hcju/isabelleps huggingface dataset:
minus_divide_le_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (uminus (divide b c)) a = (if less zero c then less_eq (uminus b) (times a c) else if less c zero then less_eq (times a c) (uminus b) else less_eq zero a)"
Fail_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "Fail s s' p = (\<exists>ip\<in>Inputs. inpt s' = (inpt s)(p := ip) \<and> phase s' = (phase s)(p := 0) \<and> dblock s' = (dblock s)(p := InitDB) \<and> outpt s' = (outpt s)(p := NotAnInput) \<and> InitializePhase s s' p \<and> disk s' = disk s)"
finite_nat_set_bounded: fixes N :: "nat set" assumes "finite N" shows "\<exists>n. \<forall>i\<in>N. i < n"
tranclp_unfold: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" shows "r\<^sup>+\<^sup>+ a b = ((a, b) \<in> {(x, y). r x y}\<^sup>+)"
simps_3: fixes l1 :: "('b \<times> nat) tree"   and a1 :: "'b"   and n1 :: "nat"   and r1 :: "('b \<times> nat) tree"   and l2 :: "('b \<times> nat) tree"   and a2 :: "'b"   and n2 :: "nat"   and r2 :: "('b \<times> nat) tree" shows "T_merge (\<langle>l1, (a1, n1), r1\<rangle> =: t1__) (\<langle>l2, (a2, n2), r2\<rangle> =: t2__) = (if a1 \<le> a2 then T_merge r1 t2__ else T_merge t1__ r2) + 1"
one_imp_one_tail: fixes x :: "'a"   and p :: "'a list" assumes "one_call_in_path x p" shows "one_call_in_path x (tl p)"
gcd_proj1_if_dvd: fixes a :: "'b"   and b :: "'b" assumes "a dvd b" shows "gcd a b = normalize a"
"case": fixes f :: "(glb \<Rightarrow> nat) \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> 'b"   and x1 :: "glb \<Rightarrow> nat"   and x2 :: "loc \<Rightarrow> nat" shows "(case st x1 x2 of st x xa \<Rightarrow> f x xa) = f x1 x2"
scale_right_diff_distrib: fixes a :: "rat"   and x :: "'a"   and y :: "'a" shows "a *R (x - y) = a *R x - a *R y"
final_NF: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" shows "relation.is_final r a = (a \<in> NF {(x, y). r x y})"
that: fixes g' :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g' \<in> G'"   and "rep_list g' \<noteq> 0"   and "lt g' = lt g"   and "punit.lt (rep_list g') = punit.lt (rep_list g)" shows "thesis"
simps_9: fixes e1 :: "exp"   and e2 :: "exp"   and C :: "ctx"   and e :: "exp" shows "plug (CIf3 e1 e2 C) e = EIf e1 e2 (plug C e)"
relation_of_CSP2: fixes x :: "('c, 'd) action" shows "action.relation_of x is CSP2 healthy"
insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "x \<notin> A" shows "sum g (insert x A) = g x + sum g A"
verit_la_disequality: fixes a :: "'b"   and b :: "'b" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
pr_simps_1: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "arr \<p>\<^sub>0[a0, a1]"
red_wf_dgrad_p_set_aux: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m" shows "wfp_on (red F)\<inverse>\<inverse> (dgrad_p_set d m)"
l7_17: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and P' :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A P P'"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B P P'" shows "A = B"
chamber_map: fixes C :: "'a set" assumes "chamber C" shows "chamber (f ` C)"
C1_b: fixes x :: "'a" shows "(-r x l\<rightarrow> (0::'a)) r\<rightarrow> x = (1::'a)"
closest_point_in_set: fixes S :: "'a set"   and a :: "'a" assumes "closed S"   and "S \<noteq> {}" shows "closest_point S a \<in> S"
subset_Un_eq: fixes A :: "'b set"   and B :: "'b set" shows "order.greater_eq B A = (A \<union> B = B)"
rec: fixes f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "old.rec_prod f1 (a, b) = f1 a b"
