hcju/isabelleps huggingface dataset:
algebraic_times: fixes x :: "'b"   and y :: "'b" assumes "algebraic x"   and "algebraic y" shows "algebraic (x * y)"
ring_simprules_4: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<ominus>\<^bsub>P\<^esub> y \<in> carrier P"
adds_except_fun_singleton: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b"   and v :: "'a" shows "(s adds t) = (except_fun s {v} adds except_fun t {v} \<and> s v adds t v)"
comm_add_exps: fixes r :: "'b list"   and u :: "'b list"   and m :: "nat"   and k :: "nat" assumes "r \<cdot> u = u \<cdot> r" shows "r \<^sup>@ m \<cdot> u \<^sup>@ k = u \<^sup>@ k \<cdot> r \<^sup>@ m"
MaxR_opt_is_greatest: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "MaxR r"   and "MaxR.MaxR_opt r A = Some x"   and "y \<in> A \<inter> Field r" shows "(y, x) \<in> r"
discI_2: fixes state :: "('a, 'b, 'c) state"   and x21 :: "'a"   and x22 :: "'b"   and x23 :: "'c" assumes "state = State x21 x22 x23" shows "isState state"
non_Package_instance_method_inheritance: fixes G :: "prog"   and old :: "qtname \<times> mdecl"   and C :: "qtname" assumes "G \<turnstile>Method old inheritable_in pid C "   and "accmodi old \<noteq> Package"   and "\<not> is_static old"   and "G\<turnstile>C\<prec>\<^sub>C declclass old"   and "G\<turnstile>Method old declared_in declclass old"   and "wf_prog G" shows "G \<turnstile>Method old member_of C \<or> (\<exists>new. G \<turnstile> new overrides\<^sub>S old \<and> G \<turnstile>Method new member_of C)"
plus_of_nat: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g1 :: "'b \<Rightarrow> 'b"   and gn1 :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "periodic_fun f g gm g1 gn1" shows "f (g x (of_nat n)) = f x"
IH1_4: assumes "\<not> M \<Turnstile> F" shows "\<not> M(p := False) \<Turnstile> F"
lemma1: fixes K :: "nat"   and H :: "msg set"   and X :: "msg" assumes "Key (invKey K) \<in> analz H" shows "analz (insert (Crypt K X) H) \<subseteq> insert (Crypt K X) (analz (insert X H))"
plus_list_ub2: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> A"   and "set ys \<subseteq> A"   and "length xs = length ys" shows "ys [\<sqsubseteq>\<^bsub>r\<^esub>] xs [\<squnion>\<^bsub>f\<^esub>] ys"
cts_wp_repeat: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "bd_cts (wp a)"   and "healthy (wp a)" shows "bd_cts (wp (repeat n a))"
left_commute: fixes b :: "'a \<Rightarrow> bool"   and a :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> bool" shows "(b \<and>* a \<and>* c) = (a \<and>* b \<and>* c)"
order_consistent_def: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "order_consistent r s = (\<forall>a a'. (a, a') \<in> r \<longrightarrow> (a', a) \<in> s \<longrightarrow> a = a')"
elims: fixes x :: "acom"   and y :: "com" assumes "strip x = y" obtains "x = SKIP" and "y = SKIP"   | xa :: "char list" and a :: "aexp" where "x = xa ::= a" and "y = xa ::= a"   | C\<^sub>1 :: "acom" and C\<^sub>2 :: "acom" where "x = C\<^sub>1__;; C\<^sub>2__" and "y = strip C\<^sub>1__;; strip C\<^sub>2__"   | b :: "bexp" and C\<^sub>1 :: "acom" and C\<^sub>2 :: "acom" where "x = IF b__ THEN C\<^sub>1 ELSE C\<^sub>2" and "y = IF b__ THEN strip C\<^sub>1 ELSE strip C\<^sub>2"   | uu_ :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" and uv_ :: "char list" and b :: "bexp" and C :: "acom" where "x = {uu_/uv_} WHILE b DO C" and "y = WHILE b DO strip C"
