hcju/isabelleps huggingface dataset:
Times_eq_cancel2: fixes x :: "'c"   and C :: "'c set"   and A :: "'d set"   and B :: "'d set" assumes "x \<in> C" shows "(A \<times> C = B \<times> C) = (A = B)"
Ramsey_induction: fixes YY :: "'b set"   and r :: "nat"   and s :: "nat"   and f :: "'b set \<Rightarrow> nat" assumes "infinite YY"   and "part_fn r s YY f" shows "\<exists>Y' t'. Y' \<subseteq> YY \<and> infinite Y' \<and> t' < s \<and> (\<forall>X. X \<subseteq> Y' \<and> finite X \<and> card X = r \<longrightarrow> f X = t')"
prems_4: shows "(s, y) \<in> set (tx # txs)"
sum_not_less_2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.weight plus zero less_eq less"   and "less_eq zero a" shows "\<not> less (plus a b) b"
nth_tl_if: fixes ys :: "'b list"   and a :: "'b"   and P :: "'b list \<Rightarrow> bool" assumes "ys \<noteq> []"   and "ys ! 0 = a"   and "P ys" shows "P (a # tl ys)"
Inl_Inr_False: fixes x :: "'c"   and y :: "'d" shows "(Inl x = Inr y) = False"
image_mset_filter_swap: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool"   and M :: "'b multiset" shows "image_mset f {#x \<in># M. P (f x)#} = filter_mset P (image_mset f M)"
neg_one_index: fixes i :: "'b" shows "(- 1) $ i = - (1::'a)"
pcompose_conjugates_integer: fixes p :: "complex poly"   and i :: "nat" assumes "\<And>i. coeff p i \<in> \<int>" shows "coeff (p \<circ>\<^sub>p [:0, \<i>:] * p \<circ>\<^sub>p [:0, - \<i>:]) i \<in> \<int>"
integrable_abs: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "integrable M (\<lambda>x. \<bar>f x\<bar>)"
homeomorphic_Euclidean_space_product_topology: fixes n :: "nat" shows "Euclidean_space n homeomorphic_space powertop_real {..<n}"
pre_suf_eq_distinct_list: fixes xs :: "'c list"   and ys :: "'c list"   and pre1 :: "'c list"   and suf1 :: "'c list"   and pre2 :: "'c list"   and suf2 :: "'c list" assumes "distinct xs"   and "ys \<noteq> []"   and "pre1 @ ys @ suf1 = xs"   and "pre2 @ ys @ suf2 = xs" shows "pre1 = pre2 \<and> suf1 = suf2"
LCons2: fixes ys :: "'b llist"   and y :: "'b" assumes "lfinite ys"   and "P LNil ys" shows "P LNil (LCons y ys)"
upper_dim_implies_perp2__col: assumes "UpperDimAxiom" shows "\<forall>X Y Z A B. X Y Perp A B \<and> X Z Perp A B \<longrightarrow> Col X Y Z"
sig_of_p: shows "sig_of_pair p = lt (poly_of_pair p)"
check_proof_def: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and P :: "proofterm"   and res :: "Core.term" shows "check_proof thy P res \<equiv> wf_theory thy \<and> replay thy P = Some res"
ideal_prod_assoc: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set"   and K :: "'c set" assumes "ring R"   and "ideal I R"   and "ideal J R"   and "ideal K R" shows "I \<cdot>\<^bsub>R\<^esub> J \<cdot>\<^bsub>R\<^esub> K = I \<cdot>\<^bsub>R\<^esub> (J \<cdot>\<^bsub>R\<^esub> K)"
prime_divprod_pow_nat: fixes p :: "nat"   and a :: "nat"   and b :: "nat"   and n :: "nat" assumes "prime_elem p"   and "coprime a b"   and "p ^ n dvd a * b" shows "p ^ n dvd a \<or> p ^ n dvd b"
exhaust: fixes y :: "'a ppath" obtains x1 :: "'a" where "y = Node x1"   | x21 :: "'a" and x22 :: "'a ppath" where "y = ppath.Cons x21 x22"
dom_lookup_keys: fixes al :: "('a, 'b) alist" shows "dom (DAList.lookup al) = keys al"
widen_null: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list" shows "P \<turnstile> NT \<le> Class C"
Cauchy_cols: fixes X :: "nat \<Rightarrow> 'a sq_mtx"   and i :: "'a" assumes "Cauchy X" shows "Cauchy (\<lambda>n. \<c>\<o>\<l> i (X n))"
