hcju/isabelleps huggingface dataset:
elims: fixes x :: "int \<Rightarrow> int \<Rightarrow> int"   and xa :: "value option"   and xb :: "value option"   and y :: "value option" assumes "maybe_arith_int x xa xb = y" obtains f :: "int \<Rightarrow> int \<Rightarrow> int" and x :: "int" and ya :: "int" where "xa__ = f" and "xa = Some (Num x)" and "xb = Some (Num ya)" and "y = Some (Num (f x ya))"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> int" and uw_ :: "value option" where "xa__ = uu___" and "xa = None" and "xb = uw___" and "y = None"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> int" and va :: "String.literal" and uw_ :: "value option" where "xa__ = uua___" and "xa = Some (Str va__)" and "xb = uw_" and "y = None"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> int" and uv_ :: "value option" where "xa__ = uub___" and "xa = uv___" and "xb = None" and "y = None"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> int" and uv_ :: "value option" and va :: "String.literal" where "xa__ = uu_" and "xa = uv_" and "xb = Some (Str va)" and "y = None"
\<beta>: fixes i :: "'a set" assumes "i \<in> \<I>" shows "ordertype (B ` J i) (lenlex less_than) = \<beta>"
igSubstCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c, 'd, 'e) model_scheme" shows "igSubstCls MOD \<equiv> igSubstIGVar1 MOD \<and> igSubstIGVar2 MOD \<and> igSubstIGAbs MOD \<and> igSubstIGOp MOD"
nsets_doubleton_2_eq: fixes x :: "'a"   and y :: "'a" shows "[{x, y}]\<^bsup>2\<^esup> = (if x = y then {} else {{x, y}})"
trans: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "b \<preceq> a"   and "c \<preceq> b" shows "c \<preceq> a"
rep_eq: fixes x :: "'b dual_ord"   and xa :: "'b dual_ord" shows "(x < xa) = (of_dual_ord xa < of_dual_ord x)"
InterpTy: fixes I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and A :: "'a" assumes "Sig iS\<^bsub>I\<^esub> \<triangleright> \<turnstile> A Type" shows "L\<lbrakk>\<turnstile> A Type\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IObj Ty\<lbrakk>A\<rbrakk>\<^bsub>I\<^esub>"
blocking_on_CD_on': fixes ds :: "'d set"   and X :: "'x set"   and h :: "'h"   and X'' :: "'x set"   and x :: "'x" assumes "blocking_on ds X h X''"   and "x \<in> X''" shows "x \<in> CD_on ds (X \<union> X'')"
noElementsPrecedesFirstElement: fixes a :: "'b"   and b :: "'b"   and list :: "'b list" assumes "a \<noteq> b" shows "\<not> precedes a b (b # list)"
inorder_adjust: fixes t :: "('b \<times> nat) tree" assumes "t \<noteq> \<langle>\<rangle>"   and "pre_adjust t" shows "Tree2.inorder (adjust t) = Tree2.inorder t"
fps_deriv_sub: fixes f :: "'a fps"   and g :: "'a fps" shows "fps_deriv (f - g) = fps_deriv f - fps_deriv g"
in_kD_after_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and nip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "kD (update rt nip (dsn, dsk, flag, hops, nhip, pre)) = insert nip (kD rt)"
fg: fixes w :: "complex" assumes "w \<in> S" shows "f w = g w"
complements_overlap: fixes x :: "i"   and y :: "i" assumes "x \<oplus> y \<noteq> u" shows "O (\<midarrow> x) (\<midarrow> y)"
simps_2: fixes k :: "nat" shows "ord_of (Suc k) = succ (ord_of k)"
select_convs_1: fixes asig :: "'c signature"   and start :: "'d set"   and trans :: "('d \<times> 'c \<times> 'd) set"   and more :: "'e" shows "ioa.asig \<lparr>ioa.asig = asig, start = start, trans = trans, \<dots> = more\<rparr> = asig"
converse_composition_of_tangle_diagrams: fixes x :: "wall"   and y :: "wall" assumes "is_tangle_diagram (x \<circ> y)" shows "domain_wall y = codomain_wall x"
