_def: fixes A :: "'b set" shows "eq_on A = (\<lambda>x y. x \<in> A \<and> x = y)"
mono_neutral_left: fixes T :: "'b set"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'a)" shows "prod g S = prod g T"
isTrivialPhi_asymmetric: fixes g :: "'g"   and a :: "'val"   and b :: "'val" assumes "uninst_code.ssa.isTrivialPhi phis g a b"   and "uninst_code.ssa.isTrivialPhi phis g b a" shows "False"
ltakeWhile_K_True: fixes xs :: "'c llist" shows "ltakeWhile (\<lambda>_. True) xs = xs"
determ_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "determ t \<equiv> Transformers.additive t \<and> maximal t"
elims_2: fixes b :: "nat list"   and x :: "nat list"   and xa :: "nat list"   and xb :: "nat" assumes "cond1 b x xa xb" obtains ys :: "nat list" and s :: "nat" where "x = ys__" and "xa = []" and "xb = s__"   | ys :: "nat list" and xc :: "nat" and xs :: "nat list" and s :: "nat" where "x = ys" and "xa = xc # xs" and "xb = s" and "s \<le> b \<bullet> ys \<and> xc \<le> maxne0_impl ys b"
subdomain_is_domain: fixes H :: "'a set" assumes "subdomain H R" shows "domain (R\<lparr>carrier := H\<rparr>)"
ide_char: fixes a :: "'b \<times> 'b" shows "D.VV.ide a = (D.VV.arr a \<and> D.VxV.ide a)"
red_union: fixes F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and G :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" shows "punit.red (F \<union> G) p q = (punit.red F p q \<or> punit.red G p q)"
exchange: fixes s :: "'w"   and w :: "'w"   and ss :: "'w list" assumes "s \<in> S"   and "S_reduced_for w ss"   and "\<not> S_reduced (s # ss)" shows "\<exists>t as bs. ss = as @ t # bs \<and> S_reduced_for w (s # as @ bs)"
fresh_3: fixes a :: "atom"   and tm1 :: "tm"   and tm2 :: "tm" shows "a \<sharp> Eats tm1 tm2 = (a \<sharp> tm1 \<and> a \<sharp> tm2)"
simps_6: shows "\<infinity>\<^sub>1 +\<^sub>1 [0]\<^sub>1 = \<infinity>\<^sub>1"
set_map_operator_precondition: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and ops :: "'b strips_operator list" shows "set (map (\<lambda>(k, op). encode_operator_precondition \<Pi> k op) (List.product [0..<t] ops)) = {encode_operator_precondition \<Pi> k op |k op. (k, op) \<in> {0..<t} \<times> set ops}"
gcd_impl_main: fixes G2 :: "'a poly"   and G1 :: "'a poly" assumes "length (coeffs G2) \<le> length (coeffs G1)" shows "gcd_impl_main G1 G2 = gcd G1 G2"
flow_implicit_function: fixes t :: "real"   and x :: "'a"   and s :: "'a \<Rightarrow> real"   and Ds :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" assumes "t \<in> existence_ivl0 x"   and "x \<in> X"   and "s (flow0 x t) = 0"   and "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"   and "isCont Ds (flow0 x t)"   and "blinfun_apply (Ds (flow0 x t)) (f (flow0 x t)) \<noteq> 0" obtains u :: "'a \<Rightarrow> real" and e :: "real" where "s (flow0 x (u x)) = 0" and "u x = t" and "\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0" and "continuous_on (cball x e) u" and "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0" and "0 < e" and "(u has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (Ds (flow0 x t)) (f (flow0 x t)))) o\<^sub>L (Ds (flow0 x t) o\<^sub>L flowderiv x t) o\<^sub>L embed1_blinfun)) (at x)"
finite_set_of_finite_funs2: fixes A :: "'d set"   and B :: "'e set"   and C :: "'f set"   and d :: "'f" assumes "finite A"   and "finite B"   and "finite C" shows "finite {f. \<forall>x y. (x \<in> A \<and> y \<in> B \<longrightarrow> f x y \<in> C) \<and> (x \<notin> A \<longrightarrow> f x y = d) \<and> (y \<notin> B \<longrightarrow> f x y = d)}"
strong_get_put: fixes \<sigma> :: "'b"   and \<rho> :: "'b" shows "put \<sigma> (get \<rho>) = \<rho>"
sum_list_map_power2_poly: fixes ls :: "real poly list"   and x :: "real" shows "0 \<le> poly (sum_list (map power2 ls)) x"
