hcju/isabelleps huggingface dataset:
index_last: fixes xs :: "'e list" assumes "xs \<noteq> []"   and "distinct xs" shows "index xs (last xs) = length xs - 1"
hta_has_idx_f_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_has_idx_f H \<equiv> hta_idx_f H \<noteq> None"
iH: shows "length ss = (\<Sum> | case  of (x, y) \<Rightarrow> x \<in> set init \<and> y \<in> set init \<and> x < y. case  of (x, y) \<Rightarrow> ALG_P ss x y init)"
well\<^sub>h_M: shows "local.well_base\<^sub>h M"
prefix_subsume: fixes v :: "'b llist"   and w :: "'b llist"   and u :: "'b llist" assumes "v \<le> w"   and "u \<le> w"   and "llength v \<le> llength u" shows "v \<le> u"
A: fixes x :: "'a" assumes "cont g" shows "cont (\<lambda>y. y \<lbrakk>A\<rbrakk> g x)"
le_add_same_cancel2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># b + a) = (\<Empt> \<subseteq># b)"
ide_char: fixes f :: "nat" shows "J.ide f = (f \<in> {m})"
msb_word_iff_sless_0: fixes w :: "'b word" shows "msb w = (w <s 0)"
undef_vec_def: fixes i :: "nat" shows "undef_vec i \<equiv> [] ! i"
divide_minus_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 "divide a (uminus b) = uminus (divide a b)"
ih: fixes ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "{#hsize ta, hsize sa#} < {#hsize t, hsize s#}"   and "ground ta"   and "ground sa" shows "ta >\<^sub>t sa \<or> sa >\<^sub>t ta \<or> ta = sa"
iszero_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.iszero zero zero"
label_2: fixes i :: "'a"   and x :: "'a" assumes "i \<in> Basis"   and "x \<in> cbox (0::'a) One \<and> x \<bullet> i = 0" shows "label x i = 0"
monadic_filter_rev_aux_def: fixes a :: "'a list"   and P :: "'a \<Rightarrow> bool nres"   and l :: "'a list" shows "monadic_filter_rev_aux a P l \<equiv> (rec\<^sub>T D. (\<lambda>(l, a). case l of [] \<Rightarrow> return a | v # l \<Rightarrow> P v \<bind> (\<lambda>c. let a = if c then v # a else a in D (l, a)))) (l, a)"
List_hd_not_in_tl: fixes h :: "ref \<Rightarrow> ref"   and p :: "ref"   and ps :: "ref list" assumes "List (h p) h ps" shows "p \<notin> set ps"
ys1: shows "ys1 = drop (Suc n div 2) xs"
Zp_char_0: fixes p :: "int"   and n :: "int" assumes "padic_integers p"   and "0 < n" shows "[n] \<cdot>\<^bsub>padic_int p\<^esub> \<one>\<^bsub>padic_int p\<^esub> \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"
H_ia_least_H: shows "H $ ia $ (LEAST n. H $ ia $ n \<noteq> (0::'a)) \<noteq> (0::'a)"
sum_power2_ge_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (plus (power.power one times x 2) (power.power one times y 2))"
L_transform_Tree\<^sub>\<alpha>_Pred\<^sub>\<alpha>: fixes f :: "'d"   and \<phi> :: "'b" shows "L_transform_Tree\<^sub>\<alpha> (FL_Formula.Pred\<^sub>\<alpha> f \<phi>) = Formula.Act\<^sub>\<alpha> (Eff f) (Formula.Pred\<^sub>\<alpha> \<phi>)"
fun_relD: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'f"   and A :: "('c \<times> 'e) set"   and B :: "('d \<times> 'f) set"   and x :: "'c"   and x' :: "'e" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
store_delete_ids_imply_messages_same: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e1 :: "'b"   and mo :: "'a"   and ir :: "'a"   and e2 :: "'b" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e2) \<in> set (node_deliver_messages xs)" shows "e1 = e2"
sel_2: fixes x1 :: "bool"   and x2 :: "nat" shows "var (Lit x1 x2) = x2"
notDom: fixes x :: "'a"   and A :: "'a \<Rightarrow> 'b option" assumes "x \<in> dom A" shows "A x \<noteq> \<bottom>"
simps: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'e"   and a2 :: "'d" shows "r\<inverse>\<inverse> a1 a2 = (\<exists>a b. a1 = b \<and> a2 = a \<and> r a b)"
p_coprime_right_int: fixes p :: "nat"   and a :: "int" assumes "residues_prime p" shows "coprime a (int p) = (\<not> int p dvd a)"
bot_entailed: shows "fst ` snd (lhd Ns) \<Turnstile>\<inter>\<G> {B}"
