hcju/isabelleps huggingface dataset:
mC_C_mC_ZOC: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "mC_C theta c d s t P F" shows "mC_ZOC theta c d s t {} (P \<union> {{}}) (\<lambda>I. if I \<in> P then F I else {})"
aux1: fixes u :: "'a \<Rightarrow> real"   and R :: "('a \<Rightarrow> real) set"   and U :: "('a \<Rightarrow> real) set set" assumes "u \<in> R"   and "R \<in> \<R>"   and "U \<subseteq> \<R>"   and "u \<in> \<Union> U" shows "R \<subseteq> \<Union> U"
not_outside_1: fixes x :: "'b pmf"   and y :: "'b pmf" assumes "Pareto (SD \<circ> R) x y" shows "x \<in> lotteries_on alts"
zero_less_log_cancel_iff: fixes a :: "real"   and x :: "real" assumes "1 < a"   and "0 < x" shows "(0 < log a x) = (1 < x)"
reachable_nodes_def: fixes g :: "'a ugraph"   and r :: "'a" shows "reachable_nodes g r \<equiv> (edges g)\<^sup>* `` {r}"
coerce_errorT_abs: fixes x :: "'h\<cdot>'j error\<cdot>'i" shows "COERCE('h\<cdot>('i, 'j) errorT, 'e\<cdot>('f, 'g) errorT)\<cdot>(errorT_abs\<cdot>x) = errorT_abs\<cdot>(COERCE('h\<cdot>'j error\<cdot>'i, 'e\<cdot>'g error\<cdot>'f)\<cdot>x)"
box_equation: fixes a :: "'e"   and b :: "'e"   and c :: "'e" assumes "a = b"   and "a = c" shows "b = c"
of_nat_diff: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and n :: "nat"   and m :: "nat" assumes "class.semiring_1_cancel plus minus zero one times"   and "n \<le> m" shows "semiring_1.of_nat one plus zero (m - n) = minus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
inv'_4: fixes i :: "nat" assumes "i < m" shows "fs' ! i \<in> carrier_vec n"
rename_image_stable: fixes h :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "bij h" shows "rename h ` stable A = stable (h ` A)"
hext_trans: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and h :: "'b"   and h' :: "'b"   and h'' :: "'b" assumes "heap_base.hext typeof_addr h h'"   and "heap_base.hext typeof_addr h' h''" shows "heap_base.hext typeof_addr h h''"
cases: fixes a :: "'b"   and r :: "('b \<times> 'c) set" assumes "a \<in> Domain r" obtains a :: "'b" and b :: "'c" where "aa__ = a" and "(a, b) \<in> r"
res1: shows "max r1 r2 = real_of_int (sint (wmax w1 w2))"
gdd_index_options: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> point_set"   and "card ps = 2" shows "block_collection index ps = 0 \<or> int (block_collection index ps) = index"
ctor2_def: shows "ctor2 = Abs_IF2 \<circ> str_init2 undefined \<circ> F2map id Rep_IF1 Rep_IF2"
P0: shows "deg R (Cring_Poly.compose R f g) = deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g)"
pIH_1: shows "a \<noteq> (0::'b) \<or> p \<noteq> 0"
r_refl: fixes x :: "'b" shows "x \<sqsubseteq> x"
homeomorphic_compact: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and t :: "'d set" assumes "compact s"   and "continuous_on s f"   and "f ` s = t"   and "inj_on f s" shows "s homeomorphic t"
term_brackets_ConsI: fixes vs :: "nat list"   and t :: "dB"   and t' :: "dB"   and v :: "nat"   and t'' :: "dB" assumes "term_brackets vs t = Some t'"   and "term_bracket v t' = Some t''" shows "term_brackets (v # vs) t = Some t''"
sscan_scons: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and xs :: "'e stream"   and a :: "'d" shows "sscan f (x ## xs) a = f x a ## sscan f xs (f x a)"
distinct_set_conv_list: fixes xs :: "'b list"   and g :: "'b \<Rightarrow> 'c" assumes "distinct xs" shows "sum g (set xs) = sum_list (map g xs)"
P_def: fixes n :: "nat" shows "P n = set (primes_upto n)"
