hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "cmd"   and a2 :: "nat"   and a3 :: "cmd" assumes "labels a1 a2 a3" obtains c :: "cmd" where "a1 = c__" and "a2 = 0" and "a3 = c__"   | V :: "char list" and e :: "expr" where "a1 = V:=e" and "a2 = 1" and "a3 = Skip"   | c\<^sub>1 :: "cmd" and l :: "nat" and c :: "cmd" and c\<^sub>2 :: "cmd" where "a1 = c\<^sub>1__;; c\<^sub>2__" and "a2 = lb__" and "a3 = ca__;; c\<^sub>2__" and "labels c\<^sub>1__ lb__ ca__"   | c\<^sub>2 :: "cmd" and l :: "nat" and c :: "cmd" and c\<^sub>1 :: "cmd" where "a1 = c\<^sub>1'__;; c\<^sub>2'__" and "a2 = lc__ + #:c\<^sub>1'__" and "a3 = cb__" and "labels c\<^sub>2'__ lc__ cb__"   | c\<^sub>1 :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" and c\<^sub>2 :: "cmd" where "a1 = if (ba__) c\<^sub>1''__ else c\<^sub>2''__" and "a2 = ld__ + 1" and "a3 = cc__" and "labels c\<^sub>1''__ ld__ cc__"   | c\<^sub>2 :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" and c\<^sub>1 :: "cmd" where "a1 = if (bb__) c\<^sub>1 else c\<^sub>2" and "a2 = le__ + #:c\<^sub>1 + 1" and "a3 = cd__" and "labels c\<^sub>2 le__ cd__"   | c' :: "cmd" and l :: "nat" and c :: "cmd" and b :: "expr" where "a1 = while (bc__) c'__" and "a2 = l + 2" and "a3 = c;; while (bc__) c'__" and "labels c'__ l c"   | b :: "expr" and c' :: "cmd" where "a1 = while (b) c'" and "a2 = 1" and "a3 = Skip"
rSim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "\<Psi> \<rhd> P \<leadsto><{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<approx> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}> Q"
inverse_mod_coprime: fixes x :: "int" assumes "prime p"   and "comm_monoid_mult_class.coprime x p" shows "M (inverse_mod x p * x) = 1"
linepath_image0_right_open_real: fixes t :: "real"   and t' :: "real" assumes "t < t'" shows "linepath t t' ` {0..<1} = {t..<t'}"
reciprocal_one: shows "reciprocal 1\<^sub>h = 1\<^sub>h"
nat_filtration_AE_eq: fixes p :: "real"   and M :: "bool stream measure"   and f :: "bool stream \<Rightarrow> real"   and g :: "bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "infinite_coin_toss_space p M"   and "AEeq M f g"   and "0 < p"   and "p < 1"   and "f \<in> borel_measurable (infinite_coin_toss_space.nat_filtration M n)"   and "g \<in> borel_measurable (infinite_coin_toss_space.nat_filtration M n)" shows "f w = g w"
inv_solve_right: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<in> carrier H"   and "b \<in> carrier H"   and "c \<in> carrier H" shows "(a = b \<oplus>\<^bsub>H\<^esub> \<ominus>\<^bsub>H\<^esub> c) = (b = a \<oplus>\<^bsub>H\<^esub> c)"
d\<^sub>3_inject: fixes x :: "\<Pi>\<^sub>3"   and y :: "\<Pi>\<^sub>3" assumes "d\<^sub>3 x = d\<^sub>3 y" shows "x = y"
inverse_swap: fixes a :: "'b"   and b :: "'b" shows "inverse (swap a b) = swap a b"
In1_mono: fixes M :: "('f, 'g) Old_Datatype.node set"   and N :: "('f, 'g) Old_Datatype.node set" assumes "M \<subseteq> N" shows "In1 M \<subseteq> In1 N"
convex_def: fixes s :: "'b set" shows "convex s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
le_less_linear: fixes y :: "'a"   and x :: "'a" shows "y \<le> x \<or> x < y"
subformulas\<^sub>\<mu>_semantics: fixes \<phi> :: "'b ltln" shows "subformulas\<^sub>\<mu> \<phi> = {\<psi> \<in> subfrmlsn \<phi>. \<exists>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi> = \<psi>\<^sub>1 U\<^sub>n \<psi>\<^sub>2 \<or> \<psi> = \<psi>\<^sub>1 M\<^sub>n \<psi>\<^sub>2}"