rep_eq: fixes x :: "'k"   and xa :: "'t \<Rightarrow>\<^sub>0 'c" shows "lookup (proj_poly x xa) = (\<lambda>t. lookup xa (term_of_pair (t, x)))"
cases: fixes x :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" obtains cl :: "(String.literal \<times> String.literal) set" and uu_ :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" where "x = (cl, uu_)"
assms_4: shows "ys \<noteq> zs \<or> y \<noteq> z"
conga_obtuse__obtuse: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "Obtuse A B C"   and "A B C CongA D E F" shows "Obtuse D E F"
VDMCall: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" assumes "({A} \<union> G) \<rhd> body : A" shows "G \<rhd> Call : A"
submonoid_pr_ideal: shows "submonoid (R\<setminus>I) R (\<cdot>) \<one>"
hconf_def: fixes P :: "'b prog"   and h :: "heap" shows "P \<turnstile> h \<surd> \<equiv> (\<forall>a obj. h a = \<lfloor>obj\<rfloor> \<longrightarrow> P,h \<turnstile> obj \<surd>) \<and> preallocated h"
val_noninf_n_val_noninf: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "(v x \<noteq> \<infinity>) = (n_val K v x \<noteq> \<infinity>)"
rep_eq: fixes x :: "'c set"   and xa :: "('c, 'd) fmap" shows "($$) (fmdrop_set x xa) = map_drop_set x (($$) xa)"
SigmaE2: fixes a :: "'a"   and b :: "'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "(a, b) \<in> Sigma A B" obtains "a \<in> A" and "b \<in> B a"
close_antimono: fixes g' :: "graph"   and p :: "nat"   and g :: "graph"   and u :: "nat"   and v :: "nat" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g"   and "u \<in> \<V> g"   and "finalVertex g u"   and "close g' u v" shows "close g u v"
check_extension_l_check_extension: fixes A :: "(nat, (char list list \<times> int) list) fmap"   and B :: "(nat, int mpoly) fmap"   and r :: "(char list list \<times> int) list"   and r' :: "int mpoly"   and i :: "nat"   and i' :: "nat"   and \<V> :: "char list set"   and \<V>' :: "nat set"   and x :: "char list"   and x' :: "nat"   and spec :: "(char list list \<times> int) list" assumes "(A, B) \<in> \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"   and "(r, r') \<in> sorted_poly_rel O mset_poly_rel"   and "(i, i') \<in> nat_rel"   and "(\<V>, \<V>') \<in> \<langle>var_rel\<rangle>set_rel"   and "(x, x') \<in> var_rel" shows "check_extension_l spec A \<V> i x r \<le> \<Down> {(st, b). (\<not> is_cfailed st) = b \<and> (is_cfound st \<longrightarrow> spec = r)} (check_extension B \<V>' i' x' r')"
interaction_bound_map_gpv'_le: shows "interaction_bound consider (map_gpv' f g h gpv) \<le> interaction_bound (consider \<circ> g) gpv"
succs_shift: fixes p :: "int"   and n :: "int"   and P :: "instr list" shows "(p - n \<in> succs P 0) = (p \<in> succs P n)"
map_fst_auxlist_drop: fixes t :: "nat" assumes "t \<in> set (map fst (drop (length done) auxlist))" shows "t \<le> nt"
simps: fixes "consider" :: "'b \<Rightarrow> bool"   and gpv :: "('d, 'b, 'e) gpv"   and n :: "enat" shows "interaction_bounded_by consider gpv n = (interaction_bound consider gpv \<le> n)"
that: fixes xs :: "'a list" assumes "walk xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "thesis"
Carmichael_prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "Carmichael (p ^ k) = (if p = 2 \<and> 2 < k then 2 ^ (k - 2) else p ^ (k - 1) * (p - 1))"
terminal_is_nil: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and v :: "'b \<times> nat list" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "v |\<in>| Solved_Task.vertices freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "(Abstract_Task.outPorts anyP antecedent consequent (Solved_Task.nodeOf freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions v) = {||}) = (snd v = [])"
remzero_replicate: fixes n :: "nat"   and l :: "bit list" shows "remzero (replicate n \<zero> @ l) = remzero l"
singleton_sub: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "{a} \<subseteq> A"
diff_monom: fixes a :: "'a"   and n :: "nat"   and b :: "'a" shows "Polynomial.monom a n - Polynomial.monom b n = Polynomial.monom (a - b) n"