sets_eq_countable: fixes A :: "'d set"   and M :: "'d measure" assumes "countable A"   and "space M = A"   and "\<And>x. x \<in> A \<Longrightarrow> {x} \<in> sets M" shows "sets M = Pow A"
LAssThrow: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and r :: "nat \<times> char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P,E \<turnstile> \<langle>V:=Throw r,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
is_generalisation_of_def: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" shows "is_generalisation_of t' t i r = (t' = remove_guard_add_update t i r \<and> i < Arity t \<and> (\<exists>v. Eq (V (I i)) (L v) \<in> set (Guards t)) \<and> r \<notin> set (map fst (Updates t)))"
invar_Some: fixes h :: "'b hp" shows "invar (Some h) = php h"
sound_monitor: fixes i :: "nat"   and v :: "'a option list"   and \<pi> :: "(char list \<times> 'a list) prefix"   and \<sigma> :: "(char list \<times> 'a list) trace" assumes "(i, v) \<in> M \<pi>"   and "prefix_of \<pi> \<sigma>" shows "MFOTL.sat \<sigma> (map the v) i \<phi>"
one_app: fixes t1 :: "lam"   and t2 :: "lam"   and t' :: "lam" assumes " App t1 t2 \<longrightarrow>\<^sub>1 t'" shows "(\<exists>s1 s2. t' = App s1 s2 \<and>  t1 \<longrightarrow>\<^sub>1 s1 \<and>  t2 \<longrightarrow>\<^sub>1 s2) \<or> (\<exists>a s s1 s2. t1 = Lam [a].s \<and> t' = s1[a::=s2] \<and>  s \<longrightarrow>\<^sub>1 s1 \<and>  t2 \<longrightarrow>\<^sub>1 s2 \<and> a \<sharp> (t2, s2))"
assms_3: shows "v \<in> local.writes (\<pi> j)"
eFresh_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and s :: "'sort"   and X :: "'a"   and ys :: "'varSort"   and y :: "'var" assumes "igWls MOD s X" shows "eFresh MOD ys y (OK X) = igFresh MOD ys y X"
star_of_simps_2: fixes x :: "'a"   and y :: "'a" shows "star_of (x - y) = star_of x - star_of y"
zero: fixes n :: "nat" assumes "n \<notin> range g" shows "f n = (0::'a)"
mult_oner: fixes x :: "'a" shows "x ; 1' = x"
case_ivl_zero: fixes l1 :: "real"   and u1 :: "real"   and l2 :: "real"   and u2 :: "real" assumes "l1 \<le> u1"   and "l2 \<le> u2" obtains "l1 \<le> 0 \<and> 0 \<le> u1 \<and> l2 \<le> 0 \<and> 0 \<le> u2"   | "l1 \<le> 0 \<and> 0 \<le> u1 \<and> 0 \<le> l2"   | "l1 \<le> 0 \<and> 0 \<le> u1 \<and> u2 \<le> 0"   | "0 \<le> l1 \<and> l2 \<le> 0 \<and> 0 \<le> u2"   | "u1 \<le> 0 \<and> l2 \<le> 0 \<and> 0 \<le> u2"   | "u1 \<le> 0 \<and> u2 \<le> 0"   | "u1 \<le> 0 \<and> 0 \<le> l2"   | "0 \<le> l1 \<and> u2 \<le> 0"   | "0 \<le> l1 \<and> 0 \<le> l2"
IH: assumes "local.path g ns"   and "2 \<le> length ns"   and "2 \<le> length ns" shows "local.path g (butlast ns) \<and> last (butlast ns) \<in> set (predecessors g (last ns))"
assms: shows "(GREATEST m. \<not> is_zero_row_upt_k m k A) < i"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b msllist \<Rightarrow> 'a"   and msllist :: "'b msllist" shows "(case msllist of MSLNil \<Rightarrow> f1 | MSLCons x xa \<Rightarrow> f2 x xa) = (if msllist = MSLNil then f1 else f2 (un_MSLCons1 msllist) (un_MSLCons2 msllist))"
h3: shows "h2 \<turnstile> set_mode new_shadow_root_ptr new_mode \<rightarrow>\<^sub>h h3"
Un_left_commute: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "(A::'b set) \<union> ((B::'b set) \<union> (C::'b set)) = B \<union> (A \<union> C)"
G_nested_propos_Only_G: fixes \<phi> :: "'a ltl" shows "Only_G (\<^bold>G \<phi>)"