Z_not_strict_def: shows "local.Z_not_strict = {f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> \<not> strict_mono_on f {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}}"
in_smallomega_zero: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "f \<in> \<omega>[F](\<lambda>x. 0::'c)"
poly_as_sum_eq_monoms: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b"   and a :: "nat" assumes "(\<Sum>i<n. monom (f i) i) = (\<Sum>i<n. monom (g i) i)"   and "a < n" shows "f a = g a"
\<tau>red1t_preserves_len: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "length xs' = length xs"
intros_2: fixes C :: "String.literal" shows "is_refT (Class C)"
geometric_series_closed: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "geometric_series n a b \<in> carrier R"
asymmetric_axioms: shows "asymmetric Y (\<sqsubset>)"
alt_group_inv_closed: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p \<in> carrier (alt_group n)" shows "inv' p \<in> carrier (alt_group n)"
induct: fixes P :: "Core.term \<Rightarrow> bool"   and a0 :: "Core.term" assumes "\<And>uu_ t. P t \<Longrightarrow> P (Abs uu_ (t $ Bv 0))"   and "\<And>uv_ v va. P (Ct v va) \<Longrightarrow> P (Abs uv_ (Ct v va))"   and "\<And>uv_ v va. P (Fv v va) \<Longrightarrow> P (Abs uv_ (Fv v va))"   and "\<And>uv_ v. P (Bv v) \<Longrightarrow> P (Abs uv_ (Bv v))"   and "\<And>uv_ v va. P (Abs v va) \<Longrightarrow> P (Abs uv_ (Abs v va))"   and "\<And>uv_ v vb vc. P (v $ Ct vb vc) \<Longrightarrow> P (Abs uv_ (v $ Ct vb vc))"   and "\<And>uv_ v vb vc. P (v $ Fv vb vc) \<Longrightarrow> P (Abs uv_ (v $ Fv vb vc))"   and "\<And>uv_ v vc. P (v $ Bv (Suc vc)) \<Longrightarrow> P (Abs uv_ (v $ Bv (Suc vc)))"   and "\<And>uv_ v vb vc. P (v $ Abs vb vc) \<Longrightarrow> P (Abs uv_ (v $ Abs vb vc))"   and "\<And>uv_ v vb vc. P (v $ (vb $ vc)) \<Longrightarrow> P (Abs uv_ (v $ (vb $ vc)))"   and "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (t $ u)"   and "\<And>v va. P (Ct v va)"   and "\<And>v va. P (Fv v va)"   and "\<And>v. P (Bv v)" shows "P a0"
rep_eq: fixes x :: "'a mpoly" shows "monomials x = keys (mapping_of x)"
vconn_implies_hpar_2: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" shows "trg\<^sub>D \<mu> = trg\<^sub>D f"
subformulas\<^sub>\<mu>\<^sub>\<nu>_disjoint: fixes \<phi> :: "'b ltln" shows "subformulas\<^sub>\<mu> \<phi> \<inter> subformulas\<^sub>\<nu> \<phi> = {}"
induct: fixes P :: "('e, 'f) splice_state \<Rightarrow> bool"   and splice_state :: "('e, 'f) splice_state" assumes "\<And>x1 x2. P (splice_state.Left x1 x2)"   and "\<And>x1 x2. P (splice_state.Right x1 x2)"   and "\<And>x. P (Left_only x)"   and "\<And>x. P (Right_only x)" shows "P splice_state"
eq_fin_symp: fixes u :: "'item list"   and v :: "'item list" assumes "u =\<^sub>F v" shows "v =\<^sub>F u"
cobounded1: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<inter># b \<subseteq># a"
sys_block_sizes_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "incidence_system point_set block_collection" shows "incidence_system.sys_block_sizes block_collection \<equiv> {int (card bl) |bl. bl \<in># block_collection}"
assms_1: shows "a \<otimes> b \<sim> a' \<otimes> b"
timpl_closure_Fu_inv': fixes f :: "'a"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" assumes "Fun (Fu f) T \<in> timpl_closure t TI" shows "\<exists>S. length S = length T \<and> t = Fun (Fu f) S"
iff_simps_7: fixes p :: "fm" assumes "p \<noteq> NOT T" shows "Ferrack.iff p T = p"
map_graph_fn_id_2: fixes X :: "('a, 'b) labeled_graph" shows "map_graph (Id_on (vertices X)) X = restrict X"
