hcju/isabelleps huggingface dataset:
nonsat_push': fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "Height_Bounded_Labeling c s t f l"   and "nonsat_push_precond f l e" shows "((f, l), NONSAT_PUSH', push_effect f e, l) \<in> pr_algo_lts'"
inject: fixes x21 :: "'x"   and x22 :: "'x list"   and y21 :: "'x"   and y22 :: "'x list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
\<omega>_limit_set_positively_invariant: fixes x :: "'a" shows "positively_invariant (\<omega>_limit_set x)"
dom_simp: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VV.arr f" shows "VV.dom f = VxV.dom f"
measurable_join: fixes N :: "'c measure" shows "join \<in> subprob_algebra (subprob_algebra N) \<rightarrow>\<^sub>M subprob_algebra N"
sall_strict: fixes p :: "'a \<rightarrow> tr" shows "sall\<cdot>p\<cdot>\<bottom> = \<bottom>"
h01: fixes z :: "complex"   and n :: "nat" assumes "z \<in> ball 0 1 - {0}" shows "h n z \<noteq> 0 \<and> h n z \<noteq> 1"
sorted_add_spvec_helper1: fixes a :: "nat"   and b :: "'a"   and arr :: "(nat \<times> 'a) list"   and brr :: "(nat \<times> 'a) list"   and ab :: "nat"   and bb :: "'a"   and list :: "(nat \<times> 'a) list" assumes "add_spvec ((a, b) # arr) brr = (ab, bb) # list" shows "ab = a \<or> brr \<noteq> [] \<and> ab = fst (hd brr)"
prems: shows "op_dg \<AA> = op_dg \<BB>"
map2_mult_scalar_plus: fixes xs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and ys :: "('a \<Rightarrow>\<^sub>0 'c) list"   and zs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "map2 (\<odot>) (map2 (+) xs ys) zs = map2 (+) (map2 (\<odot>) xs zs) (map2 (\<odot>) ys zs)"
prems_2: fixes x :: "'c"   and y :: "'d" assumes "x \<in> insert x__ A"   and "y \<in> B x" shows "g x y \<in> carrier G"
normal_invE_1: fixes N :: "'a set" assumes "N \<lhd> G\<lparr>carrier := generate G {a}\<rparr>" shows "subgroup N (G\<lparr>carrier := generate G {a}\<rparr>)"
E6: shows "(\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. \<Sum>i | i < length qs \<and> (qs ! i = y \<or> qs ! i = x). ALG y qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ()) + ALG x qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ())) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. ALGxy_det Strat qs init x y)"
bilateral_substitutes_lemma_union: fixes h :: "'c"   and X :: "'a set"   and Y :: "'a set"   and d :: "'b" assumes "Xd ` Ch h X \<inter> Xd ` Y = {}"   and "d \<notin> Xd ` Ch h X"   and "d \<notin> Xd ` Y"   and "inj_on Xd Y" shows "d \<notin> Xd ` Ch h (X \<union> Y)"
vcomp_eq: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.seq \<nu> \<mu>" shows "\<nu> \<bullet> \<mu> = \<lparr>Chn = Chn \<nu> \<odot> Chn \<mu>, Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr>"
\<b>_Nil: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat" assumes "ps\<^sub>+ = []"   and "Suc 0 \<le> i" shows "\<b> ps i = 0"
path_image_compose: fixes f :: "'b \<Rightarrow> 'a"   and p :: "real \<Rightarrow> 'b" shows "path_image (f \<circ> p) = f ` path_image p"
\<alpha>s_def: shows "\<alpha>s \<equiv> map_prod \<alpha>w \<alpha>r"
X_def_5: shows "short_ch X \<longrightarrow> Q\<^sub>x \<in> X \<and> Q\<^sub>z \<in> X \<and> (\<forall>Q\<^sub>y\<in>\<E>. [[Q\<^sub>x Q\<^sub>y Q\<^sub>z]] \<longrightarrow> Q\<^sub>y \<in> \<emptyset> Q b)"
f2: shows "Ratreal p + sqrt ((Ratreal p)\<^sup>2) = 0"