prems_4: fixes k :: "nat"   and xs :: "event_data option list set list"   and z :: "'a" assumes "i \<le> k"   and "Suc k \<le> Mini i js"   and "list_all2 (\<lambda>P. P k) Ps xs"   and "U k z" shows "U (Suc k) (f xs z)"
basic_trans_rules_31: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
sees_method_mono2: fixes C' :: "char list"   and C :: "char list"   and P :: "'b prog"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list"   and b' :: "staticb"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "'b"   and D' :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' sees M, b' :  Ts'\<rightarrow>T' = m' in D'" shows "b = b' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"
directedD: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "directed A r"   and "x \<in> A"   and "y \<in> A" shows "\<exists>z\<in>A. r x z \<and> r y z"
sum_left_div_distrib: fixes f :: "'a \<Rightarrow> real"   and x :: "real"   and I :: "'a set" shows "(\<Sum>i\<in>I. f i / x) = sum f I / x"
minWait_least: fixes n :: "nat"   and rs :: "'rule stream"   and s :: "'state" assumes "enabled (shd (sdrop n rs)) s" shows "minWait rs s \<le> n"
preserves_arr: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.arr (src f)"
eq_effect_eq_vset: fixes x :: "('c, bool) fmap \<times> ('c, bool) fmap"   and y :: "('c, bool) fmap \<times> ('c, bool) fmap"   and vs :: "'c set" assumes "snd x = snd y" shows "varset_action x vs = varset_action y vs"
assms: shows "intersect_segment_xline prec seg x = Some (m, M)"
simps_1: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.meld [] bq = bq"
subsumee_not_marked: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and sub :: "('d \<times> nat) \<times> 'd \<times> nat" assumes "RedBlack prb"   and "sub \<in> subs prb" shows "\<not> marked prb (subsumee sub)"
inducts_2: fixes x5 :: "(name \<times> ty) list"   and x6 :: "trm"   and x7 :: "trm"   and x8 :: "ty"   and P2 :: "(name \<times> ty) list \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and P1 :: "(name \<times> ty) list \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool" assumes "x5 \<turnstile> x6 \<leftrightarrow> x7 : x8"   and "\<And>s p t q \<Gamma>. \<lbrakk>s \<Down> p; t \<Down> q; \<Gamma> \<turnstile> p \<leftrightarrow> q : TBase; P2 \<Gamma> p q TBase\<rbrakk> \<Longrightarrow> P1 \<Gamma> s t TBase"   and "\<And>x \<Gamma> s t T\<^sub>1 T\<^sub>2. \<lbrakk>x \<sharp> (\<Gamma>, s, t); (x, T\<^sub>1) # \<Gamma> \<turnstile> App s (Var x) \<Leftrightarrow> App t (Var x) : T\<^sub>2; P1 ((x, T\<^sub>1) # \<Gamma>) (App s (Var x)) (App t (Var x)) T\<^sub>2\<rbrakk> \<Longrightarrow> P1 \<Gamma> s t (T\<^sub>1\<rightarrow>T\<^sub>2)"   and "\<And>\<Gamma> s t. valid \<Gamma> \<Longrightarrow> P1 \<Gamma> s t TUnit"   and "\<And>\<Gamma> x T. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P2 \<Gamma> (Var x) (Var x) T"   and "\<And>\<Gamma> p q T\<^sub>1 T\<^sub>2 s t. \<lbrakk>\<Gamma> \<turnstile> p \<leftrightarrow> q : T\<^sub>1\<rightarrow>T\<^sub>2; P2 \<Gamma> p q (T\<^sub>1\<rightarrow>T\<^sub>2); \<Gamma> \<turnstile> s \<Leftrightarrow> t : T\<^sub>1; P1 \<Gamma> s t T\<^sub>1\<rbrakk> \<Longrightarrow> P2 \<Gamma> (App p s) (App q t) T\<^sub>2"   and "\<And>\<Gamma> n. valid \<Gamma> \<Longrightarrow> P2 \<Gamma> (Const n) (Const n) TBase" shows "P2 x5 x6 x7 x8"
type_substitution_lemma: fixes X :: "tyvrs"   and Y :: "tyvrs"   and L :: "ty"   and M :: "ty"   and N :: "ty" assumes "X \<noteq> Y"   and "X \<sharp> L" shows "M[X \<mapsto> N]\<^sub>\<tau>[Y \<mapsto> L]\<^sub>\<tau> = M[Y \<mapsto> L]\<^sub>\<tau>[X \<mapsto> N[Y \<mapsto> L]\<^sub>\<tau>]\<^sub>\<tau>"
universe_closure: shows "\<exists>x. \<forall>y. P y x"