hcju/isabelleps huggingface dataset:
collision_subset: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and s :: "real set"   and t :: "real set" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"   and "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o s"   and "s \<subseteq> t" shows "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o t"
body_case: shows "new_body = (case T' of Class D \<Rightarrow> \<lparr>D\<rparr>body | _ \<Rightarrow> body)"
sign_simps_5: fixes ah :: "'c"   and bh :: "'c" shows "ah * bh = bh * ah"
simps_2: fixes env :: "('d, bool) mapping"   and t1 :: "'d ifex"   and t2 :: "'d ifex" shows "normif env Falseif t1 t2 = reduce env t2"
vimageE: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "a \<in> f -` B" obtains x :: "'d" where "f a = x" and "x \<in> B"
combine_SDG_slicesp_combine_SDG_slices_eq: fixes S :: "'node SDG_node set" shows "combine_SDG_slicesp (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> combine_SDG_slices S)"
simps_3: fixes I :: "('b, 'c, 'a) interp"   and \<phi> :: "('b, 'c, 'a) formula" shows "fml_sem I (! \<phi>) = {v. v \<notin> fml_sem I \<phi>}"
rel_interior_convex_linear_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "linear f"   and "convex S" shows "f ` rel_interior S = rel_interior (f ` S)"
multi_member_this: fixes x :: "'c"   and XS :: "'c multiset" shows "x \<in># {#x#} + XS"
Set_MP: fixes x :: "tm"   and y :: "tm"   and H :: "fm set"   and z :: "tm"   and A :: "fm" assumes "x SUBS y \<in> H"   and "z IN x \<in> H"   and "insert (z IN y) H \<turnstile> A" shows "H \<turnstile> A"
sorted_snoc_le: fixes xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "ASSUMPTION (Sorted_Less.sorted (xs @ [x]))"   and "x \<le> y" shows "Sorted_Less.sorted (xs @ [y])"
continuous_on_interior: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "continuous_on s f"   and "x \<in> interior s" shows "isCont f x"
HPT_Delta: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "t \<in> TS"   and "source t \<in> States A"   and "A \<in> SAs (HA ST)" shows "t \<in> Delta A"
dom_null: shows "B.VV.dom B.VV.null = B.VV.null"
union_grouped_out_arcs: fixes G_list :: "'b list \<times> ('b \<times> 'b) list" shows "\<Union> (sset (grouped_out_arcs G_list)) = set (snd G_list)"
bind_rule_complete: fixes M :: "'c nres"   and f :: "'c \<Rightarrow> 'b nres"   and \<Phi> :: "'b \<Rightarrow> bool" shows "(M \<bind> f \<le> SPEC \<Phi>) = (M \<le> SPEC (\<lambda>x. f x \<le> SPEC \<Phi>))"
StepT: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and s'' :: "'state" assumes "cf \<rightarrow>*c cf'"   and "cf' \<rightarrow>t s''" shows "cf \<rightarrow>*t s''"
Gauss_Jordan_in_ij_iarrays_def: fixes A :: "'b iarray iarray"   and i :: "nat"   and j :: "nat" shows "Gauss_Jordan_in_ij_iarrays A i j = (let n = least_non_zero_position_of_vector_from_index (column_iarray j A) i; interchange_A = interchange_rows_iarray A i n; A' = mult_row_iarray interchange_A i ((1::'b) / interchange_A !! i !! j) in IArray.of_fun (\<lambda>s. if s = i then A' !! s else row_add_iarray A' s i (- interchange_A !! s !! j) !! s) (nrows_iarray A))"
PRelQ: fixes Tvec :: "'a list" assumes "length xvec = length Tvec" shows "(\<Psi>, P[xvec::=Tvec], Q[xvec::=Tvec]) \<in> Rel"
sdrop_simps_2: fixes n :: "nat"   and s :: "'c stream" shows "stl (sdrop n s) = sdrop (Suc n) s"
thin4: fixes A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and H :: "fm set"   and B :: "fm"   and A4 :: "fm" assumes "insert A1 (insert A2 (insert A3 H)) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 H))) \<turnstile> B"
