hcju/isabelleps huggingface dataset:
strict_order_iff: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a <s b) = (a = signed.min a b \<and> a \<noteq> b)"
U_Model: shows "CU.Model wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) \<Phi> (intT any) intF intP"
dvd_gcdD1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a (gcd b c)" shows "dvd.dvd times a b"
ide_MkIde: fixes A :: "'c \<Rightarrow> 's" assumes "A \<in> Collect (functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S))" shows "Cop_S.ide (Cop_S.MkIde A)"
resl_eq: fixes y :: "'a" shows "residual (\<lambda>x. x \<cdot> y) = (\<lambda>x. x \<leftarrow> y)"
enabled_conjE: fixes F :: "state \<times> state \<Rightarrow> bool"   and G :: "state \<times> state \<Rightarrow> bool"   and s :: "state" assumes "s \<Turnstile> Enabled (F \<and> G)" obtains "s \<Turnstile> Enabled F" and "s \<Turnstile> Enabled G"
UNION_empty_conv_2: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "(\<Union> (B ` A) = {}) = (\<forall>x\<in>A. B x = {})"
inv_M_rev: fixes c :: "int" assumes "2 * \<bar>c\<bar> < m" shows "inv_M (M c) = c"
list_comb: fixes f :: "'c"   and xs :: "'c list"   and S :: "name fset" shows "closed_except (list_comb f xs) S = (closed_except f S \<and> list_all (\<lambda>t. closed_except t S) xs)"
elims_1: fixes x :: "bool list"   and xa :: "tree"   and y :: "bool" assumes "branch x xa = y" obtains "x = []" and "xa = Leaf" and "y"   | d :: "bool" and ds :: "bool list" and l :: "tree" and r :: "tree" where "x = d # ds" and "xa = Branching l r" and "y = (if d then branch ds l else branch ds r)"   | v :: "bool" and va :: "bool list" where "x = v__ # va__" and "xa = Leaf" and "\<not> y"   | v :: "tree" and va :: "tree" where "x = []" and "xa = Branching v va" and "\<not> y"
regexp_unfold_2: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subseteq> B) = (A \<union> B = B)"
subset_diff: fixes B :: "'d set"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "B \<subseteq> A"   and "finite A" shows "sum g A = sum g (A - B) + sum g B"
surjective: fixes r :: "'c alpha_d_scheme" shows "r = \<lparr>ok = ok r, \<dots> = alpha_d.more r\<rparr>"
extremum_uniqueI: fixes a :: "'c list" assumes "suffix a []" shows "a = []"
cs_first_pd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> n pd\<rightarrow> \<pi> 0"   and "\<forall>l<n. \<pi> l \<noteq> \<pi> n" shows "cs\<^bsup>\<pi>\<^esup> n = [\<pi> n]"
\<E>_E: fixes x :: "'k"   and S :: "'k set"   and \<Gamma> :: "('k, 'l) web_scheme" assumes "x \<in> \<E>\<^bsub>\<Gamma>\<^esub> S" obtains p :: "'k list" and y :: "'k" where "path \<Gamma> x p y" and "y \<in> B \<Gamma>" and "\<And>z. \<lbrakk>x \<noteq> y; z \<in> set p\<rbrakk> \<Longrightarrow> z = x \<or> z \<notin> S"
assms_2: shows "(m, m') \<in> \<langle>Rk, Rv\<rangle>Rm"
assms_1: fixes i :: "nat" assumes "i \<le> n" shows "g i \<in> carrier P"
minit_safe_def: fixes \<phi> :: "Formula.formula" shows "minit_safe \<phi> = (if mmonitorable_exec \<phi> then minit \<phi> else undefined)"
select_convs_2: fixes verts :: "'d set"   and arcs :: "'c set"   and tail :: "'c \<Rightarrow> 'd"   and head :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "pre_digraph.arcs \<lparr>pre_digraph.verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = arcs"