Not_Until: fixes \<phi> :: "formula"   and \<psi> :: "formula"   and I :: "\<I>" assumes "fv (Neg \<phi>) \<subseteq> fv \<psi>"   and "safe_formula \<phi>"   and "\<not> safe_formula (Neg \<phi>)"   and "safe_formula \<psi>"   and "P \<phi>"   and "P \<psi>" shows "P (Until (Neg \<phi>) I \<psi>)"
diff_union_swap: fixes a :: "'k"   and b :: "'k"   and M :: "'k multiset" assumes "a \<noteq> b" shows "add_mset b (M - {#a#}) = add_mset b M - {#a#}"
farkas_assert_bound_loop: fixes as :: "('i \<times> 'a atom) list"   and t :: "(nat \<times> linear_poly) list" assumes "\<U> (assert_bound_loop as (init t))"   and "\<triangle> t" shows "\<exists>C. farkas_coefficients_atoms_tableau (snd ` set as) t C"
rtrancl_trans: fixes x :: "'e"   and y :: "'e"   and r :: "('e \<times> 'e) set"   and z :: "'e" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
in_arcs_app_iso_cases: fixes a :: "'c"   and hom :: "('e, 'f, 'd, 'c) digraph_isomorphism"   and G :: "('e, 'f) pre_digraph" assumes "a \<in> arcs (app_iso hom G)" obtains a0 :: "'f" where "a = iso_arcs hom a0" and "a0 \<in> arcs G"
bisim1While7: fixes P :: "'a prog"   and h :: "'heap"   and c :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list" shows "P,while (c) e,h \<turnstile> (unit, xs) \<leftrightarrow> ([], xs, Suc (Suc (Suc (length (compE2 c) + length (compE2 e)))), None)"
has_integral_combine_division: fixes \<D> :: "'a set set"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'a set \<Rightarrow> 'b" assumes "\<D> division_of S"   and "\<And>k. k \<in> \<D> \<Longrightarrow> (f has_integral i k) k" shows "(f has_integral sum i \<D>) S"
Fake_parts_insert: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> synth (analz H)" shows "parts (insert X H) \<subseteq> synth (analz H) \<union> parts H"
VLambda_vsingleton: fixes a :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda (set {a}) f = set {\<langle>a, f a\<rangle>}"
card_distinct: fixes xs :: "'g list" assumes "card (set xs) = length xs" shows "distinct xs"
ide_in_hom_2: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>f : f \<Rightarrow>\<^sub>B f\<guillemotright>"
SD_efficient_same_support: fixes p :: "'alt pmf"   and p' :: "'alt pmf" assumes "set_pmf p = set_pmf p'"   and "p \<in> lotteries" shows "SD_efficient R p = SD_efficient R p'"
rotate_id: fixes n :: "nat"   and xs :: "'b list" assumes "n mod length xs = 0" shows "rotate n xs = xs"
good_item_simps_55: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and y :: "'var"   and xa :: "'var" assumes "good X" shows "X #[xa \<and> y]_zs #[y \<and> xa]_zs = X"
i_def: shows "i \<equiv> \<lambda>h x. integral (BOX h x) f /\<^sub>R h ^ DIM('a)"
mkIde_set: fixes a :: "'s" assumes "ide a" shows "mkIde (local.set a) = a"
assms: shows "component_of_term (lt p) \<noteq> component_of_term (lt q)"
lfinite_ldropn: fixes n :: "nat"   and xs :: "'b llist" shows "lfinite (ldropn n xs) = lfinite xs"
update_convs_1: fixes black' :: "bool \<Rightarrow> bool"   and cx :: "int"   and cy :: "int"   and black :: "bool"   and more :: "'a" shows "black_update black' \<lparr>cx = cx, cy = cy, black = black, \<dots> = more\<rparr> = \<lparr>cx = cx, cy = cy, black = black' black, \<dots> = more\<rparr>"
Sup_Assertion_def: fixes A :: "'a Assertion set" shows "\<Squnion>A = abs_wpt (\<Squnion>(assert ` A))"
id2_1: shows "(\<lambda>x y. x) \<in> PrimRec2"
Min_singleton: fixes x :: "'c Word.word" shows "signed.Min {x} = x"