simps_2: fixes V :: "'d set"   and t :: "('e, 'd) Term.term"   and S :: "('e, 'd) strand_step list" shows "wf\<^sub>s\<^sub>t V (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S) = (fv t \<subseteq> V \<and> wf\<^sub>s\<^sub>t V S)"
boundedE: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<preceq> linordered_monoid.min b c" obtains "a \<preceq> b" and "a \<preceq> c"
smult_assoc1: fixes a :: "'a"   and b :: "'a"   and x :: "'c" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier M" shows "a \<otimes> b \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"
prefix_to_wordset_ipset_from_cidr: fixes pfx :: "'a prefix_match" assumes "valid_prefix pfx" shows "prefix_to_wordset pfx = ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
assms: shows "eval cid t t' n (\<not>\<^sup>b \<gamma>)"
step: fixes s :: "'a net_state"   and t :: "'a net_state"   and a :: "'b node_action"   and s' :: "'a net_state"   and t' :: "'a net_state" assumes "SubnetS s t \<in> reachable (closed (pnet np (p1 \<parallel> p2))) I"   and "P s t"   and "(SubnetS s t, a, SubnetS s' t') \<in> automaton.trans (closed (pnet np (p1 \<parallel> p2)))"   and "I a" shows "P s' t'"
shortestPath_def: fixes g :: "'g"   and n :: "'node" shows "shortestPath g n \<equiv> LEAST l. \<exists>ns. length ns = l \<and> g \<turnstile> Entry g-ns\<rightarrow>n"
monotone_if_bot: fixes g :: "'a \<Rightarrow> 'b"   and bound :: "'a"   and bot :: "'b"   and f :: "'a \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. g x = (if bound \<le> x then bot else f x)"   and "\<And>x y. \<lbrakk>y \<le> x; \<not> bound \<le> x\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>x. \<not> bound \<le> x \<Longrightarrow> ord bot (f x)"   and "ord bot bot" shows "monotone (\<lambda>x y. y \<le> x) ord g"
inj_on_inverseI: fixes A :: "'e set"   and g :: "'f \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'f" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
polynomial_pf: fixes w :: "nat \<Rightarrow> real" shows "insertion w polynomial_p = polynomial_f w"
S_list_image_gallery: fixes ss :: "'a permutation list" assumes "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "gallery (map (\<lambda>w. w `\<rightarrow> C0) (sums ss))"
rk_ABC\<alpha>\<beta>\<gamma>: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points"   and R :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {Q, R} = 2"   and "rk {R, A, a} = 2"   and "rk {R, B, b} = 2"   and "rk {R, C, c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {A, B, C, \<alpha>, \<beta>, \<gamma>} = 3"
is_lubI: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" assumes "class.po below"   and "po.is_ub below S x"   and "\<And>u. po.is_ub below S u \<Longrightarrow> below x u" shows "po.is_lub below S x"
swap_apply_3: fixes c :: "'b"   and a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "c \<noteq> a"   and "c \<noteq> b" shows "Fun.swap a b f c = f c"
that: fixes b :: "'a" assumes "b \<in> S"   and "open_segment a b \<inter> S = {}" shows "thesis"
ctr_2: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g21 :: "'a \<Rightarrow> 'b"   and g22 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "unfold_llist p g21 g22 a = LCons (g21 a) (unfold_llist p g21 g22 (g22 a))"
distinct_mset_add: fixes L :: "'a multiset"   and L' :: "'a multiset" shows "distinct_mset (L + L') = (distinct_mset L \<and> distinct_mset L' \<and> L \<inter># L' = {#})"