order_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.preorder (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)"   and "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) (y::'b)"   and "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (y::'b) (z::'b)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) (z::'b)"
contour_integral_trivial: fixes a :: "complex"   and f :: "complex \<Rightarrow> complex" shows "contour_integral (linepath a a) f = 0"
sup_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "sup_class.sup (sup_class.sup x y) z = sup_class.sup x (sup_class.sup y z)"
exhaust_sel: fixes llist :: "'a llist" obtains "llist = LNil"   | "llist = LCons (lhd llist) (ltl llist)"
finite_measure_axioms_def: fixes M :: "'c measure" shows "finite_measure_axioms M \<equiv> emeasure M (space M) \<noteq> top"
dgrad_setD: fixes t :: "'d"   and d :: "'d \<Rightarrow> nat"   and m :: "nat" assumes "t \<in> dgrad_set d m" shows "d t \<le> m"
A: shows "\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2 ;- M"
induct: fixes P :: "nat \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat list" assumes "\<And>uu_. P 0 uu_"   and "\<And>n. P (Suc n) []"   and "\<And>n m ms. P n ms \<Longrightarrow> P (Suc n) (m # ms)" shows "P a0 a1"
arr_iff_in_hom: fixes f :: "'a" shows "R.arr f = R.in_hom f (R.dom f) (R.cod f)"
subgroup_generated_group_carrier: shows "subgroup_generated G (carrier G) = G"
empty_set_less_eq_aux: fixes A :: "'a set" shows "({} \<sqsubseteq>' A) = finite A"
sum_mono_complex_lex: fixes K :: "'c set"   and f :: "'c \<Rightarrow> complex"   and g :: "'c \<Rightarrow> complex" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le>\<^sub>\<complex> g i" shows "sum f K \<le>\<^sub>\<complex> sum g K"
subst_pair_compose: fixes \<delta> :: "'c \<Rightarrow> ('a, 'd) Term.term"   and \<I> :: "'d \<Rightarrow> ('a, 'b) Term.term"   and d :: "('a, 'c) Term.term \<times> ('a, 'c) Term.term" shows "d \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> = d \<cdot>\<^sub>p \<delta> \<cdot>\<^sub>p \<I>"
iso_char: fixes f :: "'c \<times> 'c \<times> 'c" shows "C.VVV.iso f = (C.VxVxV.iso f \<and> C.VVV.arr f \<and> C.VVV.arr (C.VxVxV.inv f))"
assms: fixes x :: "'a" assumes "x \<in># A" shows "prime x"
are_all_operators_non_interfering_def: shows "are_all_operators_non_interfering \<equiv> rec_list True (\<lambda>op ops. (\<and>) (list_all (\<lambda>op'. \<not> are_operators_interfering op op') ops))"
hyps_3: assumes "open S"   and "connected S" shows "connected (S - F)"
assms_3: shows "\<forall>j\<in>J. \<zeta> j = \<sigma> j"
language_Reverse_nfa: fixes M :: "'b dfa" assumes "dfa M" shows "nfa.language (Reverse_nfa M) = rev ` dfa.language M"
preserves_hom: fixes f :: "'b \<times> 'b \<times> 'b"   and a :: "'b \<times> 'b \<times> 'b"   and b :: "'b \<times> 'b \<times> 'b" assumes "VVV.in_hom f a b" shows "\<guillemotleft>\<alpha>' f : HoVH a \<Rightarrow> HoHV b\<guillemotright>"
implies_itself: fixes x :: "'c" shows "- x \<squnion> x = top"
loop_spmf_while_cong: fixes guard :: "'a \<Rightarrow> bool"   and guard' :: "'a \<Rightarrow> bool"   and body :: "'a \<Rightarrow> 'a spmf"   and body' :: "'a \<Rightarrow> 'a spmf" assumes "guard = guard'"   and "\<And>s. guard' s \<Longrightarrow> body s = body' s" shows "loop_spmf.while guard body = loop_spmf.while guard' body'"
path: fixes l1 :: "'vtx"   and l2 :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and lbl :: "'lbl"   and l3 :: "'vtx" assumes "path l1 l2 xs"   and "lbl \<in>\<^sub>A weights l2 l3" shows "path l1 l3 (xs @ [(l2, lbl, l3)])"
