hcju/isabelleps huggingface dataset:
comp: fixes less_eq'' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less'' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'b set"   and g :: "'b \<Rightarrow> 'd" assumes "OrderingSetMap (\<^bold>\<le>*) (\<^bold><*) less_eq'' less'' Q g"   and "order.greater_eq Q (f ` P)" shows "OrderingSetMap (\<^bold>\<le>) (\<^bold><) less_eq'' less'' P (g \<circ> f)"
extend_set_eq_image: fixes h :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "extend_set (\<lambda>(x, u). h x) A = h ` A"
Elem_implode: fixes A :: "ZF set"   and x :: "ZF" assumes "A \<in> range explode" shows "x |\<in>| implode A = (x \<in> A)"
simps_18: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a"   and x3 :: "'c" shows "(case Fault x3 of Normal x \<Rightarrow> f1 x | Abrupt x \<Rightarrow> f2 x | Fault x \<Rightarrow> f3 x | Stuck \<Rightarrow> f4) = f3 x3"
rsquarefree_times: fixes p :: "'a poly"   and q :: "'a poly" assumes "rsquarefree (p * q)" shows "rsquarefree q"
simps_2: fixes n :: "nat" assumes "200 \<le> n" shows "foo_cost n = foo_cost (nat \<lfloor>real n / 3\<rfloor>) + foo_cost (nat \<lfloor>3 * real n / 4\<rfloor> + 42) + real n"
hom_one: fixes a :: "'a" shows "poly 1 a = (1::'a)"
notStuck_CatchD1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>Catch c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}" shows "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
get_fst_lens: fixes x :: "'a"   and y :: "'b" shows "get\<^bsub>fst\<^sub>L\<^esub> (x, y) = x"
finite_subsets_at_top_inter: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "filtermap (\<lambda>F. F \<inter> A) (finite_subsets_at_top B) \<le> finite_subsets_at_top A"
linear_injective_left_inverse: fixes f :: "'b \<Rightarrow> 'c" assumes "linear (*a) (*b) f"   and "inj f" shows "\<exists>g. linear (*b) (*a) g \<and> g \<circ> f = id"
assms_4: shows "($$) \<eta>1 \<in> extensional (obj\<^bsub>NTCatDom \<eta>1\<^esub>)"
intro: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'a \<times> 'b \<Rightarrow> 'c" assumes "category A1"   and "category A2"   and "category B"   and "product_category A1 A2"   and "functor (product_category.comp A1 A2) B F" shows "binary_functor A1 A2 B F"
sup_right_divisibility: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x y = (\<exists>z. sup z x = y)"
closure_subtopology_minimal: fixes S :: "'c set"   and T :: "'c set"   and X :: "'c set" assumes "S \<subseteq> T"   and "closedin (top_of_set X) T" shows "closure S \<inter> X \<subseteq> T"
FOREACHoci_rule: fixes S :: "'a set"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b nres"   and P :: "'b \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>c \<sigma>; x \<in> it; it \<subseteq> S; I it \<sigma>; \<forall>y\<in>it - {x}. R x y; \<forall>y\<in>S - it. R y x\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>it \<sigma>. \<lbrakk>it \<noteq> {}; it \<subseteq> S; I it \<sigma>; \<not> c \<sigma>; \<forall>x\<in>it. \<forall>y\<in>S - it. R y x\<rbrakk> \<Longrightarrow> P \<sigma>" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,I\<^esup> S c f \<sigma>0 \<le> SPEC P"
meet_binary: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = \<Sqinter>{x, y}"
at_pres: fixes x :: "'a"   and y :: "'a" shows "\<alpha> x \<cdot> \<tau> y \<le> \<tau> y \<cdot> \<alpha> x \<cdot> \<tau> y"