cases: fixes valid_edge :: "'f \<Rightarrow> bool"   and src :: "'f \<Rightarrow> 'g"   and trg :: "'f \<Rightarrow> 'g"   and knd :: "'f \<Rightarrow> 'h edge_kind"   and E :: "'g"   and X :: "'g"   and a :: "'g LDCFG_node \<times> 'h edge_kind \<times> 'g LDCFG_node" assumes "lift_valid_edge valid_edge src trg knd E X a" obtains aa :: "'f" and e :: "'g LDCFG_node \<times> 'h edge_kind \<times> 'g LDCFG_node" where "a = e__" and "valid_edge aa" and "src aa \<noteq> E \<or> trg aa \<noteq> X" and "e__ = (Node (src aa), knd aa, Node (trg aa))"   | e :: "'g LDCFG_node \<times> 'h edge_kind \<times> 'g LDCFG_node" where "a = ea__" and "ea__ = (NewEntry, (\<lambda>s. True)\<^sub>\<surd>, Node E)"   | e :: "'g LDCFG_node \<times> 'h edge_kind \<times> 'g LDCFG_node" where "a = eb__" and "eb__ = (Node X, (\<lambda>s. True)\<^sub>\<surd>, NewExit)"   | e :: "'g LDCFG_node \<times> 'h edge_kind \<times> 'g LDCFG_node" where "a = e" and "e = (NewEntry, (\<lambda>s. False)\<^sub>\<surd>, NewExit)"
sparse_row_vector_smult: fixes y :: "'a"   and a :: "(nat \<times> 'a) list" shows "sparse_row_vector (smult_spvec y a) = scalar_mult y (sparse_row_vector a)"
"3_3": assumes "x \<noteq> y"   and "\<not> rel x y" shows "set (merge_wrt rel (x # xs) ys) = set (x # xs) \<union> set ys"
cond: shows "q' \<noteq> 0 \<and> odd (order x p' - order x q') = (q' mod p' \<noteq> 0 \<and> odd (order x p' - order x (q' mod p')))"
assms: shows "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> xs"
from_hma\<^sub>v_def: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>v v = Matrix.vec CARD('c) (\<lambda>i. v $h Bij_Nat.from_nat i)"
node\<^sub>i_root_order: fixes ts :: "('a btree \<times> 'a) list"   and k :: "nat"   and t :: "'a btree" assumes "0 < length ts"   and "length ts \<le> 4 * k + 1"   and "\<forall>x\<in>set (subtrees ts). order k x"   and "order k t" shows "root_order_up\<^sub>i k (node\<^sub>i k ts t)"
OFCLASS_elementary_divisor_ring_imp_class: assumes "OFCLASS('a, elementary_divisor_ring_class)" shows "class.elementary_divisor_ring TYPE('a)"
funof_Pair_in: fixes r :: "('b \<times> 'c) set"   and x :: "'b" assumes "single_valued r"   and "x \<in> Domain r" shows "(x, funof r x) \<in> r"
simps_3: fixes \<mu> :: "'a ltln"   and \<mu>' :: "'a ltln" shows "pure_eventual (\<mu> and\<^sub>n \<mu>') = (pure_eventual \<mu> \<and> pure_eventual \<mu>')"
Block_def: shows "Class = (\<lambda>a\<in>G. THE A. a \<in> A \<and> A \<in> Partition)"
data_refinement2: fixes R :: "'e \<Rightarrow> 'f"   and p :: "'e"   and S :: "'g \<Rightarrow> 'e"   and q :: "'g"   and R' :: "'g \<Rightarrow> 'h"   and S' :: "'h \<Rightarrow> 'f" assumes "mono R"   and "\<Turnstile> p{| S |}q"   and "DataRefinement ({. p .} \<circ> S) R R' S'" shows "\<Turnstile> R p{| S' |}R' q"
assms_2: fixes p0 :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "p0 \<in> set ps" shows "j \<le> component_of_term (sig_of_pair p0)"
intro: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k" assumes "comm_monoid f z" shows "comm_monoid_mset f z"
min_top: fixes x :: "'a upred" shows "ord.min (\<le>) true x = x"
LeftDerives1_def: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" shows "LeftDerives1 u i r v = (leftmost i u \<and> Derives1 u i r v)"
simps_2: fixes x :: "'h" shows "sumbot (Inr x) = sum_bot.Right x"
absolutely_integrable_on_subinterval: fixes S :: "real set"   and f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real" assumes "f absolutely_integrable_on S"   and "{a..b} \<subseteq> S" shows "f absolutely_integrable_on {a..b}"
distinct_21: shows "IndexOfOp \<noteq> string_binop.LessEqOp"
small_step_loc_compC: fixes c :: "('g, 'h, 'i, 'j) com"   and cs :: "('g, 'h, 'i, 'j) com list"   and ls :: "'h option \<times> 'j"   and \<alpha> :: "('g, 'i) seq_label"   and ls' :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" assumes "basic_com c"   and "(c # cs, ls) \<rightarrow>\<^bsub>\<alpha>\<^esub> ls'" shows "loc_compC c cs (snd ls) = atCs (cPGM ls')"