map\<^sub>0_in_hhom: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<sigma>'.map\<^sub>0 a : map\<^sub>0 a \<rightarrow>\<^sub>D F.map\<^sub>0 a\<guillemotright>"
rebalance_middle_tree_inorder: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and t :: "'b btree"   and sub :: "'b btree"   and rs :: "('b btree \<times> 'b) list"   and k :: "nat"   and ls :: "('b btree \<times> 'b) list"   and sep :: "'b" assumes "split split"   and "height t = height sub"   and "case rs of [] \<Rightarrow> True | (rsub, rsep) # list \<Rightarrow> height rsub = height t" shows "inorder (split.rebalance_middle_tree k ls sub sep rs t) = inorder (Node (ls @ (sub, sep) # rs) t)"
card_of_cardSuc_finite: fixes A :: "'b set" shows "finite (Field (cardSuc |A|)) = finite A"
a: fixes Y :: "'a set"   and x :: "'a" assumes "Y = {}" shows "\<Squnion> ((\<cdot>) x ` Y) \<le> x \<cdot> \<Squnion> Y"
mult_Limit: fixes \<gamma> :: "V"   and x :: "V" assumes "Limit \<gamma>" shows "x * \<gamma> = \<Squnion> ((*) x ` elts \<gamma>)"
res_JNF: shows "P' \<in> carrier_mat m m \<and> Q' \<in> carrier_mat n n \<and> S' \<in> carrier_mat m n \<and> invertible_mat P' \<and> invertible_mat Q' \<and> isDiagonal_mat S' \<and> S' = P' * A' * Q'"
prime_harmonic_series_unbounded: shows "\<not> Bseq (\<lambda>n. \<Sum>p\<leftarrow>primes_upto n. 1 / real p)"
punit_pp_of_term: shows "punit.pp_of_term = (\<lambda>x. x)"
T_det: fixes P :: "'c process"   and Q :: "'c process" shows "T (P [+] Q) = T P \<union> T Q"
pp_inf_sup_p: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.pd_algebra inf less_eq less sup bot top uminus" shows "inf (uminus (uminus x)) (sup x (uminus x)) = x"
sqmatrix_mult_assoc: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix"   and C :: "('c, 'd) sqmatrix" shows "A \<cdot> B \<cdot> C = A \<cdot> (B \<cdot> C)"
config_embed: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s0 :: "'a list"   and qs :: "'a list" shows "Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed A) (fst (Partial_Cost_Model.embed A) s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs = return_pmf (config\<^sub>p A s0 qs)"
append_cols_def: fixes A :: "'b mat"   and B :: "'b mat" shows "A @\<^sub>c B = (A\<^sup>T @\<^sub>r B\<^sup>T)\<^sup>T"
valid_path_join_D2: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "valid_path (g1 +++ g2)"   and "pathfinish g1 = pathstart g2" shows "valid_path g2"
sigma_sets_singleton: fixes X :: "'c set"   and S :: "'c set" assumes "X \<subseteq> S" shows "sigma_sets S {X} = {{}, X, S - X, S}"
typing_cond_prot_impl_typing_cond_stateful: fixes \<P> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list" assumes "typing_cond_prot_stateful \<P>"   and "\<A> \<in> \<P>" shows "typing_cond\<^sub>s\<^sub>s\<^sub>t (unlabel \<A>)"
src_simps_1: fixes F :: "('a E.term, 'a) arr" assumes "arr F" shows "Dom (src F) = E.Src (Dom F)"
simps_1: fixes z :: "octo" shows "Ree (Octonions.cnj z) = Ree z"
fps_euclidean_size_def: fixes f :: "'a fps" shows "euclidean_size f = (if f = 0 then 0 else 2 ^ subdegree f)"
intro: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "module_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1"   and "module_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2" shows "module_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2"
extends_subst_extends: fixes \<sigma> :: "'a \<Rightarrow> 'b option"   and \<tau> :: "'a \<Rightarrow> 'b option"   and x :: "'a" assumes "extends_subst \<sigma> \<tau>"   and "x \<in> dom \<sigma>" shows "\<tau> x = \<sigma> x"