Abs_freelist_in_FreeGroup: fixes ss :: "'b list"   and S :: "'b set" assumes "ss \<in> lists S" shows "Abs_freelist ss \<in> FreeGroup S"
update_convs_8: fixes dests' :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> nat option"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "dests_update dests' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests' dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr>"
consec_v1_vn: shows "consec (lan v1) (lan vn)"
IH: fixes i :: "nat"   and j :: "nat" shows "len M i j xs + len M i j xs = len (\<lambda>i j. M i j + M i j) i j xs"
inv_def: fixes C :: "nat set"   and R :: "'a set" shows "local.inv C R = (sc C (U - R) \<and> R \<subseteq> U \<and> c_exists C R)"
S5: shows "mod7 (c_fst (c_pair 4 (c_pair (sch_to_nat sch1) (sch_to_nat sch2)))) = 4"
nat_dvd_1_iff_1: fixes m :: "nat" shows "is_unit m = (m = 1)"
prime_sum_upto_altdef1: fixes f :: "nat \<Rightarrow> 'a"   and x :: "real" shows "prime_sum_upto f x = sum_upto (\<lambda>p. ind prime p * f p) x"
assms_3: shows "\<forall>\<^sub>F x in net. b \<le> f x $ i"
assms_2: shows "almost_everywhere (restr_to_subalg M F) P"
countable_measurable_PiFI: fixes I :: "'d set set"   and A :: "'d \<Rightarrow>\<^sub>F 'e \<Rightarrow> 'f"   and M :: "'d \<Rightarrow> 'e measure"   and N :: "'f measure" assumes "\<And>J. \<lbrakk>J \<in> I; finite J\<rbrakk> \<Longrightarrow> A \<in> Pi\<^sub>F {J} M \<rightarrow>\<^sub>M N" shows "A \<in> Pi\<^sub>F I M \<rightarrow>\<^sub>M N"
assms: shows "x\<^sub>A \<in> \<C>\<^sub>A"
rsquarefree: shows "rsquarefree [:- 1 / 6, - 1 / 2, - 1, 1:]"
rep_def: fixes r :: "'c \<Rightarrow>\<^sub>0 'c" shows "ideal.rep r = (\<Sum>v\<in>keys r. lookup r v * v)"
join_trees: fixes g\<^sub>1 :: "'b ugraph"   and g\<^sub>2 :: "'b ugraph"   and u :: "'b"   and v :: "'b" assumes "tree g\<^sub>1"   and "tree g\<^sub>2"   and "nodes g\<^sub>1 \<inter> nodes g\<^sub>2 = {}"   and "u \<in> nodes g\<^sub>1"   and "v \<in> nodes g\<^sub>2" shows "tree (ins_edge (u, v) (graph_join g\<^sub>1 g\<^sub>2))"
Cons: fixes i :: "nat" shows "i < length v * row_length N \<and> mat nr (length N) N \<longrightarrow> row (vec_mat_Tensor v N) i = local.times (v ! (i div row_length N)) (row N (i mod row_length N))"
Mn1f_def: shows "Mn1f = Mn 1 f"
eval: fixes z :: "complex" shows "poly (poly_of_list (map g [0..<k])) z = (\<Sum>n<k. g n * z ^ n)"
Cons_3: shows "fmrestrict_set vs s = fmrestrict_set vs s'"
diagonal_to_Smith_i_preserves_previous_diagonal: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and xs :: "nat list"   and b :: "'b" assumes "is_bezout_ext bezout"   and "i < min (nrows A) (ncols A)"   and "to_nat a \<notin> set xs"   and "to_nat a = to_nat b"   and "to_nat a \<noteq> i"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)" shows "diagonal_to_Smith_i xs A i bezout $ a $ b = A $ a $ b"
cond_3: 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" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"   and "s\<^sub>o \<le> movement.s a\<^sub>e v\<^sub>e s\<^sub>e (movement.t_stop a\<^sub>e v\<^sub>e)"   and "movement.s a\<^sub>e v\<^sub>e s\<^sub>e (movement.t_stop a\<^sub>e v\<^sub>e) < movement.s a\<^sub>o v\<^sub>o s\<^sub>o (movement.t_stop a\<^sub>o v\<^sub>o)" shows "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o {0..} = (a\<^sub>e < a\<^sub>o \<and> v\<^sub>o < v\<^sub>e \<and> 0 \<le> (v\<^sub>o - v\<^sub>e)\<^sup>2 - 2 * (a\<^sub>o - a\<^sub>e) * (s\<^sub>o - s\<^sub>e) \<and> v\<^sub>e - a\<^sub>e / a\<^sub>o * v\<^sub>o < sqrt ((v\<^sub>o - v\<^sub>e)\<^sup>2 - 2 * (a\<^sub>o - a\<^sub>e) * (s\<^sub>o - s\<^sub>e)))"