iprefix_i_append: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "(xs \<sqsubseteq> ys \<frown> f) = (prefix xs ys \<or> (\<exists>zs. xs = ys @ zs \<and> zs \<sqsubseteq> f))"
wf_ts_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and ts :: "nat list" shows "wf_ts \<sigma> P j \<phi> \<psi> ts = list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)..<j] ts"
fst_transfer: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "(rel_prod A B ===> A) fst fst"
fun_comp_eq_conv: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and fg :: "'a \<Rightarrow> 'b" shows "(f \<circ> g = fg) = (\<forall>x. f (g x) = fg x)"
weak_diamond_def: fixes r :: "('b \<times> 'b) set" shows "w\<diamond> r = (r\<inverse> O r - Id \<subseteq> r O r\<inverse>)"
dgrad_p_set_closed_red_single: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and m :: "nat"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "dickson_grading d"   and "p \<in> dgrad_p_set d m"   and "f \<in> dgrad_p_set d m"   and "red_single p q f t" shows "q \<in> dgrad_p_set d m"
Trg_Src: fixes t :: "'a local.term" assumes "Arr t" shows "Trg (Src t) = Src t"
map2_empty_iff: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'d list" shows "(Map2.map2 f xs ys = []) = (xs = [] \<or> ys = [])"
assms: fixes y :: "'a" shows "adm (\<lambda>x. y \<in> A x \<longrightarrow> P x y)"
state_rank_upper_bound: fixes q :: "'b"   and n :: "nat"   and i :: "nat" assumes "state_rank q n = Some i" shows "i < max_rank"
b_least2_less_impl_nz: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "b_least2 f x y < y" shows "f x (b_least2 f x y) \<noteq> 0"
boundOutputOpenDest: fixes xvec1 :: "name list"   and x :: "name"   and xvec2 :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*(xvec1 @ x # xvec2)\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q"   and "x \<sharp> xvec1"   and "x \<sharp> yvec"   and "x \<sharp> N"   and "x \<sharp> Q"   and "distinct yvec" obtains yvec1 :: "name list" and y :: "name" and yvec2 :: "name list" where "yvec = yvec1 @ y # yvec2" and "length xvec1 = length yvec1" and "length xvec2 = length yvec2" and "\<lparr>\<nu>*(xvec1 @ xvec2)\<rparr>M \<prec>' P = \<lparr>\<nu>*(yvec1 @ yvec2)\<rparr>([(x, y)] \<bullet> N) \<prec>' ([(x, y)] \<bullet> Q)"
size_18: fixes x8 :: "'c ltl" shows "size (G x8) = size x8 + Suc 0"
ssubseqs_subset: fixes as :: "'b list"   and bs :: "'b list" assumes "as \<in> ssubseqs bs" shows "ssubseqs as \<subseteq> ssubseqs bs"
ereal_divide_eq: fixes b :: "ereal"   and a :: "ereal"   and c :: "ereal" assumes "b \<noteq> 0"   and "\<bar>b\<bar> \<noteq> \<infinity>" shows "(a / b = c) = (a = b * c)"
primitive_def: fixes f :: "'b poly" shows "primitive f = (\<forall>x. (\<forall>y\<in>set (coeffs f). x dvd y) \<longrightarrow> x dvd (1::'b))"
not_inf_Stuck': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "nat \<Rightarrow> ('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate"   and k :: "nat"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"   and "f k = (cs, css, Stuck)" shows "False"
crypts_insert: fixes X :: "msg"   and H :: "msg set" shows "crypts (insert X H) = crypts_msg X \<union> crypts H"
in_renamed_in_unlabelled_val_1: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and r :: "'d"   and \<beta> :: "'e \<Rightarrow> 'e"   and v :: "('d, 'e, 'f) val" assumes "bij \<alpha>" shows "(\<alpha> r \<in> RID\<^sub>V (\<R>\<^sub>V \<alpha> \<beta> v)) = (r \<in> RID\<^sub>V v)"