carrier_update_5: fixes S :: "'a set" shows "(\<otimes>\<^bsub>R\<lparr>carrier := S\<rparr>\<^esub>) = (\<otimes>)"
trivial_subalg_cond_expect_eq: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "prob_space M"   and "subalgebra M N"   and "sets N = {{}, space M}"   and "integrable M f" shows "\<forall>x\<in>space M. real_cond_exp M N f x = integral\<^sup>L M f"
wf_no_infinite_down_chainE: fixes r :: "('a \<times> 'a) set"   and f :: "nat \<Rightarrow> 'a" assumes "wf r" obtains k :: "nat" where "(f (Suc k), f k) \<notin> r"
pt_delta: fixes x :: "'a" assumes "x \<in> S - U" shows "delta0 \<le> p x"
simps_3: fixes x :: "(int \<times> int) set set"   and y :: "(int \<times> int) set set"   and p :: "int" shows "CRotate (AllowPortFromTo x y p) = C (AllowPortFromTo x y p)"
blinfun_eqI: fixes x :: "'b \<Rightarrow>\<^sub>L 'c"   and y :: "'b \<Rightarrow>\<^sub>L 'c" assumes "\<And>i. x *\<^sub>v i = y *\<^sub>v i" shows "x = y"
set_integrable_def: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'c" shows "set_integrable M A f \<equiv> integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
frechet_vec_nth: fixes x\<^sub>0 :: "real"   and f :: "real \<Rightarrow> ('c, 'd) vec"   and f' :: "real \<Rightarrow> ('c, 'd) vec"   and t :: "real"   and T :: "real set"   and i :: "'d" assumes "((\<lambda>x. (f x - f x\<^sub>0 - (x - x\<^sub>0) *\<^sub>R f' t) /\<^sub>R (\<parallel>x - x\<^sub>0\<parallel>)) \<longlongrightarrow> 0) (at t within T)" shows "((\<lambda>x. (f x $ i - f x\<^sub>0 $ i - (x - x\<^sub>0) *\<^sub>R f' t $ i) /\<^sub>R (\<parallel>x - x\<^sub>0\<parallel>)) \<longlongrightarrow> (0::'c)) (at t within T)"
minus_divide_right: 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 a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "uminus (divide a b) = divide a (uminus b)"
reachable_target_elim: fixes q :: "'c"   and A :: "('d, 'c) nba"   and p :: "'c" assumes "q \<in> reachable A p" obtains r :: "('d \<times> 'c) list" where "path A r p" and "q = target r p"
subresultant_prs: fixes div_exp :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and Gk :: "'a poly"   and hk :: "'a"   and i :: "nat" assumes "\<And>x y n. ff x ^ n / ff y ^ (n - 1) \<in> range ff \<Longrightarrow> ff (div_exp x y n) = ff x ^ n / ff y ^ (n - 1)"   and "subresultant_prs div_exp G1 G2 = (Gk, hk)" shows "F k = ffp Gk \<and> h k = ff hk \<and> (i \<noteq> 0 \<longrightarrow> F i \<in> range ffp) \<and> (3 \<le> i \<longrightarrow> i \<le> Suc k \<longrightarrow> \<beta> i \<in> range ff)"
delete_edge_def: fixes v :: "'c"   and e :: "'d"   and v' :: "'c"   and g :: "('c, 'd, 'e) graph_scheme" shows "delete_edge v e v' g \<equiv> \<lparr>nodes = nodes g, edges = edges g - {(v, e, v')}\<rparr>"
C0: shows "f (nat (padic_val p f) + 1) \<noteq> \<zero>\<^bsub>residue_ring (p ^ (nat (padic_val p f) + 1))\<^esub>"
waux2: fixes xs :: "('b, 'c) Combinators list" assumes "DenyAll \<notin> set xs" shows "wellformed_policy1 xs"
Linear_order_in_diff_Id: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "Linear_order r"   and "a \<in> Field r"   and "b \<in> Field r" shows "((a, b) \<in> r) = ((b, a) \<notin> r - Id)"
continuous_on_open_gen: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and T :: "'c set" assumes "f ` S \<subseteq> T" shows "continuous_on S f = (\<forall>U. openin (top_of_set T) U \<longrightarrow> openin (top_of_set S) (S \<inter> f -` U))"
in_set_insert: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "List.insert x xs = xs"
simps_1: fixes x :: "'g" shows "sumbot (Inl x) = sum_bot.Left x"
cis_conv_exp: fixes b :: "real" shows "cis b = exp (\<i> * cor b)"