findnth_LE_def: shows "findnth_LE \<equiv> inv_image lex_pair findnth_measure"
None: fixes ord :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "'l option" shows "ord_option ord None x"
suc_p_is: shows "lnth Ns (Suc p) = (T2p, N2p)"
strict_pref_def: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and y :: "'a" shows "x \<^bsub>r\<^esub>\<prec> y \<equiv> x \<^bsub>r\<^esub>\<preceq> y \<and> (y, x) \<notin> r"
closed_rel_add: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel" shows "(s' \<otimes> r \<oplus> s \<otimes> r', s \<otimes> s') \<in> carrier rel"
intro: fixes p :: "nat"   and pm :: "nat" assumes "p = Suc pm" shows "fixed_root p pm"
antisym_conv3: fixes y :: "'b word"   and x :: "'b word" assumes "\<not> y <s x" shows "(\<not> x <s y) = (x = y)"
simps_4: fixes x2 :: "'c error_result"   and x1 :: "'b" shows "Rerr x2 \<noteq> Rval x1"
nlta: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "\<not> A B C LtA A B C"
wp_eq_intro: fixes P :: "('c, 'b) urel"   and Q :: "('c, 'b) urel" assumes "\<And>r. P wp r = Q wp r" shows "P = Q"
find_indices_snoc: fixes x :: "'b"   and ys :: "'b list"   and y :: "'b" shows "find_indices x (ys @ [y]) = find_indices x ys @ (if x = y then [length ys] else [])"
parts_spies_takeWhile_mono: fixes P :: "event \<Rightarrow> bool"   and evs :: "event list" shows "parts (knows Spy (takeWhile P evs)) \<subseteq> parts (knows Spy evs)"
infsetsum'_finite: fixes F :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite F" shows "infsetsum' f F = sum f F"
Inf_aform: fixes e :: "nat \<Rightarrow> real"   and X :: "'a \<times> 'a pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" shows "Inf_aform X \<le> aform_val e X"
factorize_polys_undo_factorize_polys_set: fixes ps :: "rat poly list"   and ftrs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and sgas :: "rat list list" assumes "factorize_polys ps = (ftrs, data)"   and "sgas = find_sgas (map real_of_rat_poly ftrs)"   and "set sgas = consistent_sign_vectors ftrs UNIV" shows "set (map (undo_factorize_polys data) sgas) = consistent_sign_vectors ps UNIV"
upd_cols_cong: fixes M :: "('c, 'd) sq_matrix"   and N :: "('c, 'd) sq_matrix"   and T :: "'d set"   and S :: "'d set"   and f :: "'d \<Rightarrow> ('c, 'd) vec"   and g :: "'d \<Rightarrow> ('c, 'd) vec" assumes "M = N"   and "T = S"   and "\<And>s. s \<in> S =simp=> f s = g s" shows "upd_cols M T f = upd_cols N S g"
P3: shows "B' Out A' A'' \<and> Cong B' A'' B A"
u1: shows "(*\<^sub>V) (Proj M) = projection (space_as_set M)"
step_Stuck_end: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c\<^sub>1' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')"   and "s' = Stuck" shows "s = Stuck \<or> (\<exists>r x. redex c\<^sub>1 = Spec r \<and> s = Normal x \<and> (\<forall>t. (x, t) \<notin> r)) \<or> (\<exists>p x. redex c\<^sub>1 = Call p \<and> s = Normal x \<and> \<Gamma> p = None)"
unilateral_substitutes_on_bilateral_substitutes_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "unilateral_substitutes_on A f" shows "bilateral_substitutes_on A f"
rel_fun_def_butlast: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and T :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e \<Rightarrow> 'f" shows "rel_fun R (rel_fun S T) f g = (\<forall>x y. R x y \<longrightarrow> rel_fun S T (f x) (g y))"
fps_inverse_one': assumes "inverse (1::'a) = (1::'a)" shows "inverse 1 = 1"
