hcju/isabelleps huggingface dataset:
SUP_ereal_sum_directed: fixes I :: "'a set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "\<And>N i j. \<lbrakk>N \<subseteq> A; i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> \<exists>k\<in>I. \<forall>n\<in>N. f n i \<le> f n k \<and> f n j \<le> f n k"   and "\<And>n i. \<lbrakk>i \<in> I; n \<in> A\<rbrakk> \<Longrightarrow> 0 \<le> f n i" shows "(SUP i\<in>I. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. Sup (f n ` I))"
heap_wellformed_induct: fixes h :: "(_) heap"   and P :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "\<And>parent. (\<And>children child. \<lbrakk>h \<turnstile> Shadow_DOM.get_child_nodes parent \<rightarrow>\<^sub>r children; child \<in> set children\<rbrakk> \<Longrightarrow> P (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 child)) \<Longrightarrow> P parent" shows "P ptr"
is_monoid: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
Un_Int_eq_1: fixes S :: "'k set"   and T :: "'k set" shows "(S \<union> T) \<inter> S = S"
dom_comp: fixes g :: "'a2"   and f :: "'a2" assumes "C2.seq g f" shows "C2.dom (g \<cdot>\<^sub>2 f) = C2.dom f"
elims_3: fixes x :: "'a list" assumes "\<not> incr x" obtains xa :: "'a" and y :: "'a" and xs :: "'a list" where "x = xa # y # xs" and "\<not> (xa < y \<and> incr (y # xs))"
tauFreshDerivative: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "x \<sharp> P" shows "x \<sharp> P'"
nxt_def: fixes \<gamma> :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.nxt \<gamma> \<equiv> \<lambda>t n. \<gamma> t (Suc n)"
wf_mstate_def: fixes \<phi> :: "'a MFOTL.formula"   and \<pi> :: "(char list \<times> 'a list) prefix"   and R :: "'a list set"   and st :: "'a mstate" shows "wf_mstate \<phi> \<pi> R st = (mstate_n st = MFOTL.nfv \<phi> \<and> (\<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> mstate_i st = Monitor.progress \<sigma> \<phi> (plen \<pi>) \<and> wf_mformula \<sigma> (plen \<pi>) (mstate_n st) R (mstate_m st) \<phi>))"
weakly_bisimilar_is_S_transform_weak_bisimulation: shows "S_transform.is_weak_bisimulation (\<approx>\<cdot>)"
quasisym_not_less: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b" assumes "comparator cmp"   and "cmp a b \<noteq> Less"   and "cmp b a \<noteq> Less" shows "cmp a b = Equiv"
simps_2: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "General.filter_min_aux rel (x # xs) ys = (if \<exists>y\<in>set xs \<union> set ys. rel y x then General.filter_min_aux rel xs ys else General.filter_min_aux rel xs (x # ys))"
assms_1: shows "\<forall>i<length ys. snd (f (xs ! i) (ys ! i))"
member_rec_2: fixes y :: "'f" shows "List.member [] y = False"
ex_finite_subset_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'a set \<Rightarrow> bool" shows "(\<exists>B. finite B \<and> B \<subseteq> f ` A \<and> P B) = (\<exists>B. finite B \<and> B \<subseteq> A \<and> P (f ` B))"
supto_snoc: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>j) = supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>j - 1) :@ [:MkI\<cdot>j:]"
Compl1_set_less_eq_aux_conv_Compl_set_less_eq_aux: fixes xs :: "'a list"   and ys :: "'a list" assumes "finite UNIV"   and "sorted xs"   and "distinct xs"   and "sorted ys"   and "distinct ys" shows "(- set xs \<sqsubseteq>' set ys) = Compl_set_less_eq_aux None xs ys"
inext_singleton: fixes n :: "nat"   and a :: "nat" shows "inext n {a} = n"