diff_fract_def: fixes q :: "'b fract"   and r :: "'b fract" shows "q - r = q + - r"
integ_exis: shows "\<forall>(k, \<gamma>)\<in>two_chain_boundary two_chain. line_integral_exists F {i} \<gamma>"
lt_monomial: fixes c :: "'c"   and t :: "'a" assumes "c \<noteq> (0::'c)" shows "punit.lt (monomial c t) = t"
chine_simps_3: shows "C.cod f.chine = f.cod.apex"
map_pmf_rep_eq: fixes f :: "'e \<Rightarrow> 'd"   and M :: "'e pmf" shows "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
invertible_lf_def: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" shows "invertible_lf s1 s2 f \<equiv> linear s1 s2 f \<and> invertible_lf_axioms f"
LIMSEQ_le_const2: fixes X :: "nat \<Rightarrow> 'c"   and x :: "'c"   and a :: "'c" assumes "X \<longlonglongrightarrow> x"   and "\<exists>N. \<forall>n\<ge>N. X n \<le> a" shows "x \<le> a"
column_QR_decomposition: fixes A :: "((real, 'a) vec, 'b) vec"   and k :: "'a" assumes "rank A = ncols A" shows "column k (Gram_Schmidt_matrix A) = column k A - (\<Sum>x\<in>{column i (fst (QR_decomposition A)) |i. i < k}. (x \<bullet> column k A / (x \<bullet> x)) *\<^sub>R x)"
Top_in_Lin: fixes L :: "(alt \<times> alt) set"   and S :: "alt set" assumes "L \<in> Lin" shows "Top S L \<in> Lin"
split_sel_asm: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and prod :: "'d \<times> 'e" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<not> (prod = (fst prod, snd prod) \<and> \<not> P (f (fst prod) (snd prod))))"
mem_PiCD: fixes x :: "'c \<Rightarrow>\<^sub>C 'd"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd set" assumes "x \<in> PiC I X" shows "apply_bcontfun x \<in> Pi I X"
trancl_into_trancl2: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set"   and c :: "'e" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>+" shows "(a, c) \<in> r\<^sup>+"
run_sdrop: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and r :: "(nat \<times> 'd) stream"   and p :: "nat \<times> 'd"   and k :: "nat" assumes "grun A w V r p" shows "grun A w V (sdrop k r) (gtarget (stake k r) p)"
isomorphic_implies_hpar_2: fixes f :: "'b"   and f' :: "'b" assumes "f \<cong> f'" shows "ide f'"
freshTauTransition: fixes P :: "pi"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
top_edge_smooth: shows "(\<lambda>x. twoC (x, 1)) piecewise_C1_differentiable_on {0..1}"
Crypt_imp_keysFor: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "K \<in> symKeys"   and "Crypt K X \<in> H" shows "K \<in> keysFor H"
mono_limit: fixes f :: "'b set \<Rightarrow> 'b set" assumes "mono f" shows "mono (limit f)"
inj_onD: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d"   and y :: "'d" assumes "inj_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
dom_closed: fixes f :: "'a" assumes "left \<omega> f" shows "left \<omega> (local.dom f)"
base: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and s :: "'a"   and S :: "'a set" assumes "class.ab_group_add plus zero minus uminus"   and "s \<in> insert zero S" shows "s \<in> ab_group_add.group_closure zero minus S"
ring_ops_def: fixes ops :: "'c arith_ops_record"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "ring_ops ops R \<equiv> ((bi_unique R \<and> right_total R) \<and> R (arith_ops_record.zero ops) (0::'d) \<and> R (arith_ops_record.one ops) (1::'d) \<and> (R ===> R ===> R) (arith_ops_record.plus ops) (+)) \<and> ((R ===> R ===> R) (arith_ops_record.minus ops) (-) \<and> (R ===> R) (arith_ops_record.uminus ops) uminus) \<and> (R ===> R ===> R) (arith_ops_record.times ops) (*) \<and> (R ===> R ===> (=)) (=) (=) \<and> Domainp R = arith_ops_record.DP ops"