axioms_1: fixes G :: "('e, 'f) monoid_scheme" assumes "finite_group G" shows "Group.group G"
sel_4: fixes x :: "quat" shows "Im3 (inverse x) = - Im3 x / ((quat.Re x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2)"
snd_comp_apsnd: fixes f :: "'b \<Rightarrow> 'c" shows "snd \<circ> apsnd f = f \<circ> snd"
interior_of_maximal_eq: fixes X :: "'d topology"   and T :: "'d set"   and S :: "'d set" assumes "openin X T" shows "(T \<subseteq> X interior_of S) = (T \<subseteq> S)"
prime_gauss_int_norm_squareD: fixes z :: "gauss_int"   and p :: "nat" assumes "prime z"   and "gauss_int_norm z = p\<^sup>2" shows "prime p \<and> z = of_nat p"
prefix_split_first: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" obtains as :: "'c list" where "prefix (as @ [x]) xs" and "x \<notin> set as"
map_to_nat_loc_def: fixes s_ops :: "('a, 'b, 'c) set_ops_scheme"   and m_ops :: "('a, nat, 'd, 'e) map_ops_scheme" shows "map_to_nat_loc s_ops m_ops \<equiv> StdSet s_ops \<and> StdMap m_ops"
nt_inv: fixes a :: "'a"   and n :: "int" assumes "sfg A a" shows "-\<^sub>a n\<triangleright>a\<^bsub>A\<^esub> = (- n)\<triangleright>a\<^bsub>A\<^esub>"
WHATWHERE_Secure_def: fixes V :: "('exp, 'id) MWLsCom list" shows "WHATWHERE_Secure V \<equiv> \<forall>d PP. \<exists>R. SdlHPPB d PP R \<and> (V, V) \<in> R"
iso_incl_backwards: fixes A :: "'c ring"   and B :: "'c ring" assumes "A \<lesssim> B" shows "id \<in> ring_hom A B"
insertAssertionExtractFrameFresh: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>" shows "insertAssertion (extractFrame (\<lparr>\<nu>*xvec\<rparr>P)) \<Psi> \<simeq>\<^sub>F \<lparr>\<nu>*xvec\<rparr>insertAssertion (extractFrame P) \<Psi>"
regions_part_ex_2: fixes u :: "'c \<Rightarrow> real" assumes "u \<in> V" shows "[u]\<^sub>\<R> \<in> \<R>"
leq_def: shows "leq = restrict_rel F {(f, f'). f' \<le> f}"
collapse_2: fixes operation :: "'a operation" assumes "\<not> is_INS operation" shows "DEL (pos operation) (op_idx operation) (op_int operation) = operation"
floor_numeral_power: fixes x :: "num"   and n :: "nat" shows "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
V_constotal_2_def: shows "V_constotal_2 \<equiv> {f. \<exists>j a k. f = j # a @ [k] \<odot> 0\<^sup>\<infinity> \<and> 2 \<le> j \<and> (\<forall>i<length a. a ! i \<le> 1) \<and> 2 \<le> k \<and> \<phi> j = j # a \<odot> \<up>\<^sup>\<infinity> \<and> \<phi> k = f}"
combine_code_dist_eq: fixes \<delta>\<^sub>L :: "int"   and p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and \<delta>\<^sub>R :: "int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and \<delta> :: "int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list" assumes "\<delta>\<^sub>L = dist_code p\<^sub>0\<^sub>L p\<^sub>1\<^sub>L"   and "\<delta>\<^sub>R = dist_code p\<^sub>0\<^sub>R p\<^sub>1\<^sub>R"   and "(\<delta>, c\<^sub>0, c\<^sub>1) = combine_code (\<delta>\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (\<delta>\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps" shows "\<delta> = dist_code c\<^sub>0 c\<^sub>1"
big_mono: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. norm (g x) \<le> norm (f x)" shows "f \<in> \<Omega>[F](g)"
static_cruel: fixes s :: "'b list"   and A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and "is" :: "'c"   and n :: "nat" assumes "s \<noteq> []" shows "static s (cruel A (s, is) n)"