word_of_int_eq_iff: fixes k :: "int"   and l :: "int" shows "(word_of_int k = word_of_int l) = (take_bit LENGTH('c) k = take_bit LENGTH('c) l)"
finite_nat_iff_bounded_le2: fixes S :: "nat set" shows "finite S = (\<exists>k. \<forall>n\<in>S. n \<le> k)"
fin: shows "finite {p. prime p \<and> real p \<le> x}"
bigtheta_trans2: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)"   and "g \<in> \<omega>[F](h)" shows "f \<in> \<omega>[F](h)"
term_ok'_mk_all: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and B :: "Core.term"   and \<tau> :: "typ"   and x :: "variable" assumes "wf_theory \<Theta>"   and "term_ok' (sig \<Theta>) B"   and "typ_of B = Some propT"   and "typ_ok \<Theta> \<tau>" shows "term_ok' (sig \<Theta>) (mk_all x \<tau> B)"
typeof_conf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val"   and T :: "ty" assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> v :\<le> T"
bij_inv_eq_iff: fixes p :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e" assumes "bij p" shows "(x = inv p y) = (p x = y)"
mono: fixes c :: "'a Word.word"   and a :: "'a Word.word"   and d :: "'a Word.word"   and b :: "'a Word.word" assumes "c \<le>s a"   and "d \<le>s b" shows "signed.max c d \<le>s signed.max a b"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "{#size ta, size sa#} < {#size t, size s#}"   and "ground ta"   and "ground sa" shows "ta >\<^sub>t sa \<or> sa >\<^sub>t ta \<or> ta = sa"
WR_ord_of: fixes i :: "nat"   and e :: "(name, hf) finfun" shows "WR (ord_of i) \<lbrakk>\<guillemotleft>ORD_OF i\<guillemotright>\<rbrakk>e"
reaches: fixes a :: "'a"   and b :: "'a" assumes "G'.reaches a b" shows "a \<rightarrow>* b"
Un_UNIV_right: fixes A :: "'b set" shows "A \<union> UNIV = UNIV"
Tgs_Issues_A: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "evs \<in> kerbV" shows "Tgs Issues A with Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace> on evs"
map_comp: fixes g1 :: "'g \<Rightarrow> 'd"   and g2 :: "'h \<Rightarrow> 'e"   and g3 :: "'i \<Rightarrow> 'f"   and f1 :: "'j \<Rightarrow> 'g"   and f2 :: "'k \<Rightarrow> 'h"   and f3 :: "'l \<Rightarrow> 'i"   and v :: "('j, 'k, 'l) F1" shows "F1map g1 g2 g3 (F1map f1 f2 f3 v) = F1map (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) v"
solution_uminus_left: fixes x :: "'a"   and y :: "'a" shows "solution (- x, y) = solution (x, y)"
kth_state_suc: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" shows "\<sigma>\<^bsup>Suc k\<^esup> = snd (local.step (path \<sigma> k, \<sigma>\<^bsup>k\<^esup>))"
sum_dcharacter_antimono_bound: fixes x0 :: "real"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "\<chi> \<noteq> \<chi>\<^sub>0"   and "0 \<le> x0"   and "x0 \<le> a"   and "a < b"   and "\<And>x. x0 \<le> x \<Longrightarrow> (f has_real_derivative f' x) (at x)"   and "\<And>x. x0 \<le> x \<Longrightarrow> 0 \<le> f x"   and "\<And>x. x0 \<le> x \<Longrightarrow> f' x \<le> 0" shows "cmod (\<Sum>n\<in>real -` {a<..b}. \<chi> n * complex_of_real (f (real n))) \<le> 2 * real (totient n) * f a"
horizontal_composition_axioms: shows "horizontal_composition (\<bullet>) (\<star>) src trg"
ide_char: fixes a :: "'a" shows "ide a = (arr a \<and> local.dom a = a \<and> cod a = a)"
pow_eq_id: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G" shows "(x [^] n = \<one>) = (ord x dvd n)"
proots_sphere_card_code1: fixes p :: "complex poly"   and z0 :: "complex"   and r :: "real" shows "proots_sphere_card p z0 r = (if r = 0 then if poly p z0 = 0 then 1 else 0 else if r < 0 \<or> p = 0 then 0 else (if poly p (z0 - complex_of_real r) = 0 then 1 else 0) + proots_unbounded_line_card (fcompose (p \<circ>\<^sub>p [:z0, complex_of_real r:]) [:\<i>, - 1:] [:\<i>, 1:]) 0 1)"