choose_arc_below: fixes x :: "'a" shows "choose_arc x \<le> - - x"
graph_def: fixes X :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "graph X f = {(x, f x) |x. x \<in> X}"
DERIV_ident: fixes F :: "'b filter" shows "((\<lambda>x. x) has_field_derivative (1::'b)) F"
negate_dfa_def: shows "negate_dfa = (\<lambda>(t, a). (t, map Not a))"
WT_gpvD_2: fixes \<Gamma> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and out :: "'d"   and c :: "('f, 'd, 'e) rpv"   and input :: "'e" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out" shows "\<Gamma> \<turnstile>g c input \<surd>"
mIH_1: fixes x :: "nat"   and y :: "nat \<times> nat"   and ya :: "nat"   and n2 :: "nat ifex" assumes "(x, y) = pm_pth (dpm bdd) n"   and "(fst (snd (pm_pth (dpm bdd) n)), ya) = y"   and "Rmi_g (fst (snd (pm_pth (dpm bdd) n))) n2 bdd" shows "t = n2"
finite_measure_def: fixes M :: "'d measure" shows "finite_measure M \<equiv> sigma_finite_measure M \<and> finite_measure_axioms M"
vsubst_eq_swap: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and y1 :: "'var"   and y2 :: "'var"   and ys :: "'varSort" assumes "good X"   and "y1 = y2 \<or> fresh ys y1 X" shows "X #[y1 // y2]_ys = X #[y1 \<and> y2]_ys"
ground_heads_var_nonempty: fixes x :: "'v" shows "ground_heads_var x \<noteq> {}"
tm_shift_eq_steps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and A :: "(action \<times> nat) list"   and stp :: "nat"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list"   and off :: "nat" assumes "steps0 (s, l, r) A stp = (s', l', r')"   and "s' \<noteq> 0" shows "steps (s + off, l, r) (shift A off, off) stp = (s' + off, l', r')"
u_def: shows "u = (\<lambda>n. dist basepoint ((f ^^ n) basepoint))"
bij_imp_permutes: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "bij_betw p S S"   and "\<And>x. x \<notin> S \<Longrightarrow> p x = x" shows "p permutes S"
step_def: fixes s :: "'c list"   and r :: "'c"   and a :: "nat \<times> nat list" shows "step s r a = (let (k, sws) = a in mtf2 k r (swaps sws s))"
add_0_right: fixes a :: "'a" shows "a + (zero_class.zero::'a) = a"
3: shows "trace (ioa.asig A) e_B = (if a \<in> externals (ioa.asig A) then Cons a (trace (ioa.asig A) e_B') else trace (ioa.asig A) e_B')"
splus_zero: fixes v :: "'a" shows "local.punit.splus (0::'a) v = v"
INF_mono: 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 B :: "'d set"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. less_eq (f n) (g m)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
round_robin_reschedule_invar_Some: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and I :: "(('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set) set"   and t0 :: "'b"   and queue :: "'b list"   and n0 :: "nat"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and x' :: "'a"   and m' :: "'c"   and \<sigma>' :: "'b list \<times> nat" assumes "multithreaded_base.deterministic final r convert_RA I"   and "multithreaded_base.round_robin_reschedule final r convert_RA t0 queue n0 s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"   and "Round_Robin.round_robin_invar (queue, n0) (dom (thr s))"   and "t0 \<in> set queue"   and "s \<in> I" shows "Round_Robin.round_robin_invar \<sigma>' (dom (thr s) \<union> {t. \<exists>x m. NewThread t x m \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>})"