Int_iff: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
Sup_multiset_def: fixes A :: "'a multiset set" shows "Sup A = (if A \<noteq> {} \<and> subset_mset.bdd_above A then Abs_multiset (\<lambda>i. SUP X\<in>A. count X i) else {#})"
tab_simps_2: shows "src \<rho>\<sigma>.tab = src (tab\<^sub>0 s \<star> p\<^sub>0)"
Step1_2_2: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> ImpNext p \<and> \<not> unchanged (e p, c p, r p, m p, rmhist!p) \<and> $S2 rmhist p \<longrightarrow> S3 rmhist p$ \<and> MClkFwd memCh crCh cst p \<and> unchanged (e p, r p, m p, rmhist!p)"
ratfps_has_asymptotics_def: fixes q :: "complex poly"   and k :: "nat"   and R :: "real" shows "ratfps_has_asymptotics q k R = (q \<noteq> 0 \<and> 0 < R \<and> (let cs = snd (yun_factorization gcd q) in list_all (\<lambda>cl. proots_ball_card (fst cl) 0 (1 / R) = 0) cs \<and> list_all (\<lambda>cl. proots_sphere_card (fst cl) 0 (1 / R) = 0) (filter (\<lambda>cl. k < snd cl) cs)))"
priv_mem\<^sub>C_def: shows "priv_mem\<^sub>C \<equiv> map priv_mem cms"
Disj: fixes A :: "nat \<Rightarrow> 'c set"   and \<Omega> :: "'c set"   and M :: "'c set set" assumes "range A \<in> Pow (smallest_ccdi_sets \<Omega> M)"   and "disjoint_family A" shows "\<Union> (range A) \<in> smallest_ccdi_sets \<Omega> M"
ccpo: shows "class.ccpo \<Or> (\<sqsubseteq>) lt"
kernel_assoc_elem: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"   and "y \<in> carrier (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"   and "finprod G x I = finprod G y I" obtains z :: "'c \<Rightarrow> 'a" where "x = y \<otimes>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub> z" and "z \<in> kernel (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) (G\<lparr>carrier := J\<rparr>) (\<lambda>x. finprod G x I)"
unop_i: fixes t :: "t"   and \<C> :: "t_context"   and uu :: "unop_i" assumes "is_int_t t" shows "\<C> \<turnstile> [Unop_i t uu] : [t] _> [t]"
app_subst_Nil: fixes S :: "nat \<Rightarrow> typ" shows "$ S [] = []"
ide_char': fixes a :: "'a \<times> 'a" shows "B.VV.ide a = (B.VV.arr a \<and> (B.VV.dom a = a \<or> B.VV.cod a = a))"
yield_def: fixes xq :: "'a lazy_sequence" shows "yield xq = (case list_of_lazy_sequence xq of [] \<Rightarrow> None | x # xs \<Rightarrow> Some (x, lazy_sequence_of_list xs))"
axioms_1: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "RModuleEnd R smult M T" shows "RModuleHom R smult M smult T"
Store_eqI: fixes s1 :: "Store"   and s2 :: "Store" assumes "\<forall>x. alive x s1 = alive x s2"   and "\<forall>l. s1@@l = s2@@l" shows "s1 = s2"
adjacent_def: fixes v :: "nat" shows "adjacent v \<equiv> incoming v \<union> outgoing v"
pp_ub1: fixes ls :: "'a list"   and y :: "'a"   and x :: "'a" assumes "set ls \<subseteq> A"   and "y \<in> A"   and "x \<in> set ls" shows "x \<sqsubseteq>\<^bsub>r\<^esub> ls \<Squnion>\<^bsub>f\<^esub> y"
subsums_trans: fixes c1 :: "('d, 'e) conf"   and c2 :: "('d, 'e) conf"   and c3 :: "('d, 'e) conf" assumes "c1 \<sqsubseteq> c2"   and "c2 \<sqsubseteq> c3" shows "c1 \<sqsubseteq> c3"
OrdP_IN_SUCC: fixes H :: "fm set"   and x :: "tm"   and y :: "tm" assumes "H \<turnstile> OrdP x"   and "H \<turnstile> y IN x" shows "H \<turnstile> SUCC y IN SUCC x"
vmc_path_ldropn: fixes n :: "nat" assumes "enat n < llength P'" shows "vmc_path G (ldropn n P') (P' $ n) p \<sigma>"
rcosets_part_G: fixes H :: "'a set" assumes "subgroup H (add_monoid G)" shows "\<Union> (rcosets\<^bsub>add_monoid G\<^esub> H) = carrier G"