simps_41: fixes x61 :: "msg"   and x62 :: "msg"   and x4 :: "key" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Key x4"
tensor_mat_mult: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat"   and m3 :: "'a mat"   and m4 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m3 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)"   and "m4 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)" shows "tensor_mat dims vars (m1 * m2) (m3 * m4) = tensor_mat dims vars m1 m3 * tensor_mat dims vars m2 m4"
p_grid: shows "p \<in> grid (child b dir d) {d}"
lm02: fixes a :: "(integer \<times> 'c set) set"   and N :: "integer set"   and \<Omega> :: "'c set" assumes "a \<in> soldAllocations'' N \<Omega>" shows "a \<in> allocationsUniverse \<and> Domain a \<subseteq> N - {seller} \<and> \<Union> (Range a) \<subseteq> \<Omega>"
assms_3: shows "L \<mu> = (\<iota> \<star> f) \<cdot>\<^sub>S local.inv \<a>[b, b, f]"
aux115: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" assumes "x \<noteq> DenyAll" shows "removeShadowRules1_alternative (xs @ [x]) = removeShadowRules1_alternative xs @ [x]"
induct: fixes P :: "('b, 'c) Combinators list \<Rightarrow> bool"   and a0 :: "('b, 'c) Combinators list" assumes "P []"   and "\<And>xs. P xs \<Longrightarrow> P (DenyAll # xs)"   and "\<And>x y xs. P xs \<Longrightarrow> P (DenyAllFromTo x y # xs)"   and "\<And>x y p xs. P xs \<Longrightarrow> P (AllowPortFromTo x y p # xs)"   and "\<And>x y xs. \<lbrakk>P [x]; P [y]; P xs\<rbrakk> \<Longrightarrow> P (x \<oplus> y # xs)" shows "P a0"
assert_def: fixes p :: "'b Assertion" shows "{\<cdot> p } = Rep_Assertion p"
bin_sc_eq_1: fixes n :: "nat" shows "bin_sc n False = unset_bit n"
induct: fixes P :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'b list" assumes "\<And>Pa xs. P Pa [] xs"   and "\<And>Pa v va. P Pa (v # va) []"   and "\<And>Pa x xs y ys. P Pa xs ys \<Longrightarrow> P Pa (x # xs) (y # ys)" shows "P a0 a1 a2"
bvars\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('e, 'd) stateful_strand_step"   and A :: "('e, 'd) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t (a # A) = set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a) \<union> bvars\<^sub>s\<^sub>s\<^sub>t A"
preimages_le_k: shows "\<forall>x\<in>f ` {(i, x). i \<in> I \<and> x \<in> A i}. card (f -` {x} \<inter> {(i, x). i \<in> I \<and> x \<in> A i}) \<le> k"
maintainedD2: fixes A :: "('e, 'f) labeled_graph"   and B :: "('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" assumes "maintained (A, B) G"   and "graph_homomorphism A G f" obtains g :: "('f \<times> 'g) set" where "graph_homomorphism B G g" and "f \<subseteq> g"
simulation: fixes i1 :: "'index"   and s1 :: "'state1"   and s2 :: "'state2"   and s2' :: "'state2" assumes "match i1 s1 s2"   and "step2 s2 s2'" shows "(\<exists>i2 s1'. step1\<^sup>+\<^sup>+ s1 s1' \<and> match i2 s1' s2') \<or> (\<exists>i2. match i2 s1 s2' \<and> i2 \<sqsubset> i1)"
Bit_B0: fixes k :: "int" shows "k BIT False = k + k"
is_binqueue_add_Some: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list"   and t :: "('c, 'd) bintree" assumes "is_binqueue l xs"   and "is_bintree_list l (children t)"   and "is_heap_list (priority t) (children t)" shows "is_binqueue l (add (Some t) xs)"
simps: fixes put :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'f"   and m :: "'e" shows "put_option put s (OptionT m) = OptionT (put s m)"