elims: fixes x :: "'b aexp"   and y :: "nat" assumes "height x = y" obtains l2 :: "value" where "x = L l2" and "y = 1"   | v2 :: "'b" where "x = V v2" and "y = 1"   | e1 :: "'b aexp" and e2 :: "'b aexp" where "x = Plus e1__ e2__" and "y = 1 + max (height e1__) (height e2__)"   | e1 :: "'b aexp" and e2 :: "'b aexp" where "x = Minus e1a__ e2a__" and "y = 1 + max (height e1a__) (height e2a__)"   | e1 :: "'b aexp" and e2 :: "'b aexp" where "x = Times e1 e2" and "y = 1 + max (height e1) (height e2)"
complex_mod_sqrt_Re_mult_cnj: fixes z :: "complex" shows "cmod z = sqrt (Re (z * cnj z))"
progress: fixes n :: "nat"   and a :: "nat \<times> form"   and list :: "(nat \<times> form) list" assumes "infinite (deriv s)" shows "snd (f n) = a # list \<longrightarrow> (\<exists>zs'. snd (f (Suc n)) = list @ zs')"
max_zero_streak_def: fixes xs :: "16 word list" shows "max_zero_streak xs \<equiv> foldr (\<lambda>xs. max (length xs)) (goup_by_zeros xs) 0"
bounded_linear_axioms_def: fixes f :: "'c \<Rightarrow> 'd" shows "bounded_linear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
canonical_empty_zone: fixes n :: "nat"   and v :: "'b \<Rightarrow> nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c"   and "canonical M n" shows "([M]\<^bsub>v,n\<^esub> = {}) = (\<exists>i\<le>n. M i i < \<one>)"
simps_1: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list"   and z :: "'b"   and zs :: "'b list" shows "x # xs \<simeq> {y # ys, z # zs, P} = (if P x xs then x = y \<and> xs \<simeq> {ys, z # zs, P} else x = z \<and> xs \<simeq> {y # ys, zs, P})"
succ_self_eq: shows "\<forall>(q, \<tau>)\<in>set (step (hd b) (a ! hd b)). q = hd b \<longrightarrow> ss ! q = a ! q"
core_chain_is_chain: fixes C :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring set"   and R :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring"   and S :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring" assumes "law_restrict ` C \<in> Chains (relation_of (\<lesssim>) \<S>)"   and "R \<in> C"   and "S \<in> C" shows "R \<lesssim> S \<or> S \<lesssim> R"
nonzero_closed: fixes a :: "'a" assumes "a \<in> nonzero R" shows "a \<in> carrier R"
trg_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg\<^sub>B (src\<^sub>B \<mu>) = src\<^sub>B \<mu>"
rbt_def: fixes t :: "('b \<times> color) tree" shows "rbt t = (invc t \<and> invh t \<and> color t = Black)"
braun_update1: fixes t :: "'b tree"   and n :: "nat"   and x :: "'b" assumes "braun t"   and "n \<in> {1..size t}" shows "braun (update1 n x t)"
div_add1_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (plus a b) c = plus (plus (divide a c) (divide b c)) (divide (plus (modulo a c) (modulo b c)) c)"
lossless_imp_plossless_gpv: fixes \<I> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "plossless_gpv \<I> gpv"
fresh_list_append: fixes a :: "'d"   and xs :: "'e list"   and ys :: "'e list" shows "a \<sharp> (xs @ ys) = (a \<sharp> xs \<and> a \<sharp> ys)"
params_subterms: fixes X :: "('a, 'b) Term.term list"   and f :: "'a" shows "set X \<subseteq> subterms (Fun f X)"
hyps_2: shows "input \<in> responses_\<I> \<Gamma> out"
alw_ev_scons: fixes P :: "'c stream \<Rightarrow> bool"   and a :: "'c"   and w :: "'c stream" shows "alw (ev P) (a ## w) = alw (ev P) w"
distinct_77: fixes x4 :: "db"   and x10 :: "nat" shows "DBAbsV x4 \<noteq> DBNum x10"
tuple_disj: fixes x :: "'b"   and z :: "'b" shows "{y. y = x \<or> y = z} = {x, z}"