relpowp_Suc_I2: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and n :: "nat"   and z :: "'d" assumes "P x y"   and "(P ^^ n) y z" shows "(P ^^ Suc n) x z"
is_maxpref_def: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "is_maxpref P xs ys = (prefix xs ys \<and> (xs = [] \<or> P xs) \<and> (\<forall>zs. prefix zs ys \<and> P zs \<longrightarrow> prefix zs xs))"
the_pmf_return: fixes x :: "'c" shows "option_to_spmf.the_pmf (return_pmf x) = x"
nip_eq: fixes x :: "i"   and y :: "i" assumes "M" shows "PP x y = (P x y \<and> x \<noteq> y)"
l1_obs_iagreement_Init: shows "oreach l1 \<subseteq> l2_iagreement_Init"
map\<^sub>0_def: fixes a :: "'b" shows "\<tau>.G.map\<^sub>0 a \<equiv> src\<^sub>C (G a)"
show_r_zero: fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier M"   and "a \<in> carrier M" shows "(x \<oplus>\<^bsub>M\<^esub> a = x) = (a = \<zero>\<^bsub>M\<^esub>)"
wf_cdecl_def: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" shows "wf_cdecl wf_md P \<equiv> \<lambda>(C, D, fs, ms). (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (C \<noteq> Object \<longrightarrow> is_class P D \<and> \<not> P \<turnstile> D \<preceq>\<^sup>* C \<and> (\<forall>m\<in>set ms. wf_overriding P D m)) \<and> (C = Thread \<longrightarrow> (\<exists>m. (run, [], Void, m) \<in> set ms))"
QAT_Base: fixes s :: "trm"   and p :: "trm"   and t :: "trm"   and q :: "trm"   and \<Gamma> :: "(name \<times> ty) list" assumes "s \<Down> p"   and "t \<Down> q"   and "\<Gamma> \<turnstile> p \<leftrightarrow> q : TBase" shows "\<Gamma> \<turnstile> s \<Leftrightarrow> t : TBase"
simps_6: fixes v :: "Core.term"   and va :: "Core.term"   and u :: "Core.term" shows "beta_reducible (v $ va $ u) = (beta_reducible (v $ va) \<or> beta_reducible u)"
len_l: shows "length (map (\<lambda>i. f (xs ! i) (ys ! i)) [0..<length ys]) = length ys"
not_bot_demand: fixes f :: "C \<rightarrow> 'a"   and r :: "C" shows "(f\<cdot>r \<noteq> \<bottom>) = (demand f \<noteq> C\<^sup>\<infinity> \<and> demand f \<sqsubseteq> r)"
ord_eq_le_trans: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a = b"   and "lex_pm b c" shows "lex_pm a c"
rOfWax_def: fixes \<sigma> :: "'c" shows "rOfWax \<sigma> = Fn (Wit \<sigma>) []"
assms_7: shows "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c'"
F_def: fixes m :: "nat"   and n :: "nat" shows "F m n = (if m \<le> n then \<Inter> (E ` {m..n}) else space M)"
partition_set: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and yes :: "'a list"   and no :: "'a list" assumes "List.partition P xs = (yes, no)" shows "set yes \<union> set no = set xs"
inf_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "inf f g x = inf (f x) (g x)"
Domain_Un_eq: fixes A :: "('b \<times> 'c) set"   and B :: "('b \<times> 'c) set" shows "r2s (A \<union> B) = r2s A \<union> r2s B"
clock_simVal: shows "sim_val Clock.MC clock_simMC clock_sim"
bounded_iff: fixes b :: "'b::semilattice_sup"   and c :: "'b::semilattice_sup"   and a :: "'b::semilattice_sup" shows "((b::'b::semilattice_sup) \<squnion> (c::'b::semilattice_sup) \<le> (a::'b::semilattice_sup)) = (b \<le> a \<and> c \<le> a)"
elem_subsetD: fixes A :: "'a set"   and B :: "'a set"   and a :: "'a" assumes "A \<subseteq> B"   and "a .\<in> A" shows "a .\<in> B"
inverse_unique: fixes a :: "'a"   and b :: "'a" assumes "a + b = (0::'a)" shows "- a = b"
index_valid_distinct_indices: fixes as :: "('a \<times> 'b atom) set"   and s :: "('a, 'b) state" assumes "index_valid as s"   and "distinct_indices_atoms as" shows "distinct_indices_state s"