parabolic_isometryD_4: fixes f :: "'b \<Rightarrow> 'b" assumes "parabolic_isometry f" shows "\<not> elliptic_isometry f"
term_pp_rel_irrefl: fixes r :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'t \<times> 'a" assumes "irreflp r" shows "\<not> term_pp_rel r a a"
n_val_defTr: fixes v :: "'a \<Rightarrow> ant"   and w :: "'a" assumes "valuation K v"   and "w \<in> carrier K" shows "\<exists>!a. a * Lv K v = v w"
f_aggregate_nth: fixes n :: "nat"   and xs :: "'b list"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" assumes "n < length xs div k" shows "f_aggregate xs k ag ! n = ag (xs \<up> (n * k) \<down> k)"
ax_derivs_asm: fixes ts :: "'a set"   and A :: "'a set" assumes "ts \<subseteq> A" shows "P A ts"
binrelchain_Cons_reduce: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "binrelchain P (x # xs)" shows "binrelchain P xs"
rt_graphE: fixes ip :: "nat"   and ip' :: "nat"   and \<sigma> :: "nat \<Rightarrow> state"   and dip :: "nat" assumes "(ip, ip') \<in> rt_graph \<sigma> dip" shows "ip \<noteq> dip \<and> (\<exists>r. rt (\<sigma> ip) = r \<and> (\<exists>dsn dsk hops. r dip = Some (dsn, dsk, val, hops, ip')))"
bisim1Sync10: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([Addr a], xs, 8 + length (compE2 e1) + length (compE2 e2), \<lfloor>a\<rfloor>)"
assms_2: fixes n :: "nat" shows "(\<lambda>x. f x - (\<Sum>k<n. mssnth cs k * h x ^ k)) \<in> O[F](\<lambda>x. h x ^ n)"
insert_remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "\<Sqinter> insert x A = (if A - {x} = {} then x else x \<sqinter> \<Sqinter> (A - {x}))"
assoc'_simps_5: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and h :: "('a S.E.term, 'a) S.arr" assumes "S.ide f"   and "S.ide g"   and "S.ide h"   and "S.src f = S.trg g"   and "S.src g = S.trg h" shows "S.cod (S.\<a>' f g h) = (S.cod f \<star>\<^sub>S S.cod g) \<star>\<^sub>S S.cod h"
axioms_2: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" assumes "l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes" shows "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"
sigma_sets_top: fixes sp :: "'b set"   and A :: "'b set set" shows "sp \<in> sigma_sets sp A"
greaterThan_def: fixes less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and l :: "'e" shows "ord.greaterThan less l \<equiv> {x. less l x}"
ceq_vector_fraction_iff: fixes x :: "'a"   and u :: "complex"   and v :: "complex"   and a :: "'a" shows "(x = (u / v) *\<^sub>C a) = (if v = 0 then x = (0::'a) else v *\<^sub>C x = u *\<^sub>C a)"
span_legs_eq_1: shows "SPN_fgh.\<mu>.leg0 = TfHgh.R\<^sub>0"
partition_onI: fixes P :: "'a set set"   and A :: "'a set" assumes "\<Union> P = A"   and "\<And>p q. \<lbrakk>p \<in> P; q \<in> P; p \<noteq> q\<rbrakk> \<Longrightarrow> disjnt p q"   and "{} \<notin> P" shows "partition_on A P"
Frame_R: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>' {P} C { Q}"   and "Frame P' P F" shows "\<turnstile>\<^sub>3\<^sub>' {P'} C { Q \<and>* F}"
assms_1: shows "\<Gamma> \<turnstile> oper $$ e : t'"
scale_left_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "'b" shows "(a + b) *s x = a *s x + b *s x"
sel_2: fixes x1 :: "'c"   and x2 :: "'b"   and x3 :: "nat"   and x4 :: "('c, 'b) SkewBinomialTree list" shows "prio (Node x1 x2 x3 x4) = x2"
i: shows "row i A \<in> vec.span (rows A - {row i A})"
mainleq: fixes n :: "nat" shows "trace (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) (Suc n)) + trace (denote_while_n_comp M0 M1 DS n \<rho>) \<le> trace \<rho>"