assms_2: fixes i :: "'b"   and j :: "'c" assumes "(i, j) \<in> A" shows "P i j is H"
blb: shows "(SOME z. z \<notin> lesvars upds \<union> varacom ({A} WHILE b DO C)) \<notin> varacom C"
FRA: shows "FRadmit ([[x := \<theta>]]\<phi>)"
infinite_sort_of_eq: fixes s :: "atom_sort" shows "infinite {a. sort_of a = s}"
simps_2: 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 infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.GE_resOf resOf (Wit \<sigma>) = \<sigma>"
W_cmpser_is_D_gchain: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "W_cmpser G n f" shows "D_gchain G n f"
cong_equiv: fixes H :: "'a set" assumes "H \<in> normalfilters" shows "equivp (cong H)"
simps_3: fixes b\<^sub>1 :: "bexp"   and b\<^sub>2 :: "bexp"   and s :: "char list \<Rightarrow> int option" shows "pbval' (And b\<^sub>1 b\<^sub>2) s = optand (pbval' b\<^sub>1 s) (pbval' b\<^sub>2 s)"
VarP_Var: fixes H :: "fm set"   and i :: "name" shows "H \<turnstile> VarP \<guillemotleft>Var i\<guillemotright>"
tau_mpj_bilinear: fixes M :: "('e, 'a) Module"   and N :: "('f, 'a) Module"   and P :: "('e \<times> 'f, 'a) Module" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "bilinear_map (compose (M \<times>\<^sub>c N) (mpj P (TR\<^bsub>R\<^esub> M N P)) (tau R M N P)) R M N (M \<^bsub>P\<^esub>\<Otimes>\<^bsub>R\<^esub> N)"
\<phi>_simps_5: shows "cod r\<^sub>0s\<^sub>1.\<phi> = s\<^sub>1 \<star> p\<^sub>0"
of_seq_code_1: shows "set_of_seq seq.Empty = {}"
carrier_singleton_filter: fixes T :: "'b set set" assumes "carrier\<^bsub>T\<^esub> \<noteq> {}" shows "{carrier\<^bsub>T\<^esub>} \<in> Filters\<^bsub>T\<^esub>"
distinct_unique1: fixes vs :: "'a list"   and ram :: "'a" assumes "distinct vs"   and "ram \<in> set vs" shows "\<exists>!s. vs = fst s @ ram # snd s"
sorted_list_of_set_image: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "mono_on g A"   and "inj_on g A" shows "sorted_list_of_set (g ` A) = map g (sorted_list_of_set A)"
infinite_UNIV: assumes "class.alg_closed_field TYPE('c)" shows "infinite UNIV"
assms_1: shows "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) b ack i1 vc st"
enumAppend_bound_rec: fixes ls :: "nat list"   and n :: "nat"   and nmax :: "nat"   and lss :: "nat list list"   and x :: "nat" assumes "ls \<in> set ((enumAppend nmax ^^ n) lss)"   and "\<forall>y\<in>set lss. \<forall>z\<in>set y. z \<le> nmax"   and "x \<in> set ls" shows "x \<le> nmax"
mult_sm_inv_l: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" assumes "k \<noteq> 0"   and "k *\<^sub>s\<^sub>m A = B" shows "A = (1 / k) *\<^sub>s\<^sub>m B"
of_nat_neq_iff_word: fixes x :: "nat"   and y :: "nat" assumes "x mod 2 ^ LENGTH('b) \<noteq> y mod 2 ^ LENGTH('b)" shows "(word_of_nat x \<noteq> word_of_nat y) = (x \<noteq> y)"
rGamma_reflection_complex': fixes z :: "complex" shows "rGamma z * rGamma (- z) = - z * sin (complex_of_real pi * z) / complex_of_real pi"
hyps_3: fixes thesis :: "bool" assumes "\<And>P'. \<lbrakk>P \<Longrightarrow>\<^sub>\<tau> P'; (P', Q'') \<in> Rel\<rbrakk> \<Longrightarrow> thesis"   and "(P, Q) \<in> Rel" shows "thesis"
PossiblyContingentObjectExistsAxiom: shows "\<exists>x v. ConcreteInWorld x v \<and> (\<exists>w. \<not> ConcreteInWorld x w)"