mod_0_imp_i_set_mult_subset: fixes a :: "nat"   and b :: "nat" assumes "a mod b = 0" shows "i_set_mult a \<subseteq> i_set_mult b"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) G G G"
\<delta>ss_accs_sound: fixes t :: "'L tree"   and q :: "'Q" assumes "accs \<delta> t q" obtains s :: "'Q set" where "s \<subseteq> Q" and "q \<in> s" and "accs \<delta>ss t s"
\<R>_bisim: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "strong_low_bisim_mm (\<R> \<Gamma>')"
right_lang_def: fixes M :: "'b dfa"   and q :: "hf" assumes "dfa M" shows "dfa.right_lang M q \<equiv> {u. dfa.nextl M q u \<in> dfa.final M}"
count_union: fixes M :: "'b multiset"   and N :: "'b multiset"   and a :: "'b" shows "count (M + N) a = count M a + count N a"
nonzero_mult_div_cancel_right: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "a * b div b = a"
fresh_list_nil: fixes a :: "'a" shows "a \<sharp> []"
charts_submanifold_carrier: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set" assumes "submanifold charts k S" shows "manifold.charts_submanifold charts (manifold.carrier charts) = charts"
mult_id_matrix: fixes i :: "nat"   and nr :: "nat"   and v :: "real vec" assumes "i < nr" shows "(id_matrix nr (dim_vec v) *\<^sub>v v) $ i = (if i < dim_vec v then v $ i else 0)"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case deny x2 of allow x \<Rightarrow> f1 x | deny x \<Rightarrow> f2 x) = f2 x2"
ide_dom: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.ide (CC.dom f)"
InvariantNoDecisionsWhenConflict_def: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and level :: "nat" shows "InvariantNoDecisionsWhenConflict F M level = (\<forall>level'<level. \<not> formulaFalse F (elements (prefixToLevel level' M)))"
ES_valid_def: fixes ES :: "'b ES_rec" shows "ES_valid ES \<equiv> es_inputs_are_events ES \<and> es_outputs_are_events ES \<and> es_inputs_outputs_disjoint ES \<and> traces_contain_events ES \<and> traces_prefixclosed ES"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) r"
TESL_interp_unfold_stepwise_implies: fixes master :: "clock"   and slave :: "clock" shows "\<lbrakk> master implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Inter> {Y. \<exists>n. Y = \<lbrakk> master implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>}"
SMP_singleton_ex_1: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "t \<in> SMP M" shows "\<exists>m\<in>M. t \<in> SMP {m}"
FGModHom_idhom: shows "FGModuleHom G (\<cdot>) V (\<cdot>) (id \<down> V)"
timpl_closure_funs_term_subset: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "\<Union> (funs_term ` timpl_closure t TI) \<subseteq> funs_term t \<union> Abs ` snd ` TI"
Fun_tuple: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "Fun (hfsetcat.tuple f g) = (\<lambda>x\<^bold>\<in>Dom f. UP \<langle>DOWN (Fun f x), DOWN (Fun g x)\<rangle>)"
not_final_thread_wait_locks: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x"   and ln :: "'a \<Rightarrow>f nat" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "ln \<noteq> no_wait_locks" shows "not_final_thread s t"
dotdecimal_of_ipv4addr_ipv4addr_of_dotdecimal: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "a < 256"   and "b < 256"   and "c < 256"   and "d < 256" shows "dotdecimal_of_ipv4addr (ipv4addr_of_dotdecimal (a, b, c, d)) = (a, b, c, d)"
simps_129: fixes x121 :: "expr"   and x122 :: "expr"   and x51 :: "expr"   and x52 :: "bop"   and x53 :: "expr" shows "x121;; x122 \<noteq> x51 \<guillemotleft>x52\<guillemotright> x53"