coboundedI2: fixes b :: "'d upred"   and c :: "'d upred"   and a :: "'d upred" assumes "c \<sqsubseteq> b" shows "c \<sqsubseteq> (a \<and> b)"
Insert_Delete_concurrent: fixes i :: "'a"   and e :: "'a \<times> 'b \<times> bool"   and n :: "'a option"   and i' :: "'a"   and n' :: "'a"   and j :: "nat" assumes "{Deliver (i, Insert e n), Deliver (i', Delete n')} \<subseteq> set (history j)"   and "hb.concurrent (i, Insert e n) (i', Delete n')" shows "n' \<noteq> fst e"
select_convs_2: fixes worlds :: "'f set"   and relations :: "'e \<Rightarrow> ('f \<times> 'f) set"   and valuation :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and more :: "'h" shows "relations \<lparr>worlds = worlds, relations = relations, valuation = valuation, \<dots> = more\<rparr> = relations"
simps_11: fixes network' :: "('g \<times> nat \<times> 'g \<times> 'h) set \<Rightarrow> ('g \<times> nat \<times> 'g \<times> 'h) set"   and round :: "'g \<Rightarrow> nat"   and state :: "'g \<Rightarrow> 'f"   and sent :: "'g \<Rightarrow> 'g set"   and rcvd :: "'g \<Rightarrow> 'g \<Rightarrow> 'h option"   and network :: "('g \<times> nat \<times> 'g \<times> 'h) set"   and more :: "'i" shows "network_update network' \<lparr>round = round, state = state, sent = sent, rcvd = rcvd, network = network, \<dots> = more\<rparr> = \<lparr>round = round, state = state, sent = sent, rcvd = rcvd, network = network' network, \<dots> = more\<rparr>"
Disjs_def: shows "Disjs = {z. \<exists>A0 A1 Gamma. z = (FConj Neg A0 A1 # Gamma, {A0 # A1 # Gamma})}"
IH: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" assumes "as \<in> valid_plans PROB" shows "fmdom' (exec_plan s as) \<subseteq> fmdom' s \<union> prob_dom PROB"
whisker_right: fixes f :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr"   and \<mu> :: "('a E.term, 'a) arr" assumes "ide f"   and "seq \<nu> \<mu>" shows "\<nu> \<cdot> \<mu> \<star> f = (\<nu> \<star> f) \<cdot> (\<mu> \<star> f)"
GC_reduction_step: fixes Dl :: "'a literal multiset \<times> label"   and Cl :: "'a literal multiset \<times> label"   and N :: "('a literal multiset \<times> label) set" assumes "snd Dl \<noteq> Old"   and "fst Dl \<subset># fst Cl" shows "N \<union> {Cl} \<leadsto>GC N \<union> {Dl}"
R_def: shows "R = {(V', V''). (V', V'') \<in> R' \<and> set (PPV V') \<subseteq> set (PPV V) \<and> set (PPV V'') \<subseteq> set (PPV V)}"
left_inverse: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and inverse :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "group f z inverse" shows "f (inverse a) a = z"
finiteE: fixes G :: "'a graph" assumes "wf_graph G" shows "finite (edges G)"
shift_snth_ge: fixes xs :: "'a list"   and p :: "nat"   and s :: "'a stream" assumes "length xs \<le> p" shows "(xs @- s) !! p = s !! (p - length xs)"
map_tpOfV_getVars: fixes \<sigma>l :: "'tp list" shows "map tpOfV (getVars \<sigma>l) = \<sigma>l"
valid_cfg_def: shows "valid_cfg = \<Union> (cfg_on ` S)"
sorted_list_of_set_sort_remdups: fixes xs :: "'a list" shows "sorted_list_of_set (set xs) = sort (remdups xs)"
to_int: fixes y :: "int"   and z :: "'a mod_ring" assumes "mod_ring_rel y z" shows "arith_ops_record.to_int (finite_field_ops_int p) y = to_int_mod_ring z"
prim_vc_1: fixes minarc :: "'b \<Rightarrow> 'b"   and sum :: "'b \<Rightarrow> 'b"   and plus :: "'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 bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and g :: "'b"   and r :: "'b" assumes "class.m_kleene_algebra minarc sum plus inf less_eq less sup bot top uminus one times conv star"   and "m_kleene_algebra.prim_precondition less_eq top uminus one times conv g r" shows "m_kleene_algebra.prim_spanning_invariant inf less_eq top uminus one times conv star bot r g r"