degree_diff_le_max: fixes p :: "'b poly"   and q :: "'b poly" shows "degree (p - q) \<le> max (degree p) (degree q)"
visits_via_trivial: fixes W :: "'a set"   and A :: "'a set" assumes "v0 \<in> W" shows "visits_via P A W"
fps_cutoff_nth: fixes n :: "nat"   and f :: "'b fps"   and i :: "nat" shows "fps_cutoff n f $ i = (if i < n then f $ i else (0::'b))"
VDM_lowerm: fixes n :: "nat"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and m :: "nat" assumes " \<Turnstile>\<^sub>n c : A "   and "m < n" shows " \<Turnstile>\<^sub>m c : A "
less_imp_neq: fixes y :: "'b"   and x :: "'b" assumes "y < x" shows "x \<noteq> y"
proj_stoch_set_finite_range: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and n :: "nat" assumes "borel_adapt_stoch_proc F X" shows "finite (range (proj_stoch_proc X n))"
mod_div_decomp: fixes a :: "'a"   and b :: "'a" obtains q :: "'a" and r :: "'a" where "q = a div b" and "r = a mod b" and "a = q * b + r"
geq: shows "Games (USubst.adjoint (dotsubstt \<theta>) I \<omega>) = Games (repc I dotid (term_sem I \<theta> \<omega>))"
state_run: shows "r i = (foldl \<delta> (q\<^sub>0 \<phi>) (prefix i w), \<lambda>\<chi>. if \<chi> \<in> \<^bold>G \<phi> then Some (\<lambda>\<psi>. semi_mojmir_def.state_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M (theG \<chi>)) w \<psi> i) else None)"
insert_union_subset_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(add_mset a A \<subset># B) = (a \<in># B \<and> A \<subset># remove1_mset a B)"
list_fast_rec_term1: fixes ts :: "'b tree list" assumes "ts \<noteq> []"   and "\<langle>\<rangle> \<notin> set ts" shows "sum_list (map (size \<circ> left) ts) + sum_list (map (size \<circ> right) ts) < sum_list (map size ts)"
assms: fixes e :: "real" assumes "0 < e" shows "\<exists>M. \<forall>x\<in>X. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
slice_def: fixes x :: "state \<Rightarrow> 'd \<Rightarrow> 'c"   and i :: "'d"   and s :: "state" shows "(x!i) s \<equiv> x s i"
space_T: fixes s :: "'s" shows "space (T s) = space (stream_space (count_space UNIV))"
eq_refl: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and v :: "Value" assumes "ArityAnalysisLetSafe Aexp Aheap" shows "eq a v v"
compactin_PiE: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and S :: "'c \<Rightarrow> 'd set" shows "compactin (product_topology X I) (Pi\<^sub>E I S) = (Pi\<^sub>E I S = {} \<or> (\<forall>i\<in>I. compactin (X i) (S i)))"
eff_def: fixes i :: "instr"   and 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 pc :: "nat"   and et :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and t :: "(ty list \<times> ty err list) option" shows "eff i P pc et t = (case t of None \<Rightarrow> [] | \<lfloor>\<tau>\<rfloor> \<Rightarrow> norm_eff i P pc \<tau> @ xcpt_eff i P pc \<tau> et)"
simps_10: fixes \<phi> :: "Formula.formula" shows "Formula.future_bounded (formula.Exists \<phi>) = Formula.future_bounded \<phi>"
cring: fixes m :: "int" assumes "residues m" shows "cring (residue_ring m)"
prv_\<phi>R_eqv: shows "prv (eqv \<phi>R (neg (PP' \<langle>\<phi>R\<rangle>)))"
lm041: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "Outside' X P = P || (Domain P - X)"
dist_diff_1: fixes a :: "'b"   and b :: "'b" shows "dist a (a - b) = norm b"