A_3: shows "Z \<subseteq> {u \<in> V. u x < real_of_int m}"
prob_space_distr: fixes f :: "'b \<Rightarrow> 'd"   and i :: "'i"   and M' :: "'d measure" assumes "D'.random_variable i M' f" shows "prob_space (distr (distr M (M' i) (X i)) M' f)"
bigo_neg_powr_iff: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "p < 0"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "\<forall>\<^sub>F x in F. 0 \<le> g x"   and "\<forall>\<^sub>F x in F. f x \<noteq> 0"   and "\<forall>\<^sub>F x in F. g x \<noteq> 0" shows "((\<lambda>x. f x powr p) \<in> O[F](\<lambda>x. g x powr p)) = (g \<in> O[F](f))"
degree_mod_less': fixes b :: "'a poly"   and a :: "'a poly" assumes "b \<noteq> 0"   and "a mod b \<noteq> 0" shows "Polynomial.degree (a mod b) < Polynomial.degree b"
reachable_closed_reachable_pnet: fixes s :: "'a net_state"   and np :: "nat \<Rightarrow> ('a, 'b seq_action) automaton"   and n :: "net_tree" assumes "s \<in> reachable (closed (pnet np n)) TT" shows "s \<in> reachable (pnet np n) TT"
assms: shows "ibounded_by consider (\<A>1, \<A>2) q"
less_asym': fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and P :: "bool" assumes "class.preorder less_eq less"   and "less a b"   and "less b a" shows "P"
deg_le_sect_pp_aux_1_1: fixes t :: "('x, nat) pp"   and xs :: "'x list"   and n :: "nat" assumes "t \<in> keys (deg_le_sect_pp_aux xs n)" shows "deg_pp t \<le> n"
sub_num_simps_2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and l :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one num.One (num.Bit0 l) = uminus (numeral.numeral one plus (Num.BitM l))"
ivl_integral_def: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'e" shows "ivl_integral a b f = integral {a..b} f - integral {b..a} f"
nth_subdegree_zero_iff: fixes f :: "'a fps" shows "(f $ subdegree f = (0::'a)) = (f = 0)"
assms_1: shows "reparam_weak \<gamma>1 \<gamma>2"
3: fixes x :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and y :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "(x, y) \<in>\<^sub>p snd ` set ps" shows "x \<in> set (p # q # hs @ map (fst \<circ> snd) ps @ map (snd \<circ> snd) ps)"
simps_18: fixes a :: "floatarith" shows "max_Var_floatarith (Floor a) = max_Var_floatarith a"
Diff_eqvt: fixes p :: "perm"   and A :: "'a set"   and B :: "'a set" shows "p \<bullet> (A - B) = p \<bullet> A - p \<bullet> B"
sorted_list_of_fset_simps_1: fixes S :: "'x fset" shows "set (sorted_list_of_fset S) = fset S"
set_cases_3: fixes e :: "'f"   and a :: "('g, 'h, 'f) new_thread_action" assumes "e \<in> set3_new_thread_action a" obtains z1 :: "'g" and z2 :: "'h" where "a = NewThread z1 z2 e"
closedin_connected_component: fixes s :: "'b set"   and x :: "'b" shows "closedin (top_of_set s) (connected_component_set s x)"
image_eq_imp_comp: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and g :: "'f \<Rightarrow> 'd"   and B :: "'f set"   and h :: "'d \<Rightarrow> 'g" assumes "f ` A = g ` B" shows "(h \<circ> f) ` A = (h \<circ> g) ` B"
po: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "minimal_element P A" shows "po_on P A"
J: shows "\<forall>i\<in>I. J i \<in> Group.iso (Gs i) ((f \<circ> Gs) i)"
inv_image_def: fixes r :: "('l \<times> 'l) set"   and f :: "'k \<Rightarrow> 'l" shows "inv_image r f = {(x, y). (f x, f y) \<in> r}"
adjoint_works: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" assumes "linear f" shows "x \<bullet> adjoint f y = f x \<bullet> y"