emeasure_restricted: fixes S :: "'b set"   and M :: "'b measure"   and X :: "'b set" assumes "S \<in> sets M"   and "X \<in> sets M" shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
basis_in_subset_eq: fixes \<E> :: "'a set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> B\<^sub>1"   and "\<E>.basis_in \<E> B\<^sub>2"   and "B\<^sub>1 \<subseteq> B\<^sub>2" shows "B\<^sub>1 = B\<^sub>2"
HFinite_hIm: fixes x :: "complex star" assumes "x \<in> HFinite" shows "hIm x \<in> HFinite"
c2: fixes a :: "'a"   and as :: "'a list"   and x :: "'a"   and xs :: "'a list" assumes "ps = a # as"   and "ls = x # xs"   and "x = a"   and "\<not> prefix as xs" shows "thesis"
list_all2_nthD: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length xs" shows "P (xs ! p) (ys ! p)"
Inf_lower: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 x :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq (Inf A) x"
measurable_embed_measure2: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "inj f" shows "f \<in> M \<rightarrow>\<^sub>M embed_measure M f"
monotone_Sup: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a set" assumes "monotone ord (\<subseteq>) f" shows "monotone ord (\<le>) (\<lambda>x. \<Squnion> f x)"
sub1: shows "(\<Union>i\<in>{i. Inr i \<in> SIGT \<theta>}. FVT (\<sigma> i)) \<subseteq> - U"
pure_eq_conv: fixes R :: "('b \<times> 'a) set"   and R' :: "('b \<times> 'a) set" shows "(pure R = pure R') = (R = R')"
integral_add: fixes M :: "'e measure"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "integrable M f"   and "integrable M g" shows "\<integral> x. f x + g x \<partial>M = integral\<^sup>L M f + integral\<^sup>L M g"
thetaParWL1_Wbis: shows "thetaParWL1 \<subseteq> Wbis"
rbt_lookup_rbt_unionwk: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t1 :: "('a, 'b) RBT_Impl.rbt"   and t2 :: "('a, 'b) RBT_Impl.rbt"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "'a" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t1"   and "ord.rbt_sorted less t2" shows "ord.rbt_lookup less (ord.rbt_union_with_key less f t1 t2) k = (case ord.rbt_lookup less t1 k of None \<Rightarrow> ord.rbt_lookup less t2 k | Some v \<Rightarrow> case ord.rbt_lookup less t2 k of None \<Rightarrow> Some v | Some w \<Rightarrow> Some (f k v w))"
distinct_1: fixes x1 :: "('a, 'b, 'c, 'd) Tree set['a]"   and x2 :: "('a, 'b, 'c, 'd) Tree" shows "tConj x1 \<noteq> tNot x2"
list_e_eq_lel_2: fixes l1 :: "'b list"   and e' :: "'b"   and l2 :: "'b list"   and e :: "'b" shows "(l1 @ e' # l2 = [e]) = (l1 = [] \<and> e' = e \<and> l2 = [])"
ostep_invariant_weakenE: fixes A :: "('d \<times> 'e, 'f) automaton"   and PS :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and PU :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e \<Rightarrow> bool"   and Q :: "('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e \<Rightarrow> bool"   and QS :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and QU :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "A \<Turnstile>\<^sub>A (PS, PU \<rightarrow>) P"   and "\<And>t. P t \<Longrightarrow> Q t"   and "\<And>\<sigma> \<sigma>' a. QS \<sigma> \<sigma>' a \<Longrightarrow> PS \<sigma> \<sigma>' a"   and "\<And>\<sigma> \<sigma>'. QU \<sigma> \<sigma>' \<Longrightarrow> PU \<sigma> \<sigma>'" shows "A \<Turnstile>\<^sub>A (QS, QU \<rightarrow>) Q"
normalize_poly_eq_map_poly: fixes p :: "'b poly" shows "normalize p = map_poly (\<lambda>x. x div unit_factor (lead_coeff p)) p"