main: shows "\<ominus>\<^bsub>V\<^esub> v = lincomb (\<lambda>u. \<ominus>\<^bsub>K\<^esub> \<one>\<^bsub>K\<^esub> \<otimes>\<^bsub>K\<^esub> a u) A"
list_result_alt_def: fixes xa :: "('a, 'b) result" shows "list_result xa = (case xa of Rval xda \<Rightarrow> Rval [xda] | Rerr xea \<Rightarrow> Rerr xea)"
cert: shows "cert_ok c (length \<tau>s) \<top> \<bottom> A"
c: shows "c = fps_nth f (subdegree g) / fps_nth g (subdegree g)"
simps_2: fixes n :: "nat" shows "fnorm (FVar n) = (PExpr1 (PVar n), PExpr1 (PCnst 1), [])"
simps_1: fixes e :: "'c"   and a :: "'d" shows "local.insert e a [] = [Node e a 0 []]"
cases: fixes x :: "'a list list" obtains "x = []"   | xss :: "'a list list" where "x = [] # xss__"   | xa :: "'a" and xs :: "'a list" and xss :: "'a list list" where "x = (xa # xs) # xss"
word_bw_same_3: fixes x :: "'b word" shows "x XOR x = 0"
union_limited_replicates: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and k :: "nat" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "0 < k"   and "\<forall>i\<in>I. card (A i) = k"   and "\<forall>i\<in>I. \<forall>x\<in>A i. card {i \<in> I. x \<in> A i} \<le> k" shows "card I \<le> card (\<Union> (A ` I))"
Ionescu_Tulcea_axioms: shows "Ionescu_Tulcea P M"
lookup_single: fixes k :: "'c"   and v :: "'b"   and k' :: "'c" shows "lookup (single k v) k' = (v when k = k')"
higher_differentiable_on_fst: fixes S :: "'b set"   and T :: "'c set"   and k :: "nat" shows "higher_differentiable_on (S \<times> T) fst k"
OrI1: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and b :: "('a, 'b) form" assumes "G \<turnstile> a" shows "G \<turnstile> Or a b"
le_supI: fixes a :: "'a multiset"   and x :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># x"   and "b \<subseteq># x" shows "a \<union># b \<subseteq># x"
s_PrjPair1: fixes v\<^sub>1 :: "Syntax.term"   and v\<^sub>2 :: "Syntax.term"   and \<pi> :: "Syntax.term list"   and uz :: "mode" assumes "value v\<^sub>1"   and "value v\<^sub>2" shows "\<lless>\<pi>, Prj1 (Syntax.Pair v\<^sub>1 v\<^sub>2)\<ggreater> uz\<rightarrow> \<lless>\<pi>, v\<^sub>1\<ggreater>"
equiv_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sim> y"   and "y \<sim> z" shows "x \<sim> z"
le_SucI: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "m \<le> Suc n"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(In2 x2 = In2 y2) = (x2 = y2)"
nrows_move_matrix_le: fixes A :: "'b matrix"   and j :: "int"   and i :: "int" shows "nrows (move_matrix A j i) \<le> nat (int (nrows A) + j)"
bij_image_singleton: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and b :: "'a"   and a :: "'b" assumes "f ` A = {b}"   and "f a = b"   and "bij f" shows "A = {a}"
F_OUT: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "f \<in> Field leq"   and "x \<in> B \<Gamma>" shows "d_OUT f x = 0"
add_diff_assoc: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "c + (b - a) = c + b - a"
10: shows "graph A w (Suc k) \<noteq> clean A w (graph A w k)"
embed_halfcong: fixes r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and r' :: "('d \<times> 'd) set" assumes "\<And>a. a \<in> Field r \<Longrightarrow> f a = g a"   and "embed r r' f" shows "embed r r' g"
LLL_inv_wD_5: fixes fs :: "int vec list"   and i :: "nat" assumes "LLL_invariant_weak fs"   and "i < m" shows "gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i \<in> Rn"