iTILL_plus: fixes x :: "nat"   and n :: "nat"   and k :: "nat" assumes "x \<in> [\<dots>n]"   and "k \<le> n - x" shows "x + k \<in> [\<dots>n]"
gaussian_density_def: shows "gaussian_density \<equiv> \<lambda>x xa. ennreal ((case x of (m, s) \<Rightarrow> \<lambda>x. if 0 < s then exp (- (x - m)\<^sup>2 / (2 * s\<^sup>2)) / sqrt (2 * pi * s\<^sup>2) else 0) xa)"
max_depth_Suc: fixes \<phi> :: "fmla"   and n :: "nat" shows "(max_depth \<phi> = Suc n) = ((\<exists>\<psi>. \<phi> = Neg \<psi> \<and> max_depth \<psi> = n) \<or> (\<exists>\<psi>1 \<psi>2. \<phi> = Conj \<psi>1 \<psi>2 \<and> max (max_depth \<psi>1) (max_depth \<psi>2) = n))"
collinear_3: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "NO_MATCH (0::'a) x" shows "collinear {x, y, z} = collinear {0::'b, x - y, z - y}"
transfer: shows "rel_fun cr_uint16 (rel_fun cr_uint16 cr_uint16) (+) (+)"
cospan_\<mu>\<nu>: shows "C.cospan \<nu>.leg0 \<pi>.leg1"
this_wall_betw_basechambers: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>ThinChamberComplex.foldpairs X. {{f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X}}) \<and> separated_by H C0 (g ` C0)) = {f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X}"
R: shows "SN_on (S\<^sup>* O R) T"
remap_polys_l_remap_polys: fixes A :: "(nat, (char list list \<times> int) list) fmap"   and B :: "(nat, int mpoly) fmap"   and spec :: "(char list list \<times> int) list"   and spec' :: "int mpoly"   and \<V> :: "char list set"   and \<V>' :: "nat set" assumes "(A, B) \<in> \<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"   and "(spec, spec') \<in> sorted_poly_rel O mset_poly_rel"   and "(\<V>, \<V>') \<in> \<langle>var_rel\<rangle>set_rel" shows "remap_polys_l spec \<V> A \<le> \<Down> (code_status_status_rel \<times>\<^sub>r \<langle>var_rel\<rangle>set_rel \<times>\<^sub>r \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel) (remap_polys spec' \<V>' B)"
hyps: fixes ys :: "'a list" assumes "ys \<in> listset xs"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> x\<rbrakk> \<Longrightarrow> f y \<in> x" shows "map f ys \<in> listset xs"
postSetpostList: fixes xs :: "('e \<times> ('f \<Rightarrow> 'g) \<times> 'h) list"   and l :: "'e \<Rightarrow> 'g"   and s :: "'f" assumes "postList xs l s" shows "postSet (set xs) l s"
JKi: fixes x :: "'a \<Rightarrow> 'd"   and i :: "'a" assumes "x \<in> carrier (DirProds (f \<circ> Gs) I)" shows "(if i \<in> I then J i else (\<lambda>_. undefined)) ((if i \<in> I then K i else (\<lambda>_. undefined)) (x i)) = x i"
pr_strict_irrefl: fixes fn :: "'f \<times> nat" shows "\<not> pr_strict' fn fn"
le_powr_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(x \<le> b powr y) = (log b x \<le> y)"
ex_le_\<tau>: fixes i :: "nat"   and x :: "nat"   and s :: "'a trace" shows "\<exists>j\<ge>i. x \<le> \<tau> s j"
induced_subgraph_def: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" shows "induced_subgraph H G \<equiv> subgraph H G \<and> arcs H = {e \<in> arcs G. tail G e \<in> verts H \<and> head G e \<in> verts H}"
main: shows "basis_reduction_main_cost True 0 state1 0 = (state2, c2)"
permutation_mat_def: fixes n :: "nat"   and p :: "nat \<Rightarrow> nat" shows "permutation_mat n p = Matrix.mat n n (\<lambda>(i, j). if i = p j then 1::'a else (0::'a))"
prodrg_sameTr6: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme"   and B :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)"   and "\<forall>k\<in>I. A k = B k" shows "prod_one I A = prod_one I B"
h_plus_not_infty: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and i :: "nat"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g" shows "flow_network.h_plus \<Delta> g i e \<noteq> \<top>"