m_closed: fixes x :: "'a \<Rightarrow> complex"   and y :: "'a \<Rightarrow> complex" assumes "x \<in> carrier (Characters G)"   and "y \<in> carrier (Characters G)" shows "x \<otimes>\<^bsub>Characters G\<^esub> y \<in> carrier (Characters G)"
pref_same_len: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>p v"   and "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|" shows "u = v"
elims: fixes x :: "'b list"   and y :: "('b \<times> 'b) set" assumes "allConsec x = y" obtains "x = []" and "y = {}"   | a :: "'b" where "x = [a__]" and "y = {}"   | a :: "'b" and b :: "'b" and as :: "'b list" where "x = a # b # as" and "y = insert (a, b) (allConsec (b # as))"
frange_vsubset_VUnion2: fixes r :: "V" shows "\<R>\<^sub>\<bullet> r \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> r))"
that: assumes "x1 < x"   and "x < x2"   and "f analytic_on cbox (Complex x1 y1) (Complex x2 y2)" shows "thesis"
13: shows "Ind (set [a]) (set b\<^sub>1)"
list_diff_closed: fixes R :: "'a set"   and gs :: "'a list"   and hs :: "'a list" assumes "Ring1 R"   and "set gs \<subseteq> R"   and "set hs \<subseteq> R" shows "set (map2 (-) gs hs) \<subseteq> R"
bounded_linear_componentwise_iff: fixes f' :: "'a \<Rightarrow> 'b" shows "bounded_linear f' = (\<forall>i\<in>Basis. bounded_linear (\<lambda>x. f' x \<bullet> i))"
comparator_axioms: fixes cmp :: "'b comparator" shows "comparator (compare cmp)"
A: shows "\<forall>xa. (\<exists>t1 t2. x = t1 @ t2 \<and> t1 \<in> D (Y xa) \<and> tickFree t1 \<and> front_tickFree t2) \<or> (\<exists>t1 t2. x = t1 @ t2 \<and> t1 @ [tick] \<in> T (Y xa) \<and> t2 \<in> D S)"
wk: shows "- (2 ^ (k - 1)) \<le> i"
Mg: shows "(\<lambda>(x, y). g x y) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure t')"
valid_strategy_updates: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and w0 :: "'a" assumes "strategy p \<sigma>"   and "v0 \<rightarrow> w0" shows "strategy p (\<sigma>(v0 := w0))"
restrict_map_eq_2: fixes m :: "'b \<Rightarrow> 'a option"   and A :: "'b set"   and k :: "'b"   and v :: "'a" shows "((m |` A) k = Some v) = (m k = Some v \<and> k \<in> A)"
induced_morphism_def: shows "induced_morphism \<equiv> \<lambda>C\<in>stfx.carrier_stalk. let r = SOME r. r \<in> C in stx.class_of (f \<^sup>\<inverse> X (fst r)) (\<phi>\<^sub>f (fst r) (snd r))"
Inf_exp_def: fixes S :: "('b \<Rightarrow> real) set" shows "Inf_exp S = (\<lambda>s. Inf {f s |f. f \<in> S})"
simps_1: fixes s :: "('d, 'e) fmap" shows "exec_plan s [] = s"
hom_0_iff: fixes x :: "'a mod_ring poly" shows "(to_int_poly x = 0) = (x = 0)"
integrable_density: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and g :: "'d \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x" shows "integrable (density M (\<lambda>x. ennreal (g x))) f = integrable M (\<lambda>x. g x *\<^sub>R f x)"
R_def: shows "R = (\<lambda>n. real (2 * n + 1) * pi)"
\<eta>_in_terms_of_\<phi>: fixes y :: "'b" assumes "Adj.D.ide y" shows "Adj'.\<eta> y = Adj'.\<eta>\<epsilon>.\<phi> y (F y)"
iff': shows "a \<noteq> b \<longrightarrow> (\<forall>i. (b <\<^bsub>P i\<^esub> a) = (b <\<^bsub>P' i\<^esub> a))"
stable_on_allocation: fixes ds :: "'b set"   and X :: "'a set" assumes "ManiPref'.stable_on ds X" shows "inj_on Xd X"
abelian_group_rng_of_frac: shows "abelian_group rec_rng_of_frac"
vbrelation_Limit_vsubset_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<AA>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "\<FF>\<lparr>ObjMap\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat set" assumes "older_seniors x xa = y" obtains x :: "nat" and n :: "nat" where "xa__ = x" and "xa = n" and "y = {s. \<exists>y. s = senior y n \<and> s < senior x n \<and> \<not> sink (token_run s n)}"