iface_cons_less_eq_i: fixes b :: "char"   and bs :: "char list"   and i :: "iface" assumes "Iface (b # bs) \<le> i" shows "\<exists>q qs. i = Iface (q # qs) \<and> (b < q \<or> Iface bs \<le> Iface qs)"
DBnd_def: shows "DBnd = det_bound n (max (1::'a) Bnd)"
prem: shows "S\<^sup>+\<^sup>+ (f x) (f y)"
nth_Cons_Suc: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "(x # xs) ! Suc n = xs ! n"
corec_sel_2: 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 "thd (corec_tllist p g1 g21 q22 g221 g222 a) = g21 a"
cong: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f" shows "comm_monoid_add.sum plus zero \<equiv> comm_monoid_add.sum plus zero"
inject_3: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm"   and y41 :: "'b"   and y42 :: "PreSimplyTyped.type"   and y43 :: "'b ptrm" shows "(PFn x41 x42 x43 = PFn y41 y42 y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
cball_diff_eq_sphere: fixes a :: "'b"   and r :: "real" shows "cball a r - ball a r = sphere a r"
comm_monoid_set_F_and: fixes f :: "'a \<Rightarrow> bool"   and s :: "'a set" shows "comm_monoid_set.F (\<and>) True f s = (finite s \<longrightarrow> (\<forall>x\<in>s. f x))"
mono_D12: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (P `;` S) \<longrightarrow> Ra (P `;` S) s = Ra (Q `;` S) s"
finite_psubset_induct: fixes A :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "finite A"   and "\<And>A. \<lbrakk>finite A; \<And>B. B \<subset> A \<Longrightarrow> P B\<rbrakk> \<Longrightarrow> P A" shows "P A"
iterateoi_def: fixes S :: "'s" shows "iterateoi S \<equiv> it_to_it (list_ordered_it S)"
le_lt_convs_3: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y = Gt) = lt_of_comp comp y x"
assms_3: shows "hom_induced (int p) (nsphere p) {} (nsphere p) {} f a = a [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> d"
preserves_dom: fixes f :: "'j" assumes "J.arr f" shows "C.dom (A'.F.map f) = A'.F.map (J.dom f)"
env_restr_useless: fixes m :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "edom m \<subseteq> S" shows "m f|` S = m"
inj_on_inverseI: fixes A :: "'g set"   and g :: "'h \<Rightarrow> 'g"   and f :: "'g \<Rightarrow> 'h" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
tab_def: shows "\<tau>\<mu>.tab \<equiv> \<a>\<^sup>-\<^sup>1[t, u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0] \<cdot> (t \<star> \<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0]) \<cdot> (t \<star> \<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<cdot> (t \<star> t\<^sub>0u\<^sub>1.\<phi>) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1)"
inf_top_right: fixes x :: "'b" shows "x \<squnion> \<bottom> = x"
rel_witness_pmf2: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf" assumes "rel_pmf A p q" shows "rel_pmf (\<lambda>(a, b') b. b = b' \<and> A a b') (rel_witness_pmf A (p, q)) q"
IH: assumes "sorted_descending (map f as)" shows "\<exists>m n. as = m @ n \<and> (\<forall>e\<in>set m. f (hd as) = f e) \<and> (\<forall>e\<in>set n. f e < f (hd as))"
step: fixes f :: "nat \<Rightarrow> ordinal" assumes "\<forall>n. P (f n)" shows "P (oStrictLimit f)"
xor_replicate_bot_left: fixes xs :: "'b list"   and n :: "nat"   and x :: "'b" assumes "length xs \<le> n"   and "x = \<bottom>" shows "replicate n x [\<oplus>] xs = xs"