simps_24: shows "OUTfromCh data24 = {data20}"
use_iso2: fixes A :: "('a, 'b) gorder_scheme"   and B :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a" assumes "isotone A B f"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<sqsubseteq>\<^bsub>A\<^esub> y" shows "f x \<sqsubseteq>\<^bsub>B\<^esub> f y"
release_append_Prog\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and p\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b"   and mis :: "instr list"   and S :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" shows "release (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis])) S \<R> = release (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S \<R>"
P: shows "pre @ [Broadcast (r, Store e2 i r)] prefix of k"
simps_16: fixes vb :: "'a"   and vc :: "value list"   and v :: "'a gexp"   and va :: "'a gexp" shows "local.less_gexp_aux (In vb vc) (Nor v va) = True"
rel_converter_eq: shows "rel_converter (=) (=) (=) (=) = (=)"
aa_1: shows "circline_type_cmat H = - 1 \<and> cos_angle_cmat (of_circline_cmat H) unit_circle_cmat = 0"
hd_lt: shows "head_\<omega> (m1 * n2 + m2 * n1) < head_\<omega> (m1 * n1 + m2 * n2)"
rep_iso: fixes x :: "'e\<cdot>('f, 'g) errorT" shows "errorT_abs\<cdot>(errorT_rep\<cdot>x) = x"
simps_1: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.matchs unapp unconst [] [] = Some fmempty"
rrep_sqn_greater_dsn: shows "paodv i \<TTurnstile> (recvmsg rreq_rrep_sn \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRrep-:1..PRrep-:4} \<longrightarrow> 1 \<le> sqn (rt \<xi>) (dip \<xi>))"
intros_2: fixes G :: "'a monoid"   and K :: "'a monoid"   and l :: "'a monoid list"   and g :: "'a \<Rightarrow> 'a"   and q :: "('a \<Rightarrow> 'a) list"   and H :: "'a monoid"   and h :: "'a \<Rightarrow> 'a" assumes "exact_seq (G # K # l, g # q)"   and "Group.group H"   and "h \<in> hom G H"   and "kernel G H h = g ` carrier K" shows "exact_seq (H # G # K # l, h # g # q)"
simps_10: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com" shows "IF x41 THEN x42 ELSE x43 \<noteq> SKIP"
wf_pair_eqs_ineqs_map': fixes X :: "'var set"   and A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Di :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and i :: "'lbl strand_label"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term" assumes "wf\<^sub>l\<^sub>s\<^sub>t X A"   and "Di \<in> set (subseqs (dbproj i D))"   and "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<subseteq> X" shows "wf\<^sub>l\<^sub>s\<^sub>t X (map (\<lambda>d. (i, \<langle>check: pair (t, s) \<doteq> pair (snd d)\<rangle>\<^sub>s\<^sub>t)) Di @ map (\<lambda>d. (i, \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t)) (filter (\<lambda>d. d \<notin> set Di) (dbproj i D)) @ A)"
star_n_export_right: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> !y \<le> !y \<cdot> x" shows "x\<^sup>\<star> \<cdot> !y = (!y \<cdot> x)\<^sup>\<star> \<cdot> !y"
sinks_elem: fixes v :: "'c"   and I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list" assumes "v \<in> sinks I D u xs" shows "\<exists>x\<in>set xs. v = D x"
at_top_le_at_infinity: shows "at_top \<le> at_infinity"
simps_1: fixes P :: "'f \<Rightarrow> bool" shows "dropWhile P [] = []"
keys_eq_empty: fixes c :: "'c \<Rightarrow>\<^sub>0 'd" shows "(Poly_Mapping.keys c = {}) = (c = 0)"
lt_spoly_less_lcs: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "punit.spoly p q \<noteq> 0" shows "punit.lt (punit.spoly p q) \<prec> fst (lcs (punit.lp p) (punit.lp q), local.punit.component_of_term (punit.lt p))"