padic_one_simp_1: fixes n :: "nat"   and p :: "int" assumes "0 < n" shows "padic_one p n = \<one>\<^bsub>residue_ring (p ^ n)\<^esub>"
add_add_same_iff_zmset: fixes a :: "'b"   and A :: "'b zmultiset"   and B :: "'b zmultiset" shows "(add_zmset a A = add_zmset a B) = (A = B)"
power2_norm_eq_cinner': fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(norm x)\<^sup>2 = Re (cinner x x)"
push_bit_eq_mult: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n a = times a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
replicate_Suc: fixes n :: "nat"   and x :: "'c" shows "replicate (Suc n) x = x # replicate n x"
1: shows "to_fun f a \<ominus> deriv f a \<otimes> b = \<zero>"
inc_dot_rule: fixes d :: "nat"   and x :: "item" shows "item_rule (inc_dot d x) = item_rule x"
not_true: fixes x :: "trilean" shows "(x \<noteq> true) = (x = false \<or> x = invalid)"
step_1: assumes "distinct xs"   and "sorted_wrt rel xs" shows "sorted_wrt (\<lambda>x y. rel x y \<and> x \<noteq> y) xs"
2: fixes x :: "'a"   and f1 :: "'a \<Rightarrow> 'a"   and f2 :: "'a \<Rightarrow> 'a"   and g1 :: "'a \<Rightarrow> 'a"   and g2 :: "'a \<Rightarrow> 'a" assumes "x \<in> S"   and "f1 x \<in> S"   and "f2 (f1 x) \<in> S"   and "homeomorphism T T f1 g1"   and "homeomorphism T T f2 g2"   and "{x. \<not> (f1 x = x \<and> g1 x = x)} \<subseteq> S"   and "{x. \<not> (f2 x = x \<and> g2 x = x)} \<subseteq> S"   and "bounded {x. \<not> (f1 x = x \<and> g1 x = x)}"   and "bounded {x. \<not> (f2 x = x \<and> g2 x = x)}" shows "\<exists>f g. homeomorphism T T f g \<and> f x = f2 (f1 x) \<and> {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
elims: fixes x :: "'b"   and xa :: "nat"   and y :: "nat set" assumes "pre_oldest_tokens x xa = y" obtains q :: "'b" and n :: "nat" where "x = q" and "xa = n" and "y = {x. \<exists>q'. oldest_token q' n = Some x \<and> q = \<delta> q' (w n)} \<union> (if q = q\<^sub>0 then {Suc n} else {})"
nat_of_Sat: fixes n :: "nat" shows "nat_of (Sat n) = min LENGTH('b) n"
elims: fixes x :: "('a \<times> real) msllist"   and y :: "('a \<times> real) msllist" assumes "inverse_ms_aux x = y" obtains xa :: "'a \<times> real" and xs :: "('a \<times> real) msllist" where "x = MSLCons xa xs" and "y = (let c' = inverse (fst xa) in scale_shift_ms_aux (c', - snd xa) (powser_ms_aux (msllist_of_msstream (mssalternate 1 (- 1))) (scale_shift_ms_aux (c', - snd xa) xs)))"   | "x = MSLNil" and "y = undefined"
f_def: shows "f = (\<lambda>e ts. case ts of Bot \<Rightarrow> Bot | _ \<Rightarrow> check_single \<C> e ts)"
insert_sort_insert_by_perm: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and e :: "'a"   and l :: "'a list" shows "insert_sort_insert_by cmp e l <~~> e # l"
acute__not_obtuse: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Acute A B C" shows "\<not> Obtuse A B C"
ltl_step_none: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and a :: "String.literal"   and b :: "value list" assumes "possible_steps e s r a b = {||}" shows "ltl_step e (Some s) r (a, b) = (None, [], r)"
geod_dist3: fixes a :: "nat"   and n :: "nat"   and b :: "nat"   and p :: "nat" assumes "a \<le> 2 ^ n"   and "b \<le> 2 ^ p" shows "dist (geod n a) (geod p b) = \<bar>real b / 2 ^ p - real a / 2 ^ n\<bar> * dist x0 y0"
ins2_iff_ins_2: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "RBTi2 TYPE('b) bal_i e" shows "(RBTi2.ins2 bal_i n d x t = Bal2 t') = (RBTi1.ins bal_i n d x t = Bal t')"