Quotient_rep_equiv1: fixes R :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and Abs :: "'j \<Rightarrow> 'k"   and Rep :: "'k \<Rightarrow> 'j"   and T :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'j"   and b :: "'k" assumes "Quotient R Abs Rep T"   and "T a b" shows "R a (Rep b)"
upd_nested_max_tstp_fold: fixes X :: "('a \<times> 'b) set"   and m :: "('a, ('b, nat + nat) mapping) mapping"   and d :: "nat + nat" assumes "finite X" shows "upd_nested_max_tstp m d X = Finite_Set.fold (upd_nested_step d (max_tstp d)) m X"
inv_f_f: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "inj f" shows "inv f (f x) = x"
wellformed_2: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "snd e \<in> pverts G"
sign_changes_coeff_sign_changes: fixes xs :: "'a list"   and p :: "'a poly" assumes "Poly xs = p" shows "sign_changes xs = sign_changes (coeffs p)"
write_def: fixes c :: "'e \<Rightarrow> 'd"   and a :: "'e"   and P :: "'d process" shows "c`!`a \<rightarrow> P \<equiv> Mprefix {c a} (\<lambda>x. P)"
eq_iff: fixes x :: "'a list"   and y :: "'a list" shows "(x = y) = (subseq x y \<and> subseq y x)"
routing_ipassmt_wi_distinct: fixes tbl :: "'b routing_rule list" shows "distinct (map fst (routing_ipassmt_wi tbl))"
keys_proj_poly_syz: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "keys (proj_poly_syz n p) = map_component (\<lambda>k. k + n) -` keys p"
min_set_obtains: fixes A :: "'b fset"   and B :: "'b fset set" assumes "A \<in> B" obtains C :: "'b fset" where "C |\<subseteq>| A" and "C \<in> min_set B"
SSpec_def: fixes c :: "IMP"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" shows "SSpec c s t = s , c \<Down> t "
dfs_invariant': fixes S :: "'b"   and xs :: "'a list"   and I :: "'b \<Rightarrow> bool" assumes "invariant S"   and "list_all isNode xs"   and "I S"   and "\<And>S x. \<lbrakk>\<not> memb x S; isNode x; invariant S; I S\<rbrakk> \<Longrightarrow> I (ins x S)" shows "I (dfs S xs)"
fixp_unfold: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c" assumes "class.ccpo Sup less_eq less"   and "monotone less_eq less_eq f" shows "ccpo.fixp Sup less_eq f = f (ccpo.fixp Sup less_eq f)"
val_Zp_not_equal_imp_notequal_1: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "val_Zp x \<noteq> val_Zp y" shows "x \<noteq> y"
preSuc: fixes A :: "nat list" shows "\<forall>n. (Suc n \<in> set A) = (n \<in> set (preSuc A))"
assms: shows "MGU \<sigma> (Fun f X) (Fun g Y)"
strict_order_iff: fixes a :: "'c"   and b :: "'c" shows "(a < b) = (a = inf a b \<and> a \<noteq> b)"
N: shows "N \<in> null_sets (lebesgue_on S)"
Mismatch: fixes P :: "pi"   and Rs :: "Late_Semantics.residual"   and a :: "name"   and b :: "name" assumes "P \<longmapsto>\<^sub>l Rs"   and "a \<noteq> b" shows "[a\<noteq>b]P \<longmapsto>\<^sub>l Rs"
M_dim_3: shows "gs2.M (length (map of_int_hom.vec_hom fs')) \<in> carrier_mat (length (map of_int_hom.vec_hom fs')) (length (map of_int_hom.vec_hom fs'))"
Min_ex: fixes F :: "'a set" assumes "finite F"   and "F \<noteq> {}" shows "\<exists>m\<in>F. \<forall>n\<in>F. m \<le> n"
pequiv_equiv_trans: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<simeq> b = equiv_trans (wp a) (wp b)"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes> y \<otimes> z = x \<otimes> (y \<otimes> z)"
arr_lunit: fixes a :: "'a" assumes "ide a" shows "arr \<l>[a]"
1: assumes "order.greater_eq (f ` y) (f ` x)" shows "f ` y = f ` x"
arctan_upper_33_def: shows "arctan_lower_32 \<equiv> \<lambda>x. (64 * x ^ 4 + 735 * x\<^sup>2 + 945) * x / (15 * (15 * x ^ 4 + 70 * x\<^sup>2 + 63))"