SETdom: fixes A :: "ZF"   and B :: "ZF"   and f :: "ZF \<Rightarrow> ZF" assumes "isZFfun (ZFfun A B f)" shows "dom\<^bsub>SET\<^esub> ZFfun A B f = A"
Lim: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and net :: "'a filter" shows "(f \<longlongrightarrow> l) net = (net = bot \<or> (\<forall>e>0. \<forall>\<^sub>F x in net. dist (f x) l < e))"
le_multiset_Max_in_imp_Max: fixes D :: "'b literal multiset"   and A :: "'b"   and C :: "'b literal multiset" assumes "Max (atms_of D) = A"   and "C \<le> D"   and "A \<in> atms_of C" shows "Max (atms_of C) = A"
removeNones_vol_edges_not_f12: fixes x :: "nat"   and y :: "nat" assumes "x \<in> set (removeNones vol)" shows "(x, y) \<notin> \<E> f12"
Y_def: shows "Y \<equiv> \<lambda>i \<omega>. if i \<in> I then X i \<omega> else 0"
3: fixes z :: "'v"   and c :: "'f" assumes "\<delta> z = Fun c []" shows "Fun c [] \<sqsubseteq> s \<or> Fun c [] \<sqsubseteq> t"
par: shows "par (tuple (local.comp (pr1 a b) h) (local.comp (pr0 a b) h)) h"
fsmult_smult_domain_comm: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and r :: "('b, 'a) aezfun"   and v :: "'c"   and a :: "'b" assumes "FGModuleHom G smult V smult' T"   and "r \<in> aezfun_setspan G"   and "v \<in> V" shows "aezfun_scalar_mult.fsmult smult a (smult r v) = smult r (aezfun_scalar_mult.fsmult smult a v)"
simps_2: fixes vref :: "ref" shows "bdt (Node Tip vref Tip) = (\<lambda>bdtvar. if bdtvar vref = 0 then Some Zero else if bdtvar vref = 1 then Some One else None)"
sb1: shows "(*) (C * complex_of_real r) ` (\<lambda>z. f (a + complex_of_real r * z) / (C * complex_of_real r)) ` ball 0 1 \<subseteq> f ` ball a r"
qtable_bin_join: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q1 :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and B :: "nat set"   and Q2 :: "'a option list \<Rightarrow> bool"   and Y :: "'a option list set"   and b :: "bool"   and C :: "nat set"   and Q :: "'a option list \<Rightarrow> bool" assumes "qtable n A P Q1 X"   and "qtable n B P Q2 Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A"   and "C = A \<union> B"   and "\<And>x. \<lbrakk>wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> P (restrict A x) \<and> P (restrict B x)"   and "\<And>x. \<lbrakk>b; wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> Q x = (Q1 (restrict A x) \<and> Q2 (restrict B x))"   and "\<And>x. \<lbrakk>\<not> b; wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> Q x = (Q1 (restrict A x) \<and> \<not> Q2 (restrict B x))" shows "qtable n C P Q (bin_join n A X b B Y)"
ceq_simple_tycon_def: shows "CEQ(simple_tycon) = Some (=)"
unfold: shows "eval_body = (\<Lambda> r v. Fixrec.run\<cdot>(match_Val\<cdot>v\<cdot>(\<Lambda> n. Fixrec.succeed\<cdot>(Just\<cdot>n)) +++ match_Add\<cdot>v\<cdot>(\<Lambda> x y. Fixrec.succeed\<cdot>(mliftM2 (\<Lambda> a b. a + b)\<cdot>(r\<cdot>x)\<cdot>(r\<cdot>y))) +++ match_Throw\<cdot>v\<cdot>(Fixrec.succeed\<cdot>mfail) +++ match_Catch\<cdot>v\<cdot>(\<Lambda> x y. Fixrec.succeed\<cdot>(mcatch\<cdot>(r\<cdot>x)\<cdot>(r\<cdot>y)))))"
rep_abs_rsp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and x1 :: "'c"   and x2 :: "'c" assumes "Quotient3 R Abs Rep"   and "R x1 x2" shows "R x1 (Rep (Abs x2))"
semiring_normalization_rules_36: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "power.power one times x (2 * n) = times (power.power one times x n) (power.power one times x n)"