preserves_dom: fixes f :: "'c" assumes "C.arr f" shows "C.dom (trg\<^sub>C f) = trg\<^sub>C (C.dom f)"
mat_abelian_monoid: fixes n :: "nat"   and b :: "'d" shows "abelian_monoid (mat_ring n b)"
inject: fixes x1 :: "'b word"   and x2 :: "nat"   and y1 :: "'b word"   and y2 :: "nat" shows "(PrefixMatch x1 x2 = PrefixMatch y1 y2) = (x1 = y1 \<and> x2 = y2)"
mod_subset_refl: fixes c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and p :: "nat"   and X :: "'c topology" shows "(c, c) \<in> mod_subset p X"
onlyGoodsAreSold: fixes a :: "(integer \<times> 'c set) set"   and N :: "integer set"   and \<Omega> :: "'c set" assumes "a \<in> soldAllocations'' N \<Omega>" shows "\<Union> (Range a) \<subseteq> \<Omega>"
lderiv_toplevel_summands_Zero: fixes xs :: "'a list"   and r :: "'b rexp"   and s :: "'b rexp" assumes "lderivs xs (Pr r) = Pr s"   and "toplevel_summands r = {Zero}" shows "toplevel_summands s = {Zero}"
Scan_def: fixes T :: "(symbol \<times> character list) set"   and k :: "nat"   and I :: "item set" shows "Scan T k I = I \<union> {uu_. \<exists>x t c. uu_ = inc_item x (k + length c) \<and> x \<in> bin I k \<and> (t, c) \<in> T \<and> next_symbol x = Some t}"
elims_3: fixes x :: "('a, 'b) insert_message set"   and xa :: "('a, 'b) insert_message"   and xb :: "('a, 'b) insert_message" assumes "\<not> depends_on x xa xb" obtains M :: "('a, 'b) insert_message set" and xc :: "('a, 'b) insert_message" and y :: "('a, 'b) insert_message" where "x = M" and "xa = xc" and "xb = y" and "\<not> (xc \<in> M \<and> y \<in> M \<and> I xc \<in> deps (Insert y))"
closure_invariant_def: shows "closure_invariant = while_option test_invariant step_invariant"
induct: fixes P :: "(int\<^sub>\<bottom> \<rightarrow> int\<^sub>\<bottom> \<rightarrow> (int\<^sub>\<bottom>) LList) \<Rightarrow> bool" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (\<Lambda> va v. Fixrec.run\<cdot>(match_up\<cdot>va\<cdot>(\<Lambda> xa. match_up\<cdot>v\<cdot>(\<Lambda> y. Fixrec.succeed\<cdot>(if xa \<le> y then LCons\<cdot>(up\<cdot>xa)\<cdot>(x\<cdot>(up\<cdot>(xa + 1))\<cdot>(up\<cdot>y)) else LNil)))))" shows "P enumFromToL"
dghm_id_components_3: fixes \<CC> :: "V" shows "dghm_id \<CC>\<lparr>HomDom\<rparr> = \<CC>"
A: fixes xa :: "real"   and ya :: "real" assumes "\<bar>xa\<bar> < \<bar>ya\<bar>"   and "0 \<le> ya" shows "(\<lambda>n. (a gchoose n) * complex_of_real xa ^ n * complex_of_real ya powr (a - of_nat n)) sums complex_of_real (xa + ya) powr a"
distinct_1: fixes x1 :: "('a, 'b) strand_step"   and x2 :: "('a, 'b) Term.term" shows "Step x1 \<noteq> Decomp x2"
oSuc_leI: fixes x :: "ordinal"   and y :: "ordinal" assumes "x < y" shows "oSuc x \<le> y"
bit_uint_iff: fixes w :: "'a word"   and n :: "nat" shows "bit (uint w) n = (n < LENGTH('a) \<and> bit w n)"
writes_bind_pure: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and S :: "('a, 'b, 'd) prog set"   and g :: "'c \<Rightarrow> ('a, 'b, 'e) prog"   and h' :: "'a" assumes "pure f h"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> writes S (g x) h h'" shows "writes S (Heap_Error_Monad.bind f g) h h'"
swap_list_cons: fixes a :: "nat"   and b :: "nat"   and x :: "'a"   and L :: "'a list" shows "swap_list (Suc a) (Suc b) (x # L) = x # swap_list a b L"
rel_symp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "symp R" shows "symp (rel_option R)"
intro: fixes Null :: "'b"   and Obj :: "'b set" assumes "Null \<notin> Obj" shows "discrete_category Obj Null"