fps_conv_radius_cmult_left: fixes c :: "'a"   and f :: "'a fps" assumes "c \<noteq> (0::'a)" shows "fps_conv_radius (fps_const c * f) = fps_conv_radius f"
Crypt_valid_pubEK: fixes C :: "agent"   and EKi :: "nat"   and evs :: "event list" assumes "Crypt (priSK RCA) \<lbrace>Agent C, Key EKi, onlyEnc\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> set_cr" shows "EKi = pubEK C"
Eq_Zero_D: fixes H :: "fm set"   and t :: "tm"   and u :: "tm"   and A :: "fm" assumes "H \<turnstile> t EQ Zero"   and "H \<turnstile> u IN t" shows "H \<turnstile> A"
P3: shows "A B Perp D B \<and> A B OS C D"
Rep_process_cases: fixes y :: "('a list \<times> 'a set) set \<times> 'a list set" assumes "y \<in> process_set" obtains x :: "'a process" where "y = Rep_process x"
hashcode_option_def: fixes opt :: "'a option" shows "hashcode opt = (case opt of None \<Rightarrow> 0 | Some a \<Rightarrow> hashcode a + 1)"
mergesort_1: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "sorted_wrt (\<lambda>x y. f x \<le> f y) (mergesort f xs)"
hseq_leg\<^sub>0: shows "hseq t\<^sub>0 \<omega>.chine"
that: fixes ns :: "'node list" assumes "g \<turnstile> defNode g v-ns\<rightarrow>m"   and "EntryPath g ns" shows "thesis"
assms_1: shows "\<forall>(t, t')\<in>set D. (fv t \<union> fv t') \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"
simps_4: fixes t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (insert\<langle>t,t'\<rangle>) = fv t \<union> fv t'"
countable_TE_wtFsym: shows "countable (Collect TE_wtFsym)"
in_angle_def: fixes cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and betw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "TarskiAbsolute cong betw" shows "TarskiAbsolute.in_angle betw p a b c = (b \<noteq> a \<and> b \<noteq> c \<and> p \<noteq> b \<and> (\<exists>x. betw a x c \<and> x \<noteq> a \<and> x \<noteq> c \<and> TarskiAbsolute.on_ray betw p b x))"
triangle_right: shows "(g \<star> \<xi>) \<cdot> (\<zeta> \<star> g) = g"
centre_in_ball: fixes x :: "'c"   and e :: "real" shows "(x \<in> ball x e) = (0 < e)"
foo3a: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and c :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "applied_rule_rev Cp x (a @ [b] @ c) = \<lfloor>b\<rfloor>"   and "r \<in> set c"   and "b \<notin> set c" shows "x \<notin> dom (Cp r)"
exhaust: fixes y :: "tree" obtains "y = Leaf"   | x21 :: "tree" and x22 :: "tree" where "y = Branching x21 x22"
assms_1: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
R2: fixes p :: "'a"   and x :: "'a"   and q :: "'a" assumes "H p x q" shows "x \<le> Ref p q"
elims: fixes x :: "prop_rule"   and y :: "form list" assumes "consequent x = y" obtains "x = allI" and "y = [ForallX (P x)]"   | "x = allE" and "y = [P t]"   | "x = impI" and "y = [imp X Y]"   | "x = impE" and "y = [Y]"
lens_put_irr1: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and v :: "'c" assumes "X \<bowtie> Y" shows "get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>"
that: fixes c :: "'a vec" assumes "mat_of_cols nr ss *\<^sub>v v = A *\<^sub>v c"   and "dim_vec c = nc" shows "thesis"
oneBIT_step0x: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a" assumes "x \<noteq> y"   and "x \<in> {x0, y0}"   and "y \<in> {x0, y0}" shows "type0 [x0, y0] x y \<bind> (\<lambda>s. BIT_step s x \<bind> (\<lambda>(a, is'). return_pmf (step (fst s) x a, is'))) = type0 [x0, y0] x y"
Pbij_composeD: fixes l1 :: "Location"   and l3 :: "Location"   and \<beta> :: "(Location \<times> Location) set"   and \<gamma> :: "(Location \<times> Location) set" assumes "(l1, l3) \<in> Pbij_compose \<beta> \<gamma>" shows "\<exists>l2. (l1, l2) \<in> \<beta> \<and> (l2, l3) \<in> \<gamma>"
rFail_def: shows "rFail = (\<lambda>_. Fail)"
iT_div_1: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "[\<dots>n] \<oslash> k = [\<dots>n div k]"