future_consistent_def: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" shows "future_consistent P D R \<equiv> \<forall>u\<in>range D. \<forall>xs ys. (xs, ys) \<in> R u \<longrightarrow> next_dom_events P D u xs = next_dom_events P D u ys \<and> ref_dom_events P D u xs = ref_dom_events P D u ys"
step: fixes \<rho>' :: "'var \<Rightarrow> 'value" assumes "P \<rho>'" shows "P (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<rho>'\<^esub>)"
inter_set_filter: fixes A :: "'g set"   and xs :: "'g list" shows "A \<inter> set xs = set (filter (\<lambda>x. x \<in> A) xs)"
Posix_Star1: fixes s1 :: "'b list"   and r :: "'b rexp"   and v :: "'b val"   and s2 :: "'b list"   and vs :: "'b val list" assumes "s1 \<in> r \<rightarrow> v"   and "s2 \<in> Star r \<rightarrow> Stars vs"   and "flat v \<noteq> []"   and "\<nexists>s\<^sub>3 s\<^sub>4. s\<^sub>3 \<noteq> [] \<and> s\<^sub>3 @ s\<^sub>4 = s2 \<and> s1 @ s\<^sub>3 \<in> lang r \<and> s\<^sub>4 \<in> lang (Star r)" shows "(s1 @ s2) \<in> Star r \<rightarrow> Stars (v # vs)"
is_Ghost\<^sub>s\<^sub>b_simps_2: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_Ghost\<^sub>s\<^sub>b (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = False"
conf_def: fixes P :: "'a prog"   and h :: "'heap"   and v :: "'addr val"   and T :: "ty" shows "P,h \<turnstile> v :\<le> T \<equiv> \<exists>T'. typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
bintrunc_bintrunc_min: fixes m :: "nat"   and n :: "nat"   and w :: "int" shows "take_bit m (take_bit n w) = take_bit (min m n) w"
sc_sound: fixes P :: "'a \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "sound (\<lambda>s. c * P s)"
left: fixes Q :: "'b"   and Cal :: "'b processCalculus"   and Q' :: "'b"   and P' :: "'b"   and Rel :: "('b \<times> 'b) set" assumes "Q \<longmapsto>Cal* Q'"   and "(P', Q') \<in> Rel" shows "(P', Q) \<in> cSim_cs Rel Cal"
exhaust: fixes y :: "'b message_af" obtains "y = NoMsg"   | x2 :: "'b" where "y = Msg x2"
prefs_undefined': fixes i :: "'a" assumes "i \<notin> agents" shows "R2 i = (\<lambda>_ _. False)"
A_def: shows "A = four_block_mat A1 A2 A3 A4"
assms_2: shows "f n\<^sub>p < n \<and> (\<forall>k. f n\<^sub>p < k \<and> k \<le> n \<longrightarrow> (\<nexists>k\<^sub>0. f k\<^sub>0 = k))"
pseudo_proj_True_preimage_stake: fixes w :: "bool stream"   and n :: "nat" assumes "w = pseudo_proj_True n w" shows "pseudo_proj_True n -` {w} = {x. stake n x = stake n w}"
ev_one: shows "ev (holds (\<lambda>x. P x \<and> Q x)) xs"
comp_char: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" shows "g \<odot> f = (if Cop.seq (fst g) (fst f) \<and> C.seq (snd g) (snd f) then (fst g \<cdot>\<^sup>o\<^sup>p fst f, snd g \<cdot> snd f) else (Cop.null, C.null))"
ilist_slice_def: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat" shows "ilist_slice f k \<equiv> \<lambda>x. map f [x * k..<Suc x * k]"
fInf_subdistl_var: fixes f :: "'b \<Rightarrow> 'c"   and G :: "('d \<Rightarrow> 'b) set" assumes "mono f" shows "f \<circ> \<Sqinter> G \<le> \<Sqinter> ((\<circ>) f ` G)"
topological_space_def: fixes "open" :: "'d set \<Rightarrow> bool" shows "class.topological_space open \<equiv> open UNIV \<and> (\<forall>S T. open S \<longrightarrow> open T \<longrightarrow> open (S \<inter> T)) \<and> (\<forall>K. (\<forall>S\<in>K. open S) \<longrightarrow> open (\<Union> K))"