HR_is_R: shows "(\<mu> \<rho>''. (\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>''))(x := Abs_cfun (ESem_syn e)\<cdot>\<rho>')) ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>'') = (\<mu> \<rho>'. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>domA \<Gamma>\<^esub> (\<Lambda> \<rho>'. \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)\<cdot>\<rho>''))(x := Abs_cfun (ESem_syn e)\<cdot>\<rho>'))"
alpha_qOp_iff: fixes X :: "('c, 'd, 'e, 'f, 'g) qTerm"   and delta' :: "'g"   and inp' :: "'c \<Rightarrow> ('c, 'd, 'e, 'f, 'g) qTerm option"   and binp' :: "'d \<Rightarrow> ('c, 'd, 'e, 'f, 'g) qAbs option" shows "(X #= qOp delta' inp' binp') = (\<exists>inp binp. X = qOp delta' inp binp \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
fds_perzeta__left_0: shows "fds_perzeta 0 = fds_zeta"
P: shows "\<down>\<^sub>t (c1;; c2, s) = P"
sums_unique: fixes f :: "nat \<Rightarrow> 'c"   and s :: "'c" assumes "f sums s" shows "s = suminf f"
diameter_bounded_2: fixes S :: "'b set" assumes "bounded S" shows "\<forall>d>0. d < diameter S \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>S. d < dist x y)"
min_range_mono: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and l :: "float"   and u :: "float"   and p :: "nat"   and F :: "floatarith \<Rightarrow> floatarith"   and DF :: "floatarith \<Rightarrow> floatarith"   and Y :: "(real \<times> real pdevs) \<times> real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "x \<in> aform_err e X"   and "real_of_float l \<le> x"   and "x \<le> real_of_float u"   and "min_range_mono p F DF l u X = Some Y"   and "e \<in> UNIV \<rightarrow> {- 1..1}"   and "\<And>x. real_of_float x \<in> {real_of_float l..real_of_float u} \<Longrightarrow> interpret_floatarith (F (Num x)) [] = f (real_of_float x)"   and "\<And>x. real_of_float x \<in> {real_of_float l..real_of_float u} \<Longrightarrow> interpret_floatarith (DF (Num x)) [] = f' (real_of_float x)"   and "\<And>x. x \<in> {real_of_float l..real_of_float u} \<Longrightarrow> (f has_real_derivative f' x) (at x)"   and "\<And>x. x \<in> {real_of_float l..real_of_float u} \<Longrightarrow> min (f' (real_of_float l)) (f' (real_of_float u)) \<le> f' x" shows "f x \<in> aform_err e Y"
AA24: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<diamond>\<langle>\<langle>P\<rangle>_f\<rangle>_f = \<diamond>\<langle>P\<rangle>_f"
vconn_implies_hpar_2: fixes \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" shows "trg\<^sub>C \<mu> = trg\<^sub>C f"
finite_iff: fixes x :: "'a" shows "relation_algebra_rtc_class.finite x = (backward_finite x \<and> forward_finite x)"
lspasl_empl: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Gamma :: "bool"   and h :: "'a"   and Delta :: "bool" assumes "class.heap_sep_algebra plus zero sep_disj"   and "Gamma \<and> h = zero \<longrightarrow> Delta" shows "Gamma \<and> sep_algebra.sep_empty zero h \<longrightarrow> Delta"
"1_1": fixes z' :: "'a"   and buf' :: "event_data option list set list list"   and nts' :: "nat list"   and i :: "nat" assumes "\<not> ([] \<in> set buf \<or> nts = [])"   and "mbufnt_take f (f (map hd buf) (hd nts) z) (map tl buf) (tl nts) = (z', buf', nts')"   and "wf_mbufn i js Ps (map tl buf)"   and "list_all2 R [i..<j] (tl nts)"   and "\<And>k. k \<in> set js \<Longrightarrow> k \<le> j"   and "U i (f (map hd buf) (hd nts) z)"   and "\<And>k xs t z. \<lbrakk>i \<le> k; Suc k \<le> Mini j js; list_all2 (\<lambda>P. P k) Ps xs; R k t; U k z\<rbrakk> \<Longrightarrow> U (Suc k) (f xs t z)" shows "U (Mini (i + length (tl nts)) js) z'"