has_vderiv_on_zero_constant: fixes s :: "real set"   and f :: "real \<Rightarrow> 'c" assumes "convex s"   and "(f has_vderiv_on (\<lambda>h. 0::'c)) s" obtains c :: "'c" where "\<And>x. x \<in> s \<Longrightarrow> f x = c"
ext: fixes g :: "'j"   and f :: "'j" assumes "g \<cdot>\<^sub>J f \<noteq> J.null" shows "J.seq g f"
cr_id_prob_simps: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" shows "cr_id_prob A (return_id x) (return_pmf y) = A x y"
RSD_pareto_eqclass_rsd_winners_aux: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a list" assumes "pref_profile_wf agents alts R"   and "finite alts"   and "alts \<noteq> {}"   and "set agents' \<subseteq> agents" shows "RSD_pareto_eqclass (set agents') alts R (rsd_winners R alts agents')"
norm_exp_i_times: fixes y :: "real" shows "cmod (exp (\<i> * complex_of_real y)) = 1"
intros: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "encoding.indRelRp Enc (SourceTerm S) (TargetTerm (Enc S))"
has_bottom_pow: fixes A :: "'c set" shows "order.has_bottom (Pow A)"
spr_simAction_jAction: fixes t :: "'s Trace"   and ec :: "'s odlist" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = spr_jkbpCSt t" shows "set (local.spr_simAction agent ec) = set (jAction (spr_repMC (toSet ec)) (tLast t) agent)"
simps: fixes as :: "'b iarray"   and n :: "integer" shows "IArray.sub' (as, n) = as !! nat_of_integer n"
finite_UN: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and I :: "'d set"   and A :: "'d \<Rightarrow> 'c set" assumes "ring_of_sets \<Omega> M"   and "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" shows "\<Union> (A ` I) \<in> M"
countable_UN: fixes I :: "'f set"   and A :: "'f \<Rightarrow> 'g set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (\<Union> (A ` I))"
i_Exec_Stream_append: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream trans_fun (xs \<frown> input) c = f_Exec_Comp_Stream trans_fun xs c \<frown> i_Exec_Comp_Stream trans_fun input (f_Exec_Comp trans_fun xs c)"
ap_state_rev_pure_B: fixes f :: "'e \<Rightarrow> 'd"   and x :: "('c, 'e) state" shows "pure f \<diamondop> B x = B (pure f \<diamondop> x)"
eq: shows "quotient_web \<Gamma> f = quotient_web \<Gamma> h"
that_2: assumes "g \<turnstile> hd (tl ns)-tl ns\<rightarrow>m"   and "n \<in> set (predecessors g (hd (tl ns)))" shows "thesis"
b: fixes Y :: "'a set"   and x :: "'a" assumes "Y \<noteq> {}" shows "\<Squnion> ((\<cdot>) x ` Y) \<le> x \<cdot> \<Squnion> Y"
update_convs_4: fixes msg' :: "msg \<Rightarrow> msg"   and id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "msg_update msg' \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = \<lparr>state.id = id, no = no, nhid = nhid, msg = msg' msg, num = num, sid = sid, \<dots> = more\<rparr>"
set_intros_1: fixes x4 :: "'a" shows "x4 \<in> RID\<^sub>V (Rid x4)"
insert: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and A :: "'a set" assumes "MaxR r" shows "MaxR.MaxR_opt r (insert x A) = (if x \<in> Field r then Some (case MaxR.MaxR_opt r A of None \<Rightarrow> x | Some y \<Rightarrow> MaxR.maxR r x y) else MaxR.MaxR_opt r A)"
pderivs_append: fixes s1 :: "'b list"   and s2 :: "'b list"   and r :: "'b rexp" shows "pderivs (s1 @ s2) r = pderivs_set s2 (pderivs s1 r)"
range_from_nat_into: fixes A :: "'c set" assumes "A \<noteq> {}"   and "countable A" shows "range (from_nat_into A) = A"
fund_sol_minimal: fixes D :: "nat"   and a :: "nat"   and b :: "nat" assumes "pell D"   and "pell.nontriv_solution D (a, b)" shows "pell.pell_valuation D (case pell.fund_sol D of (x, y) \<Rightarrow> (int x, int y)) \<le> pell.pell_valuation D (int a, int b)"