assms_2: shows "\<forall>i. P (S i) (S (Suc i))"
assms_1: shows "fps_nth f 0 \<noteq> (0::'a)"
simps_1: fixes uu :: "'c \<Rightarrow> 'b" shows "NegPos_map uu [] = []"
POS_sint: shows "sint POS_INF = 2 ^ 31 - 1"
bij: shows "bij_betw (\<lambda>i. linorder_rank R (set xs) (xs ! i)) {..<n} {..<n}"
closure_of_Times: fixes X :: "'d topology"   and Y :: "'e topology"   and S :: "'d set"   and T :: "'e set" shows "prod_topology X Y closure_of S \<times> T = (X closure_of S) \<times> Y closure_of T"
Parallel_AllNone_lemma: fixes Ss :: "('b ann_com option \<times> 'b set) list"   and s :: "'b"   and Ts :: "('b ann_com option \<times> 'b set) list"   and t :: "'b"   and n :: "nat" assumes "All_None Ss"   and "(Parallel Ss, s) -Pn\<rightarrow> (Parallel Ts, t)" shows "Ts = Ss \<and> n = 0 \<and> s = t"
simps_1: fixes x :: "Preliminaries.var" shows "Ik.tpOf (Var x) = TE.tpOfV x"
mat_mult_right_mono: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat"   and C :: "'b mat" assumes "A \<ge>\<^sub>m 0\<^sub>m n n"   and "B \<ge>\<^sub>m C"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n" shows "A * B \<ge>\<^sub>m A * C"
until_introL: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<phi> \<pi>l"   and "until \<phi> \<psi> (map stl \<pi>l)" shows "until \<phi> \<psi> \<pi>l"
poly_to_row_map_key_PP: fixes xs :: "('c, 'd) pp list"   and p :: "('c, 'd) pp \<Rightarrow>\<^sub>0 'b" shows "poly_to_row (map pp.mapping_of xs) (Poly_Mapping.map_key PP p) = poly_to_row xs p"
subst_fv_aux: fixes v :: "exp"   and x :: "nat"   and e :: "exp" assumes "FV v = {}" shows "FV (subst x v e) \<subseteq> FV e - {x}"
rsd_winners_empty: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and alts :: "'a set" shows "rsd_winners R alts [] = alts"
map\<^sub>0_def: fixes a :: "'c" shows "H.map\<^sub>0 a \<equiv> src\<^sub>D (H a)"
allUse_implies_allDef: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and m :: "'node" assumes "v \<in> braun_ssa.allUses g m"   and "m \<in> set (\<alpha>n g)" shows "v \<in> braun_ssa.allDefs g (defNode v)"
stable_pair_on_individually_rational: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "individually_rational_on ds (match XD_XH)"
neg_sup_eq_inf: fixes a :: "'a"   and b :: "'a" shows "- sup a b = inf (- a) (- b)"
ground_imp_subst_iden: fixes s :: "('d, 'c) tm"   and \<rho> :: "'c \<Rightarrow> ('d, 'c) tm" assumes "ground s" shows "subst \<rho> s = s"
add_point_def: fixes p :: "'a" shows "sys2.add_point p \<equiv> insert p \<V>'"
exhaust: fixes y :: "'g list" obtains "y = []"   | x21 :: "'g" and x22 :: "'g list" where "y = x21 # x22"
simps: fixes a1 :: "'s"   and a2 :: "'s stream" shows "K.enabled a1 a2 = (\<exists>\<omega> s. a1 = s \<and> a2 = \<omega> \<and> K.enabled (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
Main_no_return_source: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>Main\<^esub>f'" shows "False"
eq': shows "ltake (enat (Suc r')) E [\<approx>] ltake (enat (Suc r')) E'"
cases: fixes a1 :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list"   and a2 :: "'a"   and a3 :: "String.literal"   and a4 :: "'a list" assumes "static_operation_defined' a1 a2 a3 a4" obtains operations :: "(String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option) list" and \<tau> :: "'a" and name :: "String.literal" and \<pi> :: "'a list" and oper :: "String.literal \<times> 'a \<times> (String.literal \<times> 'a \<times> param_dir) list \<times> 'a \<times> bool \<times> 'b option" where "a1 = operations" and "a2 = \<tau>" and "a3 = name" and "a4 = \<pi>" and "unique_static_operation operations \<tau> name \<pi> oper"