simps: fixes a1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and a2 :: "'d"   and a3 :: "('c \<Rightarrow> real) set"   and a4 :: "'d"   and a5 :: "('c \<Rightarrow> real) set" shows "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>a4, a5\<rangle> = ((\<exists>A l Z. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l \<and> a5 = Z) \<or> (\<exists>A l Z l' Z' l'' Z''. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l'' \<and> a5 = Z'' \<and> A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l', Z'\<rangle> \<and> A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<alpha> \<langle>l'', Z''\<rangle>))"
sum_set_simps_3: fixes x :: "'d" shows "setr (Inl x) = {}"
A9_NSources: shows "\<forall>C\<in>AbstrLevel level0. C \<noteq> sA9 \<and> C \<noteq> sA8 \<longrightarrow> sA9 \<notin> Sources level0 C"
list_nth: fixes i :: "nat"   and s :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "i < length (list s n)" shows "list s n ! i = s i"
unit_is_iso: shows "local.iso \<eta>\<^sub>0"
simps_3: fixes v :: "(int \<times> int \<times> protocol) set set"   and va :: "(int \<times> int \<times> protocol) set set"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "wellformed_policy2Pr (DenyAllFromTo v va # xs) = ((\<forall>c a b. c = DenyAllFromTo a b \<and> c \<in> set xs \<longrightarrow> dom (Cp (DenyAllFromTo v va)) \<inter> dom (Cp c) = {}) \<and> wellformed_policy2Pr xs)"
butlast_drop: fixes n :: "nat"   and xs :: "'d list" shows "butlast (drop n xs) = drop n (butlast xs)"
equivalence_in_bicategory_axioms: shows "internal_equivalence (src f) (src f) (local.inv \<i>[src f]) \<i>[src f]"
simps_1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a option" shows "option_le P None y = True"
iso_inv_iso: fixes f :: "'b" assumes "D.iso f" shows "D.iso (D.inv f)"
a_ptr_disconnected_node_rel_disconnected_node: fixes h :: "(_) heap"   and document :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr" assumes "h \<turnstile> get_disconnected_nodes document \<rightarrow>\<^sub>r disc_nodes" shows "(node_ptr \<in> set disc_nodes) = ((cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<in> local.a_ptr_disconnected_node_rel h)"
span_zero: fixes S :: "'b set" shows "(0::'b) \<in> span S"
ereal_add_strict_mono2: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal"   and d :: "ereal" assumes "a < b"   and "c < d" shows "a + c < b + d"
analz_image_priEK: fixes evs :: "event list"   and C :: "agent"   and KK :: "nat set" assumes "evs \<in> set_cr" shows "(Key (priEK C) \<in> analz (Key ` KK \<union> knows Spy evs)) = (priEK C \<in> KK \<or> C \<in> bad)"
cases: fixes a1 :: "lam"   and a2 :: "lam" assumes " a1 \<longrightarrow>\<^sub>\<beta> a2" obtains s1 :: "lam" and s2 :: "lam" and t :: "lam" where "a1 = App s1__ ta__" and "a2 = App s2__ ta__" and " s1__ \<longrightarrow>\<^sub>\<beta> s2__"   | s1 :: "lam" and s2 :: "lam" and t :: "lam" where "a1 = App t s1a__" and "a2 = App t s2a__" and " s1a__ \<longrightarrow>\<^sub>\<beta> s2a__"   | s1 :: "lam" and s2 :: "lam" and a :: "name" where "a1 = Lam [a__].s1b__" and "a2 = Lam [a__].s2b__" and " s1b__ \<longrightarrow>\<^sub>\<beta> s2b__"   | a :: "name" and s2 :: "lam" and s1 :: "lam" where "a1 = App (Lam [a].s1) s2" and "a2 = s1[a::=s2]" and "a \<sharp> s2"
dur_shift: fixes \<omega> :: "('c \<Rightarrow> real) stream"   and i :: "nat"   and j :: "nat" shows "dur \<omega> (i + j) = dur \<omega> i + dur (sdrop i \<omega>) j"