gauss_jordan_check_invertable: fixes A :: "'c mat"   and n :: "nat"   and B :: "'c mat"   and nc :: "nat"   and b :: "'d" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n nc" shows "(A \<in> Units (ring_mat TYPE('c) n b)) = (fst (gauss_jordan A B) = 1\<^sub>m n)"
take_tl: fixes n :: "nat"   and xs :: "'a list" shows "take n (tl xs) = tl (take (Suc n) xs)"
real_bounded_linear: fixes f :: "real \<Rightarrow> real" shows "bounded_linear f = (\<exists>c. f = (\<lambda>x. x * c))"
le_inf_prefix_iff: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" shows "prefix x (y \<sqinter> z) = (prefix x y \<and> prefix x z)"
simps_1: fixes x :: "'b"   and y :: "'b"   and yl :: "'b bintree"   and yr :: "'b bintree"   and ts :: "'b bintree list" shows "t_ins_aux \<lparr>t_type.folding = False, item = x, subtrees = Branch y yl yr # ts\<rparr> = (if x \<le> y then t_ins_aux \<lparr>t_type.folding = False, item = x, subtrees = yl # Branch y yl yr # ts\<rparr> else t_ins_aux \<lparr>t_type.folding = False, item = x, subtrees = yr # Branch y yl yr # ts\<rparr>)"
minus_zero: shows "\<ominus>\<^bsub>poly_ring R\<^esub> \<zero>\<^bsub>poly_ring R\<^esub> = \<zero>\<^bsub>poly_ring R\<^esub>"
simps_1: fixes x :: "'c" shows "compile (Variable x) = [Load x]"
csupport_on_mono: fixes carrier :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'f" assumes "\<And>x. \<lbrakk>x \<in> carrier; f x \<noteq> (0::'d)\<rbrakk> \<Longrightarrow> g x \<noteq> (0::'f)" shows "csupport_on carrier f \<subseteq> csupport_on carrier g"
beta: fixes s :: "dB"   and t :: "dB" shows "Abs s \<degree> t \<rightarrow>\<^sub>\<beta> s[t/0]"
valid_cfgI: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and s :: "'a"   and cfg :: "'a cfg" assumes "Markov_Decision_Process_Invariant K S"   and "s \<in> S"   and "cfg \<in> Markov_Decision_Process.cfg_on K s" shows "cfg \<in> Markov_Decision_Process_Invariant.valid_cfg K S"
integral_power: fixes a :: "real"   and b :: "real"   and k :: "nat" assumes "a \<le> b" shows "LBINT x. x ^ k * indicat_real {a..b} x = (b ^ Suc k - a ^ Suc k) / real (Suc k)"
add_right_cancel: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and b :: "'g"   and a :: "'g"   and c :: "'g" assumes "class.cancel_semigroup_add plus" shows "(plus b a = plus c a) = (b = c)"
tt: shows "Strat ! n = (Strat @ r1) ! n"
inf_bot_left: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and x :: "'b::type" assumes "class.bounded_lattice_bot (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (x::'b::type) = bot"
nonTickFree_n_frontTickFree: fixes s :: "'b event list" assumes "\<not> tickFree s"   and "front_tickFree s" shows "\<exists>t. s = t @ [tick]"
rf_def: shows "rf = Cn 2 (r_smn 1 2) [r_dummy 1 (r_const (encode r)), recf.Id 2 0, recf.Id 2 1]"
rel_of_pred_def: fixes s :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_of_pred s \<equiv> {(a, b). s a b}"
prime_imp_prime_elem: fixes p :: "'a" assumes "prime p" shows "prime_elem p"
s_learn_U0: shows "learn_bc \<phi> U\<^sub>0 s"
eq: shows "(- 1) ^ (card X - Suc j) = - ((- 1) ^ (card X - j))"
times_matrix_def: fixes A :: "'b matrix"   and B :: "'b matrix" shows "A * B = mult_matrix (*) (+) A B"
conseq: fixes P' :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<forall>s. P' s \<longrightarrow> P s"   and "\<turnstile>\<^sub>t {P} c {Q}"   and "\<forall>s. Q s \<longrightarrow> Q' s" shows "\<turnstile>\<^sub>t {P'} c {Q'}"