index_rev: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "index (rev xs) x = length xs - index xs x - 1"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>c e. P (Eq (CNP 0 c e))"   and "\<And>c e. P (NEq (CNP 0 c e))"   and "\<And>c e. P (Lt (CNP 0 c e))"   and "\<And>c e. P (Le (CNP 0 c e))"   and "P T"   and "P F"   and "\<And>va. P (Le (CP va))"   and "\<And>va. P (Le (tm.Bound va))"   and "\<And>va vb. P (Le (tm.Add va vb))"   and "\<And>va vb. P (Le (tm.Mul va vb))"   and "\<And>va. P (Le (tm.Neg va))"   and "\<And>va vb. P (Le (tm.Sub va vb))"   and "\<And>vd vb vc. P (Le (CNP (Suc vd) vb vc))"   and "\<And>va. P (Lt (CP va))"   and "\<And>va. P (Lt (tm.Bound va))"   and "\<And>va vb. P (Lt (tm.Add va vb))"   and "\<And>va vb. P (Lt (tm.Mul va vb))"   and "\<And>va. P (Lt (tm.Neg va))"   and "\<And>va vb. P (Lt (tm.Sub va vb))"   and "\<And>vd vb vc. P (Lt (CNP (Suc vd) vb vc))"   and "\<And>va. P (Eq (CP va))"   and "\<And>va. P (Eq (tm.Bound va))"   and "\<And>va vb. P (Eq (tm.Add va vb))"   and "\<And>va vb. P (Eq (tm.Mul va vb))"   and "\<And>va. P (Eq (tm.Neg va))"   and "\<And>va vb. P (Eq (tm.Sub va vb))"   and "\<And>vd vb vc. P (Eq (CNP (Suc vd) vb vc))"   and "\<And>va. P (NEq (CP va))"   and "\<And>va. P (NEq (tm.Bound va))"   and "\<And>va vb. P (NEq (tm.Add va vb))"   and "\<And>va vb. P (NEq (tm.Mul va vb))"   and "\<And>va. P (NEq (tm.Neg va))"   and "\<And>va vb. P (NEq (tm.Sub va vb))"   and "\<And>vd vb vc. P (NEq (CNP (Suc vd) vb vc))"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
SINK: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "d_OUT f x = 0" shows "x \<in> MFMC_Network.SINK f"
gen_g_impl_rel_ext_def: fixes Rm :: "('h \<times> 'j) set"   and Rv :: "('e \<times> 'i set) set"   and Re :: "('f \<times> ('i \<times> 'i) set) set"   and Rv0 :: "('g \<times> 'i set) set" shows "\<langle>Rm, Rv, Re, Rv0\<rangle>gen_g_impl_rel_ext \<equiv> {(\<lparr>gi_V = Vi, gi_E = Ei, gi_V0 = V0i, \<dots> = mi\<rparr>, \<lparr>g_V = V, g_E = E, g_V0 = V0, \<dots> = m\<rparr>) |Vi Ei V0i mi V E V0 m. (Vi, V) \<in> Rv \<and> (Ei, E) \<in> Re \<and> (V0i, V0) \<in> Rv0 \<and> (mi, m) \<in> Rm}"
mproj2_def: fixes R :: "('f, 'g) Ring_scheme"   and M1 :: "('e, 'f, 'h) Module_scheme"   and M2 :: "('e, 'f, 'h) Module_scheme"   and M :: "('e, 'f, 'h) Module_scheme" shows "mproj2 R M1 M2 M = mproj1 R M2 M1 M"
check_prefix_def: fixes xs :: "'a list"   and ys :: "'a list" shows "check_prefix xs ys \<equiv> ebind (CHECK (length xs \<le> length ys) ()) (\<lambda>_. ebind (EWHILEIT (\<lambda>i. i \<le> length xs \<and> take i xs = take i ys) (\<lambda>i. i < length xs) (\<lambda>i. ebind (EASSERT (i < length xs \<and> i < length ys)) (\<lambda>_. ebind (CHECK (xs ! i = ys ! i) ()) (\<lambda>_. ERETURN (i + 1)))) 0) (\<lambda>_. ERETURN ()))"
frechet_derivative_worksI: fixes f :: "'c \<Rightarrow> 'd"   and net :: "'c filter" assumes "f differentiable net" shows "(f has_derivative frechet_derivative f net) net"
when_monom_mult: fixes c :: "'c"   and P :: "bool"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.monom_mult (c when P) t p = (MPoly_Type_Class.punit.monom_mult c t p when P)"
Abrupt: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "'b" shows "\<Gamma>\<turnstile>c \<down> Abrupt s"