index_Fun_fv_subset: fixes i :: "nat"   and T :: "('a, 'b) Term.term list"   and f :: "'a" assumes "i < length T" shows "fv (T ! i) \<subseteq> fv (Fun f T)"
pmf_map_inj_inv: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'a pmf"   and f' :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "inj_on f (set_pmf p)"   and "\<And>x. f' (f x) = x" shows "pmf (map_pmf f p) x = (if x \<in> range f then pmf p (f' x) else 0)"
\<tau>3': shows "(fst \<tau>3', snd \<tau>3' @ snd \<tau>3'') \<in> seq ars"
acyclic_3b_def: fixes x :: "'a" shows "acyclic_3b x \<equiv> \<forall>y. y \<le> x \<and> y\<^sup>\<star> = x\<^sup>\<star> \<longrightarrow> y = x"
2097: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) \<sqinter> x = x"
norm_vec_mult_le: fixes A :: "'a sq_mtx"   and x :: "(real, 'a) vec" shows "\<parallel>A *\<^sub>V x\<parallel> \<le> (\<parallel>A\<parallel>) * (\<parallel>x\<parallel>)"
genPoly_new_face: fixes g' :: "graph"   and n :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph" assumes "g' \<in> set (generatePolygon n v f g)"   and "minGraphProps g"   and "f \<in> set (nonFinals g)"   and "v \<in> \<V> f"   and "3 \<le> n" shows "\<exists>f\<in>set (finals g') - set (finals g). |vertices f| = n"
card_bij_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "inj_on g B"   and "g ` B \<subseteq> A"   and "finite A"   and "finite B" shows "card A = card B"
complex_eq: fixes a :: "complex" shows "a = cor (Re a) + \<i> * cor (Im a)"
add: fixes f :: "'q \<Rightarrow> 'r"   and x :: "'q"   and y :: "'q" assumes "additive f" shows "f (x + y) = f x + f y"
AE_uniform_measure: fixes M :: "'b measure"   and A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "emeasure M A \<noteq> 0"   and "emeasure M A < \<infinity>" shows "almost_everywhere (uniform_measure M A) P = set_almost_everywhere A M P"
assms: fixes x :: "'a" shows "x \<in> B \<longrightarrow> x \<in> C"
sees_field_i_i_i_o_oI: fixes x :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xa :: "char list"   and xb :: "char list"   and xc :: "ty"   and xd :: "char list" assumes "x \<turnstile> xa sees xb:xc in xd" shows "pred.eval (sees_field_i_i_i_o_o x xa xb) (xc, xd)"
m_lcomm: fixes x :: "nat"   and y :: "nat"   and z :: "nat" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z) = y \<otimes>\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> z)"
meeting_paths_produce_cycle: fixes v :: "Vertices"   and xs :: "Vertices list"   and ys :: "Vertices list" assumes "path (v # xs)"   and "xs \<noteq> []"   and "path (v # ys)"   and "ys \<noteq> []"   and "last xs = last ys"   and "hd xs \<noteq> hd ys" shows "\<exists>zs. cycle zs \<and> hd zs = v"
coplanar_perm_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet A B D C"
ACeD_axioms: shows "ACeD D (\<cdot>) e"
assms_4: shows "wf_sig (map_of cto, map_of tao, translate_osig sa)"
rep_list_vec_of: shows "rep_list (vec_of sp) = snd sp"
map_poly_inj_zero_hom_axioms: shows "map_poly_inj_zero_hom to_int_mod_ring"
\<phi>_def: shows "\<phi> \<equiv> \<lambda>n x. sin ((n + 1 / 2) * x) * ((f (t + x) + f (t - x) - 2 * l) / x)"
assms: shows "atom u \<sharp> (f, k, del, a, b)"
StuckProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> =n\<Rightarrow> Stuck"
fA_def: fixes x :: "'a" shows "fA x = (if x \<in> A then f x else (0::'b))"
infront: shows "a + d *\<^sub>R l \<in> rel_frontier S"