InsInitV: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and c :: "stmt"   and v :: "Term.var"   and Q :: "'b assn" shows "G,A\<turnstile>{Normal P} InsInitV c v=\<succ> {Q}"
ide_def: fixes a :: "'b" shows "D.ide a \<equiv> a \<cdot>\<^sub>D a \<noteq> D.null \<and> (\<forall>f. (f \<cdot>\<^sub>D a \<noteq> D.null \<longrightarrow> f \<cdot>\<^sub>D a = f) \<and> (a \<cdot>\<^sub>D f \<noteq> D.null \<longrightarrow> a \<cdot>\<^sub>D f = f))"
mult_poly_full_def: fixes p :: "(char list list \<times> int) list"   and q :: "(char list list \<times> int) list" shows "mult_poly_full p q = Let (PAC_Polynomials_Operations.mult_poly_raw p q) PAC_Polynomials_Operations.normalize_poly"
cinner_simps_5: fixes r :: "complex"   and x :: "'c"   and y :: "'c" shows "\<langle>r *\<^sub>C x, y\<rangle> = cnj r * \<langle>x, y\<rangle>"
section_is_mono: fixes g :: "'a \<times> 'a" assumes "VV.section g" shows "VV.mono g"
convert_term_simps_1: fixes name :: "name" shows "convert_term (const name) = const name"
assms_1: shows "hgf = [h, g, f]\<^sub>\<circ>"
cong_sum: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "\<And>x. x \<in> A \<Longrightarrow> unique_euclidean_semiring.cong modulo (f x) (g x) a" shows "unique_euclidean_semiring.cong modulo (comm_monoid_add.sum plus zero f A) (comm_monoid_add.sum plus zero g A) a"
a_type_wf_def: fixes h :: "(_) heap" shows "l_type_wf_def\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t.a_type_wf h = True"
pareto_losers_lift: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "lowered.is_pref_profile R" shows "pareto_losers (lift R) = pareto_losers R \<union> (alts - alts')"
expand_fps_eq: fixes p :: "('c, 'd) fps"   and q :: "('c, 'd) fps" shows "(p = q) = (\<forall>n. p $ n = q $ n)"
rexp_DA_axioms: shows "rexp_DA \<sigma> wf_atom project lookup init delta final wf_state post L n"
xj_set: fixes x :: "'b"   and j :: "'b" shows "(x \<in> {a |a. (a, j) \<in> e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>}) = ((x, j) \<in> e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>)"
fds_const_def: fixes c :: "'a" shows "fds_const c = fds (\<lambda>n. if n = 1 then c else (0::'a))"
inverse_fun_def: fixes f :: "'c \<Rightarrow> 'd" shows "inverse f = inverse \<circ> f"
induct: fixes x :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) \<times> (nat \<times> ('sf, 'sc, 'sz) step) list"   and P :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) \<times> (nat \<times> ('sf, 'sc, 'sz) step) list \<Rightarrow> bool" assumes "proof_ok x"   and "\<And>D S. deriv_ok (start_proof D) S \<Longrightarrow> P (D, S)" shows "P x"
WhileAnnoFix: fixes P :: "'d set"   and I :: "'e \<Rightarrow> 'd set"   and b :: "'d set"   and Q :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'e \<Rightarrow> ('d, 'f, 'g) com"   and V :: "'e \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set" assumes "P \<subseteq> {s. \<exists>Z. s \<in> I Z \<and> I Z \<inter> - b \<subseteq> Q}"   and "\<forall>Z \<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> I Z \<inter> b) c Z ({t. (t, \<sigma>) \<in> V Z} \<inter> I Z),A"   and "\<forall>Z. c Z = c undefined"   and "\<forall>Z. wf (V Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P whileAnnoFix b I V c Q,A"
finite_subsets_at_top_neq_bot: fixes A :: "'c set" shows "finite_subsets_at_top A \<noteq> bot"