loose_bvar_iff_exist_loose_bvar1: fixes t :: "Core.term"   and lev :: "nat" shows "loose_bvar t lev = (\<exists>lev'\<ge>lev. loose_bvar1 t lev')"
qring_mem: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> carrier R" shows "a \<uplus>\<^bsub>R\<^esub> I \<in> carrier (R /\<^sub>r I)"
length_take: fixes n :: "nat"   and xs :: "'g list" shows "length (take n xs) = min (length xs) n"
IH: assumes "set as \<subseteq> carrier G"   and "a divides foldr (\<otimes>) as \<one>" shows "\<exists>i<length as. a divides as ! i"
simps_1: fixes g :: "'c \<times> 'c \<Rightarrow> 'd" shows "star_matrix' [] g = mbot"
MAX: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "Suc (max k m) \<le> n" shows "k \<le> n \<and> m \<le> n"
strict_trans: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a < b"   and "b < c" shows "a < c"
c_manifold_order_le: fixes l :: "enat" assumes "l \<le> k" shows "c_manifold charts l"
assms_6: shows "has_least_fixpoint (g \<circ> h)"
intro: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "injective f"
emeasure_spmf_of_set: fixes S :: "'c set"   and A :: "'c set" shows "emeasure (measure_spmf (spmf_of_set S)) A = ennreal (real (card (S \<inter> A)) / real (card S))"
phis_phi: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list" assumes "phis g (n, v) = Some vs" shows "phi g v = Some vs"
swap_simps_1: fixes a :: "atom"   and b :: "atom" shows "[(a, b)] \<bullet> a = b"
assms_2: shows "\<forall>i\<in>net_ips (SubnetS s1 s2). the (netmap (SubnetS s1 s2) i) = \<sigma> i"
bij_betw_ide_Obj_4: fixes a :: "('obj, 'obj) concrete_category.arr" assumes "a \<in> Collect ide" shows "MkIde (Dom a) = a"
dvd_int_poly_def: fixes q :: "int poly"   and p :: "int poly" shows "dvd_int_poly q p = (if q = 0 then p = 0 else div_int_poly p q \<noteq> None)"
weakly_fair_def3: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "weakly_fair enabled taken = (\<lambda>s. (\<diamond>\<box>enabled) s \<longrightarrow> (\<box>\<diamond>taken) s)"
ground_SMP_disjointI_aux1: fixes f :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and g :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "f \<equiv> \<lambda>M. {t \<cdot> \<delta> |t \<delta>. t \<in> M \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> fv (t \<cdot> \<delta>) = {}}"   and "g \<equiv> \<lambda>M. {t \<in> M. fv t = {}}" shows "f (SMP M) = g (SMP M)"
finite_entropy_distributed: fixes S :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and Px :: "'c \<Rightarrow> real" assumes "finite_entropy S X Px" shows "distributed M S X (\<lambda>x. ennreal (Px x))"
phiUses_def: fixes g :: "'g"   and n :: "'node" shows "phiUses g n \<equiv> \<Union>n'\<in>set (successors g n). \<Union>v'\<in>phiDefs g n'. snd ` Set.filter (\<lambda>(n'', v). n'' = n) (set (zip (predecessors g n') (the (phis g (n', v')))))"
assms: shows "[\<lparr>F,x,y,z\<rparr> in v]"
assms_4: shows "fs' = fs[i := fs ! i - c \<cdot>\<^sub>v fs ! j]"
coreflexive_comp_inf: fixes x :: "'a"   and y :: "'a" assumes "coreflexive x"   and "coreflexive y" shows "x * y = x \<sqinter> y"
intro: fixes L :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "field L"   and "subfield K L"   and "algebraic_closure_axioms L K" shows "algebraic_closure L K"
leg1_in_hom_2: shows "\<guillemotleft>g : g \<Rightarrow> g\<guillemotright>"