bheight_rbtreeify_g: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> Suc (length kvs)" shows "bheight (fst (rbtreeify_g n kvs)) = RBT_Impl.plog2 n"
PiE_dflt_def: fixes A :: "'b set"   and dflt :: "'c"   and B :: "'b \<Rightarrow> 'c set" shows "PiE_dflt A dflt B = {f. \<forall>x. (x \<in> A \<longrightarrow> f x \<in> B x) \<and> (x \<notin> A \<longrightarrow> f x = dflt)}"
final_thread_def: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "r1.final_thread s t \<equiv> case thr s t of None \<Rightarrow> False | \<lfloor>(x, ln)\<rfloor> \<Rightarrow> final1 x \<and> ln = no_wait_locks \<and> wset s t = None"
HBL1_rev: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv (wrepr.PP \<langle>\<phi>\<rangle>)" shows "prv \<phi>"
arr_Set_arr_ParD_2: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Set \<alpha> T" shows "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) = T\<lparr>ArrDom\<rparr>"
simps_1: fixes x :: "Vars.var"   and y :: "Vars.var"   and z :: "Vars.var" shows "(Var x)[y::=z] = Var x[y::v=z]"
distinct_31: fixes x4 :: "key"   and x5 :: "msg" shows "Key x4 \<noteq> Hash x5"
composable: shows "Span.src (SPN g) = Span.trg (SPN h)"
assms_2: shows "set ps = (\<lambda>p. (p, multiplicity p n - 1)) ` prime_factors n"
reflect_poly_pCons: fixes a :: "'b"   and p :: "'b poly" assumes "a \<noteq> (0::'b)" shows "reflect_poly (pCons a p) = Poly (rev (a # coeffs p))"
aHeap_thunks_nonrec: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "nonrec \<Gamma>"   and "x \<in> thunks \<Gamma>"   and "x--x\<in>\<G>\<^bsub>a\<^esub> e" shows "(Aheap \<Gamma> e\<cdot>a) x = up\<cdot>0"
fun_of_def: fixes S :: "('c \<times> 'd) set" shows "fun_of S \<equiv> \<lambda>x y. (x, y) \<in> S"
nullhomotopic_from_lemma: fixes a :: "'b"   and r :: "real"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on (cball a r - {a}) g"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> norm (g x - f a) < e"   and "\<And>x. x \<in> cball a r \<and> x \<noteq> a \<Longrightarrow> f x = g x" shows "continuous_on (cball a r) f"
snd_conv: fixes x1 :: "'e"   and x2 :: "'d" shows "snd (x1, x2) = x2"
compactin_sing: fixes X :: "'d topology"   and a :: "'d" shows "compactin X {a} = (a \<in> topspace X)"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "L.seq g f" shows "L.comp g f = g \<cdot> f"
simps_3: fixes f :: "'a \<Rightarrow> 'a fmUni" shows "map_atomUni f FalseFUni = FalseFUni"
paste_Domain: fixes P :: "('b \<times> 'c) set"   and Q :: "('b \<times> 'c) set" shows "Domain (P +< Q) = Domain P \<union> Domain Q"
simps: fixes a1 :: "'action list"   and a2 :: "'state" shows "path a1 a2 = ((\<exists>p. a1 = [] \<and> a2 = p) \<or> (\<exists>a p r. a1 = a # r \<and> a2 = p \<and> en a p \<and> path r (ex a p)))"
ccInf_less_iff: fixes S :: "'a set" assumes "countable S" shows "(Inf S < a) = (\<exists>x\<in>S. x < a)"
word_oa_dist: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "x && y || z = (x || z) && (y || z)"
f_def: shows "f = (\<lambda>z. if z = 1 then a else newman z + deriv zeta z + c * zeta z)"
ran_alt: fixes f :: "'c \<Rightarrow> 'b option" shows "ran f = (the \<circ> f) ` dom f"
card_of_image: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "|f ` A| \<le>o |A|"
e2lenas_def: fixes e :: "nat" shows "e2lenas e \<equiv> e_length (e2xs e)"
shift_back: fixes II :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat" assumes "II \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}" shows "local.shift cl n (back cl n II) = II"