hom_mset_factors: fixes F :: "'b poly multiset"   and p :: "'b poly" assumes "mset_factors F p" shows "mset_factors (image_mset poly_x_minus_y F) (poly_x_minus_y p)"
hb_broadcast_exists2: fixes m1 :: "'a"   and m2 :: "'a" assumes "hb m1 m2" shows "\<exists>i. Broadcast m2 \<in> set (history i)"
Says_Server_imp_YM2: fixes A :: "agent"   and B :: "agent"   and k :: "msg"   and na :: "msg"   and nb :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, k, na, nb\<rbrace>, X\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "Gets Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, na, nb\<rbrace>\<rbrace> \<in> set evs"
in_bound_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bound :: "'a bound"   and x :: "'a" shows "in_bound less_eq less bound x \<equiv> case bound of Bound l r \<Rightarrow> less_eq l x \<and> less x r"
tst1': shows "ts1' t = \<lfloor>(x1', ln1')\<rfloor>"
multiplicity_Lcm: fixes A :: "'a set"   and p :: "'a" assumes "finite A"   and "A \<noteq> {}"   and "prime p"   and "(0::'a) \<notin> A" shows "multiplicity p (Lcm A) = Max (multiplicity p ` A)"
B_def: shows "B = (\<lambda>e. {n. e \<le> u n})"
axioms: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and \<Phi> :: "('a, 'b) lit list set"   and D :: "univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "CU.Model wtFsym wtPsym arOf parOf \<Phi> D intF intP" shows "U.Model wtFsym wtPsym arOf parOf \<Phi> D intF intP"
not_zle: fixes n :: "int"   and m :: "int" shows "(\<not> n \<le> m) = (m < n)"
rep_eq: fixes x :: "'c \<Rightarrow> 'a upred"   and xa :: "'c \<Rightarrow> ('b, 'a) uexpr" shows "\<lbrakk>UINF x xa\<rbrakk>\<^sub>e = (\<lambda>b. \<Sqinter>{uu_. \<exists>xb. uu_ = \<lbrakk>xa xb\<rbrakk>\<^sub>e b \<and> \<lbrakk>x xb\<rbrakk>\<^sub>e b})"
not_eq_extremum: fixes a :: "'a" shows "(a \<noteq> \<top>) = (a < \<top>)"
strassen_cost_pos: fixes n :: "nat" assumes "1 \<le> n" shows "0 < strassen_cost n"
rep_Gromov_completion_limit: fixes x :: "'c Gromov_completion" shows "(\<lambda>n. to_Gromov_completion (rep_Gromov_completion x n)) \<longlonglongrightarrow> x"
Max_image_left_mult: fixes S :: "'a set"   and c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal" assumes "finite S"   and "S \<noteq> {}" shows "(MAX x\<in>S. c * f x) = c * Max (f ` S)"
SeqFormP_imp_OrdP: fixes s :: "tm"   and k :: "tm"   and t :: "tm" shows "{SeqFormP s k t} \<turnstile> OrdP k"
U0_altdef: shows "U\<^sub>0 = {prenum e |e. e \<in> UNIV}"
fold_graph_insertE: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and z :: "'e"   and x :: "'e"   and B :: "'e set"   and v :: "'e" assumes "comp_fun_commute_on f A"   and "fold_graph f z (insert x B) v"   and "x \<notin> B"   and "insert x B \<subseteq> A"   and "z \<in> A" obtains y :: "'e" where "v = f x y" and "fold_graph f z B y"
finite_gpv_rel''D1: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and gpv :: "('e, 'g, 'i) gpv"   and gpv' :: "('f, 'h, 'j) gpv"   and \<I> :: "('g, 'i) \<I>"   and \<I>' :: "('h, 'j) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "finite_gpv \<I> gpv"   and "rel_\<I> C R \<I> \<I>'" shows "finite_gpv \<I>' gpv'"
prems_4: shows "last (msgs (S t i) cid) = Marker"
normalizability_E: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> A\<^sup>!" obtains "(a, b) \<in> A\<^sup>*" and "b \<in> NF A"
intensional_simps_13: fixes P :: "'a \<Rightarrow> bool" shows "P \<longrightarrow> P \<equiv> #True"
poly_add_zero'_2: fixes p :: "'a list list" assumes "set p \<subseteq> carrier (poly_ring R)" shows "UP.poly_add [] p = UP.normalize p"