d_OUT_add: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" shows "d_OUT (\<lambda>e. f e + g e) x = d_OUT f x + d_OUT g x"
compact_imp_closed: fixes "open" :: "'e set \<Rightarrow> bool"   and s :: "'e set" assumes "class.t2_space open"   and "topological_space.compact open s" shows "topological_space.closed open s"
genideal_ideal: fixes R :: "('a, 'b) ring_scheme"   and S :: "'a set" assumes "ring R"   and "S \<subseteq> carrier R" shows "ideal (Idl\<^bsub>R\<^esub> S) R"
distinct_7: fixes x2 :: "nat"   and x3 :: "nat" shows "WAIT x2 \<noteq> HOLD x3"
hbt_insert_2: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "hbt t" shows "height (local.insert x t) \<in> {height t, height t + 1}"
is_type_def: fixes P :: "'a prog"   and T :: "ty" shows "is_type P T \<equiv> case T of Class C \<Rightarrow> is_class P C | _ \<Rightarrow> True"
SeqStTermP_Ind: fixes s :: "name"   and k :: "name"   and v :: "tm"   and t :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, v, t, i)"   and "atom k \<sharp> (v, t, i)" shows "{VarP v, IndP t} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqStTermP v i t t (Var s) (Var k)))"
decode_Op_def: fixes n :: "nat" shows "decode_Op n = (if c_fst n = 0 then Dec (c_fst (c_snd n)) (c_fst (c_snd (c_snd n))) (c_snd (c_snd (c_snd n))) else Inc (c_fst (c_snd n)) (c_snd (c_snd n)))"
poly_subst_subst_pp: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and g :: "'e \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'e \<Rightarrow>\<^sub>0 nat" shows "poly_subst f (subst_pp g t) = subst_pp (\<lambda>x. poly_subst f (g x)) t"
SigmaE: fixes c :: "'d \<times> 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "c \<in> Sigma A B" obtains x :: "'d" and y :: "'e" where "x \<in> A" and "y \<in> B x" and "c = (x, y)"
zhmsetmset_plus: fixes x :: "zhmultiset"   and xa :: "zhmultiset" shows "zhmsetmset (x + xa) = zhmsetmset x + zhmsetmset xa"
has_field_derivative_inverse_basic: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and g :: "'a \<Rightarrow> 'a"   and y :: "'a"   and t :: "'a set" assumes "(f has_field_derivative f') (at (g y))"   and "f' \<noteq> (0::'a)"   and "isCont g y"   and "open t"   and "y \<in> t"   and "\<And>z. z \<in> t \<Longrightarrow> f (g z) = z" shows "(g has_field_derivative inverse f') (at y)"
assms_1: shows "all_fulfill (in_all_resolvents_upon S A) R"
prveqlPT_prv_instInp_instInpP: fixes \<phi> :: "'fmla"   and \<tau> :: "'fmla"   and t :: "'trm" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> \<subseteq> {inp}"   and "\<tau> \<in> ptrm 0"   and "t \<in> trm"   and "FvarsT t = {}"   and "prveqlPT \<tau> t" shows "prv (imp (instInpP \<phi> 0 \<tau>) (instInp \<phi> t))"
simps: fixes a :: "'b"   and r :: "('b \<times> 'c) set" shows "(a \<in> Domain r) = (\<exists>a b. a = a \<and> (a, b) \<in> r)"
\<beta>_def: fixes fs :: "'b vec list"   and l :: "nat" shows "\<beta> fs l = Gramian_determinant fs (Suc l) / Gramian_determinant fs l"
unit_resolvent_def: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set" shows "unit_resolvent P1 P2 C = ((\<exists>L. C = P1 - {L} \<union> (P2 - {complement L}) \<and> L \<in> P1 \<and> complement L \<in> P2) \<and> Unit P1)"
i: shows "i \<le> (GREATEST m. \<not> is_zero_row_upt_k m k A)"
chamber_card: fixes C :: "'b set"   and D :: "'b set" assumes "codomain.chamber C"   and "codomain.chamber D" shows "card C = card D"
less_omegaD: fixes x :: "ordinal" assumes "x < \<omega>" shows "\<exists>n. x = ordinal_of_nat n"