filtermap_at_bot_linear_eq: fixes c :: "'c"   and b :: "'c" assumes "c \<noteq> (0::'c)" shows "filtermap (\<lambda>x. x * c + b) at_bot = (if (0::'c) < c then at_bot else at_top)"
M_3: fixes vs :: "nat \<Rightarrow> nat" shows "\<exists>vs'. ((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>* \<and> P vs vs'"
cases: fixes a1 :: "('state, 'effect) L_state"   and a2 :: "('state, 'effect) L_state" assumes "L_bisimilar a1 a2" obtains P :: "'state" and F :: "'effect fs_set" and Q :: "'state" where "a1 = EF (F__, P__)" and "a2 = EF (F__, Q__)" and "P__ \<sim>\<cdot>[F__] Q__"   | P :: "'state" and F :: "'effect fs_set" and Q :: "'state" and f :: "'effect" where "a1 = AC (f, F, \<langle>f\<rangle>P)" and "a2 = AC (f, F, \<langle>f\<rangle>Q)" and "P \<sim>\<cdot>[F] Q" and "f \<in>\<^sub>f\<^sub>s F"
i_shrink_0: fixes f :: "nat \<Rightarrow> 'b message_af" shows "f \<div> 0 = (\<lambda>n. NoMsg)"
odd_of_bool_self: fixes p :: "bool" shows "odd (of_bool p) = p"
pred_inject_1: fixes P :: "'c \<Rightarrow> bool" shows "pred_option P None"
F1_def: fixes X' :: "'x set" shows "F1 X' = - Rf CH X'"
assms_1: shows "proj_poly_syz n p = proj_poly_syz n q"
lead_coeff_min_int_poly_pos: fixes x :: "'a" shows "0 < Polynomial.lead_coeff (min_int_poly x)"
take_0: fixes xs :: "'c list" shows "take 0 xs = []"
assms_2: fixes a' :: "'a" assumes "a' \<in> A"   and "a .= a'" shows "P"
real_root_le_iff: fixes n :: "nat"   and x :: "real"   and y :: "real" assumes "0 < n" shows "(root n x \<le> root n y) = (x \<le> y)"
mono_ndet_FD_right: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q" shows "(S |-| P) \<sqsubseteq>\<^sub>F\<^sub>D Q"
update_ls_delete_map_of: fixes l :: "('a \<times> 'b) list list"   and k :: "'a"   and x :: "'a" assumes "ht_hash l"   and "ht_distinct l"   and "ht_hash (l[bounded_hashcode_nat (length l) k := fst (ls_delete k (l ! bounded_hashcode_nat (length l) k))])"   and "ht_distinct (l[bounded_hashcode_nat (length l) k := fst (ls_delete k (l ! bounded_hashcode_nat (length l) k))])"   and "1 < length l" shows "map_of (concat (l[bounded_hashcode_nat (length l) k := fst (ls_delete k (l ! bounded_hashcode_nat (length l) k))])) x = (map_of (concat l) |` (- {k})) x"
assms: shows "m = map_of (map (\<lambda>v. (v, d)) vs)"
monic_factorization_unique_mset: fixes P :: "'a poly multiset"   and Q :: "'a poly multiset" assumes "\<Prod>\<^sub># P = \<Prod>\<^sub># Q"   and "set_mset P \<subseteq> {q. irreducible q \<and> monic q}"   and "set_mset Q \<subseteq> {q. irreducible q \<and> monic q}" shows "P = Q"
assms: fixes e :: "ennreal" assumes "0 < e" shows "\<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. eNorm N (u n - u m) < e"
plus_list_ub1: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> A"   and "set ys \<subseteq> A"   and "length xs = length ys" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] xs [+f] ys"
assms_7: shows "has_greatest_postfixpoint (h \<circ> t)"
lprefix_coinduct: fixes P :: "'c llist \<Rightarrow> 'c llist \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'c llist" assumes "P xs ys"   and "\<And>xs ys. P xs ys \<Longrightarrow> (lnull ys \<longrightarrow> lnull xs) \<and> (\<not> lnull xs \<longrightarrow> \<not> lnull ys \<longrightarrow> lhd xs = lhd ys \<and> (P (ltl xs) (ltl ys) \<or> ltl xs \<sqsubseteq> ltl ys))" shows "xs \<sqsubseteq> ys"