rel_R_def: fixes P :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" shows "rel_R P Q = \<Union> {X. rel_aka.dom_op P \<subseteq> rel_aka.wp X Q}"
id_stake_snth_sdrop: fixes s :: "'d stream"   and i :: "nat" shows "s = stake i s @- s !! i ## sdrop (Suc i) s"
distinct_insert: fixes x :: "'d"   and xs :: "'d list" shows "distinct (List.insert x xs) = distinct xs"
swapPoly\<^sub>f_def: shows "swapPoly\<^sub>f \<equiv> map_fun id (map_fun id (map_fun fmlookup Abs_fmap)) (\<lambda>b i mp mon. mp (swap0 b i mon))"
Ker_image: fixes a :: "'a" assumes "a \<in> additive.Ker" shows "\<eta> a = \<zero>'"
m_ac_1: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
greatest_postfixpoint_unique: fixes f :: "'a \<Rightarrow> 'a" assumes "has_greatest_postfixpoint f" shows "\<exists>!x. is_greatest_postfixpoint f x"
BOresChainSimps_3: fixes N' :: "'d"   and P' :: "('d, 'e, 'f) psi"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "(N' \<prec>' P' = N \<prec>' P) = (N = N' \<and> P = P')"
ex1I: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "\<exists>!x. P x"
empty_suffix_eq: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "emptyseq s" shows "s |\<^sub>s n = s"
cases: fixes x :: "Core.term" obtains uu_ :: "String.literal" and T :: "typ" where "x = Ct uu_ T__"   | uv_ :: "variable" and T :: "typ" where "x = Fv uv_ Ta__"   | uw_ :: "nat" where "x = Bv uw_"   | T :: "typ" and t :: "Core.term" where "x = Abs T t__"   | t :: "Core.term" and u :: "Core.term" where "x = t $ u"
mset_distrib: fixes A :: "'a multiset"   and B :: "'a multiset"   and M :: "'a multiset"   and N :: "'a multiset" assumes "A + B = M + N" obtains Am :: "'a multiset" and An :: "'a multiset" and Bm :: "'a multiset" and Bn :: "'a multiset" where "A = Am + An" and "B = Bm + Bn" and "M = Am + Bm" and "N = An + Bn"
set_mset_inter: fixes A :: "'k multiset"   and B :: "'k multiset" shows "set_mset (A \<inter># B) = set_mset A \<inter> set_mset B"
ivl_disj_un_19: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l < m"   and "m \<le> u" shows "{l<..<m} \<union> {m..u} = {l<..u}"
\<theta>\<delta>_support_1: shows "\<theta> supports \<I>"
map_nth_eq_self: fixes fas :: "'e list"   and l :: "nat" assumes "length fas = l" shows "map ((!) fas) [0..<l] = fas"
dghm_dag_Rel_ArrMap_app_is_arr: fixes \<alpha> :: "V"   and b :: "V"   and a :: "V"   and T :: "V" assumes "T : b \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> a" shows "\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ArrMap\<rparr>\<lparr>T\<rparr> : \<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> \<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
pOp_assocTr41: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c \<plusminus>\<^bsub>A\<^esub> d = a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> (c \<plusminus>\<^bsub>A\<^esub> d)"
assms: fixes u :: "'t" assumes "v \<prec>\<^sub>t u" shows "lookup p u = (0::'b)"
weakStatImp_def: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<lessapprox><Rel> Q \<equiv> \<forall>\<Psi>''. \<exists>Q' Q''. \<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q' \<and> insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi> \<and> \<Psi> \<otimes> \<Psi>'' \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'' \<and> (\<Psi> \<otimes> \<Psi>'', P, Q'') \<in> Rel"
IH: shows "dnf y = Finite_Set.fold (\<lambda>\<phi>. (\<otimes>) (dnf \<phi>)) {{||}} A"