sel_2: fixes x1 :: "'a interval"   and x2 :: "nat" shows "idx (IdxInterval x1 x2) = x2"
add_le_imp_le_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c + a \<le> c + b" shows "a \<le> b"
complex_poly_eq: fixes r :: "complex poly" shows "r = map_poly complex_of_real (map_poly Re r) + smult \<i> (map_poly complex_of_real (map_poly Im r))"
F'_6: shows "ms_exp_gt e (ms_aux_hd_exp F')"
a2_eq': fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> d y) = ad (x \<cdot> y)"
smap_def: shows "smap \<equiv> \<mu> smap. \<Lambda> f v. Fixrec.run\<cdot>(match_stcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(f\<cdot>x && smap\<cdot>f\<cdot>xs)))"
cases: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a2 :: "'Var \<Rightarrow> 'Val"   and a3 :: "nat"   and a4 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a5 :: "'Var \<Rightarrow> 'Val" assumes "(a1, a2) \<leadsto>\<^bsub>a3\<^esub> (a4, a5)" obtains cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list" and n :: "nat" and mem :: "'Var \<Rightarrow> 'Val" and cm' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)" and mem' :: "'Var \<Rightarrow> 'Val" where "a1 = cms" and "a2 = mem" and "a3 = n" and "a4 = cms[n := cm']" and "a5 = mem'" and "(cms ! n, mem) \<leadsto> (cm', mem')" and "n < length cms"
shift_2: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs" shows "standard_decomp k (local.shift k m qs)"
r: shows "(replicate n False)[q := True] = (replicate n False)[p := True]"
d_1: shows "roots_of_rat_poly_main p = (let n = degree p in if n = 0 then [] else if n = 1 then [roots1 p] else if n = 2 then rat_roots2 p else case rational_root_test p of None \<Rightarrow> [] | Some x \<Rightarrow> x # roots_of_rat_poly_main (p div [:- x, 1:]))"
i_set_min_def: shows "i_set_min \<equiv> iFROM_set \<union> iIN_set \<union> iMOD2_set \<union> iMODb2_set"
Sup_upto_llist_subset_Sup_llist: fixes Xs :: "'a set llist"   and j :: "enat" shows "Sup_upto_llist Xs j \<subseteq> Sup_llist Xs"
qfree_FR\<^sub>1: fixes \<phi> :: "atom fm" assumes "nqfree \<phi>" shows "qfree (FR\<^sub>1 \<phi>)"
2139: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (x \<squnion> y) = - (x \<squnion> y)"
Span_subgroup_props_2: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R"   and "subfield K R"   and "set Us \<subseteq> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<in> ring.Span R K Us"
axioms_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "MonotModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "MonotProblem wtFsym wtPsym arOf resOf parOf \<Phi>"
that: fixes \<pi> :: "'b \<Rightarrow> 'b" assumes "\<pi> permutes agents"   and "R2 \<circ> \<pi> = R1" shows "thesis"
wp2_alt: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "wp\<^sub>2 c Q = (\<lambda>s. if \<down> (c, s) then enat (\<down>\<^sub>t (c, s)) + Q (\<down>\<^sub>s (c, s)) else \<infinity>)"
distinct_1: fixes x2 :: "bool" shows "Unit \<noteq> Bool x2"
next_action_after_step: shows "\<forall>s a d aseqs. current s \<noteq> d \<longrightarrow> fst (CISK_control (int.step s a) d aseqs) = fst (CISK_control s d aseqs)"
distributed_real_measurable: fixes N :: "'c measure"   and f :: "'c \<Rightarrow> real"   and M :: "'d measure"   and X :: "'d \<Rightarrow> 'c" assumes "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> f x"   and "distributed M N X (\<lambda>x. ennreal (f x))" shows "f \<in> borel_measurable N"