inter_filter: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool" assumes "finite A" shows "sum g {x \<in> A. P x} = (\<Sum>x\<in>A. if P x then g x else (0::'d))"
sep_add_ac_2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "plus x y = plus y x"
coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<le> c" shows "inf a b \<le> c"
seqE: fixes g :: "'s"   and f :: "'s" assumes "S.seq g f" obtains "S.arr f" and "S.arr g" and "S.dom g = S.cod f"
llength_lfilter_ile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "llength (lfilter P xs) \<le> llength xs"
wf_graph_union_edges: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "wf_graph \<lparr>nodes = V, edges = E'\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E \<union> E'\<rparr>"
mem_set_mr_cos: fixes N :: "'a set"   and x :: "'a set" assumes "submodule R M N"   and "x \<in> set_mr_cos M N" shows "\<exists>m\<in>carrier M. x = m \<uplus>\<^bsub>M\<^esub> N"
tla_defs_4: fixes s :: "nat \<Rightarrow> 'a" shows "first s \<equiv> s 0"
mapping_Trie_def: shows "mapping_Trie = Mapping_IMPL"
formal_in_THE: 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 r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and ins :: "'c list"   and outs :: "'c list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs" shows "(THE ins. \<exists>outs. (p, ins, outs) \<in> set procs) = ins"
inv_cancelL_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> Units R"   and "y \<in> carrier R"   and "z \<in> carrier R"   and "y = z \<otimes> x" shows "inv x \<otimes> y = z"
gt_Max_notin: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}"   and "Max A < x" shows "x \<notin> A"
in_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b list"   and b :: "'c list" shows "list_all2 R a b = (\<exists>z. z \<in> {x. set x \<subseteq> {(x, y). R x y}} \<and> map fst z = a \<and> map snd z = b)"
from_pre_digraph_def: fixes G :: "('c, 'd) pre_digraph" shows "from_pre_digraph G \<equiv> \<lparr>Graph.verts = pre_digraph.verts G, arcs = arcs_ends G\<rparr>"
verit_prod_simplify_1: fixes a :: "'a" shows "(1::'a) * a = a"
help1: shows "btw = fst (splitAt (hd vs) btw) @ [hd vs] @ snd (splitAt (hd vs) btw)"
induct: fixes P :: "('c, 'd) SkewBinomialTree list \<Rightarrow> ('c, 'd) SkewBinomialTree list \<Rightarrow> bool"   and a0 :: "('c, 'd) SkewBinomialTree list"   and a1 :: "('c, 'd) SkewBinomialTree list" assumes "\<And>bq. P [] bq"   and "\<And>v va. P (v # va) []"   and "\<And>t1 bq1 t2 bq2. \<lbrakk>rank t1 < rank t2 \<Longrightarrow> P bq1 (t2 # bq2); \<lbrakk>\<not> rank t1 < rank t2; rank t2 < rank t1\<rbrakk> \<Longrightarrow> P (t1 # bq1) bq2; \<lbrakk>\<not> rank t1 < rank t2; \<not> rank t2 < rank t1\<rbrakk> \<Longrightarrow> P bq1 bq2\<rbrakk> \<Longrightarrow> P (t1 # bq1) (t2 # bq2)" shows "P a0 a1"
borel_measurable_const: fixes c :: "'d"   and M :: "'c measure" shows "(\<lambda>x. c) \<in> borel_measurable M"
lookup_monom_mult_plus: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and c :: "'d"   and t :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.monom_mult pair_of_term term_of_pair c t p) (term_powerprod.splus pair_of_term term_of_pair t v) = c * lookup p v"
s1: shows "space M = space (Pi\<^sub>M J (\<lambda>_. N))"
S5_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "S5 rmhist p = S rmhist True True False clkB rpcB histB histB p"
inj_selfbutter_upto_phase: fixes x :: "'b"   and y :: "'b" assumes "selfbutter x = selfbutter y" shows "\<exists>c. cmod c = 1 \<and> x = c *\<^sub>C y"
lookup_tuple_in': fixes as :: "'a option list" shows "Mapping.lookup tuple_in' as = safe_max (fst ` {tas \<in> ts_tuple_rel (set (linearize data_in)). valid_tuple tuple_since' tas \<and> as = snd tas})"
folding: fixes C :: "'a set" assumes "folding.chamber C"   and "C \<in> f \<turnstile> X" shows "\<exists>!D. folding.chamber D \<and> D \<notin> f \<turnstile> X \<and> f ` D = C"
prj_simps_3: shows "C.cod \<mu>\<nu>.prj\<^sub>1 = \<mu>.apex"