wcode_backto_standard_pos_2_exists: fixes ires :: "cell list"   and rs :: "nat"   and b :: "cell list"   and list :: "cell list" assumes "wcode_backto_standard_pos_2 ires rs (b, Bk # list)" shows "(\<exists>ln. b = Bk # Bk \<up> ln @ Oc # ires) \<and> (\<exists>rn. list = Oc \<up> Suc (Suc rs) @ Bk \<up> rn)"
p1: shows "Esigma m = Esigma g * Esigma M"
in_segment_fst_le: fixes x0 :: "real"   and x1 :: "real"   and x :: "real"   and b :: "real"   and y0 :: "real"   and y1 :: "real" assumes "x0 \<le> x1"   and "(x, b) \<in> closed_segment (x0, y0) (x1, y1)" shows "x \<le> x1"
of_nat_mult: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m * n) = times (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
renameLCs_cong: fixes f :: "'form"   and f1 :: "'var \<Rightarrow> 'var"   and f2 :: "'var \<Rightarrow> 'var" assumes "\<And>x. x \<in> lconsts f \<Longrightarrow> f1 x = f2 x" shows "renameLCs f1 f = renameLCs f2 f"
intros_2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r\<^sup>+\<^sup>+ a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
antisym: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "ord (PP x) (PP y)"   and "ord (PP y) (PP x)" shows "x = y"
Class_cong: fixes a :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "Class a = Class a'"   and "Class b = Class b'"   and "a \<in> M"   and "a' \<in> M"   and "b \<in> M"   and "b' \<in> M" shows "Class (a \<cdot> b) = Class (a' \<cdot> b')"
preserves_reflects_arr: fixes f :: "'b" shows "C.arr (G f) = B.arr f"
rel_1: shows "0 \<le> int_of_uint32 x"
M_is_borel: fixes M :: "real measure" assumes "finite_borel_measure M" shows "sets M = sets borel"
assign_abrupt_propagation: fixes f :: "'a \<Rightarrow> state \<Rightarrow> state"   and n :: "'a"   and s :: "state"   and x :: "abopt" assumes "fst (f n s) \<noteq> x"   and "fst (assign f n s) = x" shows "fst s = x"
simps_1: fixes r :: "'a pfm" shows "reduce' r \<^bold>\<bottom>\<^sub>! = (r \<^bold>\<longrightarrow>\<^sub>! \<^bold>\<bottom>\<^sub>!)"
real_le_mult_sustain: fixes r :: "real"   and y :: "real" assumes "\<And>z. \<lbrakk>0 < z; z < 1\<rbrakk> \<Longrightarrow> z * r \<le> y" shows "r \<le> y"
Un_absorb1: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "A \<union> B = B"
VA: shows "Vagree \<nu> \<omega> (\<Union>y\<in>{y. Inl (Inr y) \<in> SIGO ODE}. FVT (\<sigma> y))"
ext: fixes g :: "'d"   and f :: "'d" assumes "g \<cdot>\<^sub>D f \<noteq> D.null" shows "D.seq g f"
integral_eq_zero_AE: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "AE x in M. f x = (0::'b)" shows "integral\<^sup>L M f = (0::'b)"
WCR_onD: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "WCR_on r A"   and "a \<in> A"   and "(a, b) \<in> r"   and "(a, c) \<in> r" shows "(b, c) \<in> r\<^sup>\<down>"
strict_trans2: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "strict_prefix b a"   and "??.Sublist.prefix c b" shows "strict_prefix c a"
unlabel_in: fixes l :: "'e"   and x :: "'f"   and A :: "('e \<times> 'f) list" assumes "(l, x) \<in> set A" shows "x \<in> set (unlabel A)"
simps: fixes a :: "['b]" shows "finite_list a = (a = [] \<or> (\<exists>x xs. a = x : xs \<and> finite_list xs))"
sorted_group_sort: fixes f :: "'a \<Rightarrow> 'b"   and merge :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a list \<Rightarrow> 'c"   and xs :: "'a list" assumes "groupsort f merge g" shows "sorted (map f (groupsort.group_sort f merge xs))"
prems_1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set (x # xs)"   and "b \<in> set (x # xs)" shows "a \<le> b \<or> b \<le> a"