edge_preserving_unionI: fixes f :: "('e \<times> 'f) set"   and A :: "('g \<times> 'e \<times> 'e) set"   and G :: "('g \<times> 'f \<times> 'f) set"   and B :: "('g \<times> 'e \<times> 'e) set" assumes "edge_preserving f A G"   and "edge_preserving f B G" shows "edge_preserving f (A \<union> B) G"
alw_ev_holds_mp: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and Q :: "'b \<Rightarrow> bool" assumes "alw (holds P) xs"   and "ev (holds Q) xs" shows "ev (holds (\<lambda>x. P x \<and> Q x)) xs"
map_matrix_def: fixes f :: "'d \<Rightarrow> 'a"   and x :: "(('d, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.map_matrix f x = (\<chi>i j. f (x $h i $h j))"
return_def: fixes v :: "'a" shows "return v = TM v 0"
conj_one_left: fixes x :: "'c" shows "- (1::'c) AND x = x"
in_dom: fixes i :: "'agent" assumes "i \<in> agents" shows "Order_Predicates.preorder_on (lotteries_on alts) ((SD \<circ> R) i)"
iso_runit: fixes f :: "'d" assumes "D.ide f" shows "D.iso \<r>\<^sub>D[f]"
fps_integral_def: fixes a :: "'b fps"   and a0 :: "'b" shows "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else inverse (of_nat n) * a $ (n - 1))"
split: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b"   and unit :: "unit" shows "P (case unit of () \<Rightarrow> f) = (unit = () \<longrightarrow> P f)"
integrable': shows "set_integrable lborel {a..b} (\<lambda>x. f (g x) * g' x)"
path_image_def: fixes g :: "real \<Rightarrow> 'c" shows "path_image g = g ` {0..1}"
poly_monom: fixes a :: "'b"   and n :: "nat"   and x :: "'b" shows "poly (Polynomial.monom a n) x = a * x ^ n"
identity_3: fixes g :: "'a" assumes "g \<in> G" shows "binop e g = g"
map_cons: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "map f (x # xs) = f x # map f xs"
iwlsFSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "iwlsFSw MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWls MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igFreshCls MOD \<and> igSwapCls MOD \<and> igAbsCongS MOD"
rep_props_2: fixes r :: "'a" assumes "ide r" shows "local.iso (rep r)"
assms: shows "fmdom' (snd a) \<subseteq> fmdom' s"
is_subset_2: fixes c :: "'a" assumes "c \<in> \<delta>_states ta.\<delta>" shows "c \<in> ta.Q"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_on_left_moving_O m rs (l, r) = (\<exists>rn. l = Oc \<up> Suc rs @ Bk # Oc \<up> Suc m \<and> r = Oc # Bk \<up> rn)"
dwd: shows "Sig iS\<^bsub>CI T\<^esub> \<triangleright> Vx : A \<turnstile> d : B"
simps_6: fixes ux :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)"   and b :: "bexp"   and C :: "acom" shows "strip ({ux} WHILE b DO C) = WHILE b DO strip C"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and p :: "('psym, 'tp) epsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "satA \<xi> (Pr p Tl) = eintP p (map (int \<xi>) Tl)"
psimps: fixes x :: "'a" assumes "recfun_aux_dom x" shows "recfun_aux x = (if Q x then g x else h' (recfun_aux (h x)))"
quasi_inverses_symmetric: fixes f :: "'a"   and g :: "'a" assumes "quasi_inverses f g" shows "quasi_inverses g f"
wqo_extensions_wf_conv: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "qo_on P A" shows "wqo_on P A = (\<forall>Q. (\<forall>x\<in>A. \<forall>y\<in>A. P x y \<longrightarrow> Q x y) \<and> qo_on Q A \<longrightarrow> wfp_on (strict Q) A)"
eSuc_max: fixes x :: "eint"   and y :: "eint" shows "max x y + 1 = max (x + 1) (y + 1)"