hn_refine_alt: fixes Fpre :: "assn"   and c :: "'g Heap"   and Fpost :: "assn"   and P :: "'h \<Rightarrow> 'g \<Rightarrow> assn"   and m :: "'h nres" shows "hn_refine Fpre c Fpost P m \<equiv> nofail m \<longrightarrow> <Fpre> c <\<lambda>r. hn_rel P m r * Fpost>\<^sub>t"
rel_generat_reflI: fixes x :: "('h, 'i, 'j) generat"   and A :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and C :: "'j \<Rightarrow> 'j \<Rightarrow> bool" assumes "\<And>y. y \<in> generat_pures x \<Longrightarrow> A y y"   and "\<And>out. out \<in> generat_outs x \<Longrightarrow> B out out"   and "\<And>cont. cont \<in> generat_conts x \<Longrightarrow> C cont cont" shows "rel_generat A B C x x"
trancl_induct: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (a, y) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>+; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
asAbs_pick: fixes A :: "('i, 'j, 'varSort, 'var, 'k) qAbs set" assumes "goodAbs A" shows "asAbs (pick A) = A"
index_f: shows "bij_betw index_f (UNIV // eq_app_right language) (hfset (ord_of min_states))"
tendsto_componentwise1: fixes b :: "'e \<Rightarrow> 'd \<Rightarrow>\<^sub>L 'f"   and a :: "'d \<Rightarrow>\<^sub>L 'f"   and F :: "'e filter" assumes "\<And>j. j \<in> Basis \<Longrightarrow> ((\<lambda>n. blinfun_apply (b n) j) \<longlongrightarrow> blinfun_apply a j) F" shows "(b \<longlongrightarrow> a) F"
comm_monoid_axioms: shows "Group.comm_monoid (add_monoid G)"
comparator_proj_comp: fixes cmp :: "(('c, 'd) pp \<times> nat) nat_term_order" shows "comparator (proj_comp cmp)"
span_explicit: fixes b :: "'b set" shows "span b = {\<Sum>a\<in>t. r a *s a |t r. finite t \<and> t \<subseteq> b}"
distr_plus_real: fixes M :: "real measure"   and f :: "real \<Rightarrow> ennreal"   and c :: "real" assumes "has_density M lborel f" shows "has_density (distr M borel ((+) c)) lborel (\<lambda>x. f (x - c))"
QTrans: shows "Q \<longmapsto> \<alpha> \<prec> Q'"
numeral_eq_ereal: fixes w :: "num" shows "numeral w = ereal (numeral w)"
add_mset_lt_left_lt: fixes a :: "'b"   and b :: "'b"   and A :: "'b multiset" assumes "a < b" shows "add_mset a A < add_mset b A"
isFault_simps_1: fixes s :: "'e" shows "isFault (Normal s) = False"
zip_assoc: fixes xs :: "'a list"   and ys :: "'b list"   and zs :: "'c list" shows "zip xs (zip ys zs) = map2 (\<lambda>(x, y) z. (x, y, z)) (zip xs ys) zs"
min_set_Union_image: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c fset set" assumes "finite X" shows "min_set (\<Union>x\<in>X. min_set (f x)) = min_set (\<Union> (f ` X))"
callee_invariant: fixes y :: "'b"   and s' :: "'s"   and s :: "'s"   and x :: "'a" assumes "(y, s') \<in> set_spmf (left s x)"   and "I s"   and "x \<in> outs_\<I> \<I>l" shows "I s'"
integrable_continuous_interval: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
simps_5: fixes A :: "'b ptrm"   and B :: "'b ptrm" shows "ptrm_fvs (PPair A B) = ptrm_fvs A \<union> ptrm_fvs B"
einterpret_euler_fas1: fixes CX :: "floatarith list"   and vs :: "real list"   and X0 :: "floatarith list"   and h :: "floatarith" assumes "einterpret CX vs \<in> Csafe"   and "length X0 = DIM('c)"   and "length CX = DIM('c)"   and "DIM('c) \<le> length vs" shows "einterpret (take DIM('c) (euler_fas X0 h CX)) vs = einterpret X0 vs + interpret_floatarith h vs *\<^sub>R ode (einterpret X0 vs) + ((interpret_floatarith h vs)\<^sup>2 / 2) *\<^sub>R blinfun_apply (ode_d 0 (einterpret CX vs) (ode (einterpret CX vs))) (ode (einterpret CX vs))"
emb_transfer: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "rel_fun (rel R) (rel' R) emb emb"
bind_mono: fixes B :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'c option"   and C :: "'c \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'f option" assumes "monotone (flatf_ord None) option_ord B"   and "\<And>y. monotone (flatf_ord None) option_ord (C y)" shows "monotone (flatf_ord None) option_ord (\<lambda>f. B f \<bind> (\<lambda>y. C y f))"