measure_nested_Union: fixes S :: "nat \<Rightarrow> 'c set"   and B :: "real" assumes "\<And>n. S n \<in> lmeasurable"   and "\<And>n. Sigma_Algebra.measure lebesgue (S n) \<le> B"   and "\<And>n. S n \<subseteq> S (Suc n)" shows "(\<lambda>n. Sigma_Algebra.measure lebesgue (S n)) \<longlonglongrightarrow> Sigma_Algebra.measure lebesgue (\<Union> (range S))"
wf_ruleset_def: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list" shows "wf_ruleset \<gamma> p rs \<equiv> \<forall>r\<in>set rs. \<not> matches \<gamma> (get_match r) (get_action r) p \<or> (\<nexists>chain. get_action r = Call chain) \<and> get_action r \<noteq> Return \<and> (\<nexists>chain. get_action r = Goto chain) \<and> get_action r \<noteq> Unknown"
eucl_less_def: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "euclidean_space.eucl_less inner Basis a b = (\<forall>i\<in>Basis. inner a i < inner b i)"
ramanujan_sum_conv_gen_ramanujan_sum: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "c k n = s (\<lambda>x. of_nat (id x)) moebius_mu k n"
QmpTr3: fixes H :: "'a set"   and N :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "G \<triangleright> H"   and "G \<triangleright> N"   and "H \<subseteq> N"   and "X \<in> carrier (G / H)"   and "Y \<in> carrier (G / H)" shows "(Qm\<^bsub>G H,N\<^esub>) (c_top G H X Y) = c_top G N ((Qm\<^bsub>G H,N\<^esub>) X) ((Qm\<^bsub>G H,N\<^esub>) Y)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L S.map \<ll>"
root_real_alg_def: shows "root_real_alg \<equiv> (id ---> rep_real_alg ---> Real_Alg_Quotient) root_3"
assms: shows "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. x islimpt t)"
downset_set_subset: fixes S :: "'b set" shows "downset {xs. set xs \<subseteq> S}"
assms: shows "x = y \<or> x = z \<or> x = t \<or> y = z \<or> y = t \<or> z = t"
Gram_Schmidt_column_k_iarrays_efficient_def: fixes A :: "real iarray iarray"   and k :: "nat" shows "Gram_Schmidt_column_k_iarrays_efficient A k = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>a b. let column_b_A = column_iarray b A in (if b = k then column_b_A - (\<Sum>x\<leftarrow>map (\<lambda>n. column_iarray n A) [0..<b]. (column_b_A \<bullet>i x / (x \<bullet>i x)) *\<^sub>R x) else column_b_A) !! a)"
allowed_insert_deliver_in_set: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "'a"   and e :: "'a \<times> 'b \<times> bool"   and m :: "'a option"   and j :: "nat" assumes "es @ [Deliver (i, Insert e m)] prefix of j" shows "m = None \<or> (\<exists>i' m' n v b. m = Some m' \<and> Deliver (i', Insert (m', v, b) n) \<in> set es)"
preserves_comp: fixes g :: "'a1"   and f :: "'a1" assumes "A1.seq g f" shows "F1 (g \<cdot>\<^sub>A\<^sub>1 f) = F1 g \<cdot>\<^sub>B\<^sub>1 F1 f"
base: fixes xs :: "'a list"   and x :: "'a" shows "P (xs @ [x]) []"
comp_pres_conj: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> conjunctive.conjunctive inf inf times"   and "y \<in> conjunctive.conjunctive inf inf times" shows "times x y \<in> conjunctive.conjunctive inf inf times"
sorted_list_subset_fusion_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and g1 :: "('b, 'c) generator"   and g2 :: "('b, 'd) generator"   and s1 :: "'c"   and s2 :: "'d" shows "ord.sorted_list_subset_fusion less eq g1 g2 s1 s2 = ord.sorted_list_subset less eq (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
until_introR: fixes \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<psi> \<pi>l" shows "until \<phi> \<psi> \<pi>l"