simps_1: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" shows "removeAllDuplicates (x # xs) = removeDuplicates x # removeAllDuplicates xs"
eval_ms_def': fixes F :: "'a ms" shows "eval F = (case F of MS x f \<Rightarrow> f)"
abs_neq_infinity_cases: fixes x :: "ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>" obtains r :: "real" where "x = ereal r"
point_indices_def: fixes t :: "int" shows "source.point_indices t \<equiv> {int (\<B> index ps) |ps. int (card ps) = t \<and> ps \<subseteq> \<V>}"
DBM_val_bounded_len'1: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and vs :: "nat list"   and c :: "'c" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "0 \<notin> set vs"   and "v c \<le> n"   and "\<forall>k\<in>set vs. 0 < k \<longrightarrow> k \<le> n \<and> (\<exists>c. v c = k)" shows "dbm_entry_val u (Some c) None (len m (v c) 0 vs)"
exhaust: fixes y :: "residual_Rep" obtains x11 :: "name" and x12 :: "name \<Rightarrow> pi noption" where "y = BoundOutputR_Rep x11 x12"   | x21 :: "freeRes" and x22 :: "pi" where "y = FreeR_Rep x21 x22"
RedStaticUpDynCast: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list"   and E :: "char list \<Rightarrow> ty option"   and a :: "nat"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> Path last Cs to C unique"   and "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "P,E \<turnstile> \<langle>Cast C (ref (a, Cs)),s\<rangle> \<rightarrow> \<langle>ref (a, Ds),s\<rangle>"
S1MemUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RNext rmCh mm ires p]_(m p) \<and> $S1 rmhist p \<longrightarrow> unchanged m p"
th: shows "fps_cos c + fps_cos c = fps_cos c * fps_const 2"
smult_cancel: fixes c :: "'b"   and p :: "'b poly"   and q :: "'b poly" assumes "c \<noteq> (0::'b)"   and "smult c p = smult c q" shows "p = q"
lookup_single_eq: fixes k :: "'c"   and v :: "'b" shows "lookup (Poly_Mapping.single k v) k = v"
rot_diamond_cube_valid_two_cube: shows "valid_two_cube rot_diamond_cube"
IH1: shows "real (fib n) = (\<phi> ^ n - \<psi> ^ n) / sqrt 5"
Collect_cong: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
group_hom_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" shows "group_hom_axioms G H h \<equiv> h \<in> hom G H"
times_Box_strict1: fixes y :: "'a Box" shows "\<bottom> * y = \<bottom>"
bounded_C0: fixes \<psi> :: "'a \<Rightarrow> complex" assumes "has_ell2_norm \<psi>" shows "ell2_norm (C0 \<psi>) \<le> ell2_norm \<psi>"
left_commute: fixes b :: "'c multiset"   and a :: "'c multiset"   and c :: "'c multiset" shows "b \<union># (a \<union># c) = a \<union># (b \<union># c)"
cdot_square_norm: fixes x :: "'a" shows "cinner x x = complex_of_real ((norm x)\<^sup>2)"
delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_ok: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "ptr |\<in>| object_ptr_kinds h" shows "delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h \<noteq> None"
cInf_singleton: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type" assumes "class.conditionally_complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) {x::'b::type} = x"
Receive: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> (S @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)])"
ptensor_mat_add: fixes m1 :: "'b mat"   and m2 :: "'b mat"   and m3 :: "'b mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d1 d1"   and "m3 \<in> carrier_mat d2 d2" shows "ptensor_mat (m1 + m2) m3 = ptensor_mat m1 m3 + ptensor_mat m2 m3"