hcju/isabelleps huggingface dataset:
simps: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.tree_to_multiset (BinomialHeapStruc.Node e a r ts) = {#(e, a)#} + BinomialHeapStruc.queue_to_multiset ts"
on_curvep_def: fixes a :: "'a"   and b :: "'a" shows "on_curvep a b = (\<lambda>(x, y, z). x \<in> carrier R \<and> y \<in> carrier R \<and> z \<in> carrier R \<and> (z \<noteq> \<zero> \<longrightarrow> y [^] 2 \<otimes> z = x [^] 3 \<oplus> a \<otimes> x \<otimes> z [^] 2 \<oplus> b \<otimes> z [^] 3))"
assoc_add_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat"   and C :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "A + B + C = A + (B + C)"
antisym_basic_type: fixes \<tau> :: "'a basic_type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<le> \<sigma>"   and "\<sigma> \<le> \<tau>" shows "\<tau> = \<sigma>"
op_list_singleton_def: fixes x :: "'c" shows "op_list_singleton x \<equiv> [x]"
term_ok_eq_ax_7: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "is_std_sig (sig \<Theta>)" shows "term_ok \<Theta> eq_abstract_rule_ax"
hyps: fixes m :: "'a \<Rightarrow> 'b option" assumes "distinct (map fst l)" shows "(map_of l = m) = (set l = map_to_set m)"
sorted_by_cons_trans: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "transp cmp" shows "sorted_by cmp (x # xs) = ((\<forall>x'\<in>set xs. cmp x x') \<and> sorted_by cmp xs)"
cobounded2: fixes a :: "'t"   and b :: "'t" shows "ord_term_lin.min a b \<preceq>\<^sub>t b"
sup_ge2: fixes y :: "'a"   and x :: "'a" shows "y \<le> x \<squnion>1 y"
lift_dom_trm: fixes t :: "'a trm"   and \<theta> :: "('a \<times> 'a trm) list"   and C :: "'a literal set"   and \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "dom_trm (t \<lhd> \<theta>) (subst_cl C \<theta>)"   and "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>" shows "dom_trm (t \<lhd> \<sigma>) (subst_cl C \<sigma>)"
stable_pair_on_Cd: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and XD_XH :: "'d set \<times> 'd set"   and d :: "'e" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_pair_on Pd Ch ds XD_XH"   and "d \<in> ds" shows "Contracts.Cd Pd d (fst XD_XH) = fst XD_XH \<inter> snd XD_XH \<inter> Field (Pd d)"
mem_fset_from_list: fixes x :: "'a"   and l :: "'a list" shows "(x |\<in>| fset_from_list l) = (x \<in> set l)"
vars_mset_fun: fixes t :: "('b, 'a) tm" shows "vars_mset (fun t) \<subseteq># vars_mset t"
float_of_inverse: fixes y :: "real" assumes "y \<in> float" shows "real_of_float (float_of y) = y"
subEq: fixes s :: "real" assumes "s \<in> {0..<1}"   and "s = 0 \<or> Re (g s) = Re z"   and "\<forall>t\<in>{s<..<1}. Re (g t) = Re z"   and "finite_ReZ_segments (subpath 0 s g) z" shows "P"
Domainp_conversep: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "Domainp R\<inverse>\<inverse> = Rangep R"
circ_separate_1: fixes y :: "'a"   and x :: "'a" assumes "y * x \<le> x * y" shows "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> * y\<^sup>\<circ>"
prems01: shows "p * a + q * b = d \<and> d dvd a \<and> d dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd d)"
vcpowerE: fixes f :: "V"   and A :: "V"   and n :: "V" assumes "f \<in>\<^sub>\<circ> A ^\<^sub>\<times> n" obtains "f \<in>\<^sub>\<circ> (\<Prod>\<^sub>\<circ>i\<in>\<^sub>\<circ>n. A)"
assms_2: fixes z :: "'a" assumes "z \<in> results y" shows "f z = g z"
EmptyMap_ran_override: fixes S :: "'d set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" assumes "S \<noteq> {}"   and "S \<inter> dom G = {}" shows "ran (G ++ EmptyMap S) = insert {} (ran G)"
intro: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "total_preorder_on carrier le"   and "finite_total_preorder_on_axioms carrier" shows "finite_total_preorder_on carrier le"
interior_subset: fixes S :: "'a set" shows "interior S \<subseteq> S"
hoare_total_correct_def: fixes P :: "complex mat"   and S :: "com"   and Q :: "complex mat" shows "(\<Turnstile>\<^sub>t {P} S {Q}) = (\<forall>\<rho>\<in>density_states. trace (P * \<rho>) \<le> trace (Q * denote S \<rho>))"
restriction_stable1: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u\<parallel>w \<longrightarrow> restrict v f c = restrict u f c"
simps_1: fixes f :: "'effect"   and F :: "'effect fs_set"   and P :: "'state"   and \<alpha>P' :: "(('act, 'effect) L_action, ('state, 'effect) L_state) residual" shows "AC (f, F, P) \<rightarrow>\<^sub>L \<alpha>P' = (\<exists>\<alpha> P'. P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<and> \<alpha>P' = \<langle>L_action.Act \<alpha>,EF (L (\<alpha>, F, f), P')\<rangle> \<and> bn \<alpha> \<sharp>* (F, f))"
four: fixes xs :: "('b com \<times> 'b set \<times> ('b \<times> 'b) set \<times> ('b \<times> 'b) set \<times> 'b set) list"   and rely :: "('b \<times> 'b) set"   and guar :: "('b \<times> 'b) set"   and pre :: "'b set"   and x :: "('b com option list \<times> 'b) list"   and s :: "'b"   and i :: "nat" assumes "xs \<noteq> []"   and "\<forall>i<length xs. rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j)) \<subseteq> Rely (xs ! i)"   and "(\<Union>j\<in>{j. j < length xs}. Guar (xs ! j)) \<subseteq> guar"   and "pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i))"   and "\<forall>i<length xs. \<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)]"   and "x \<in> par_cp (ParallelCom xs) s"   and "x \<in> par_assum (pre, rely)"   and "Suc i < length x"   and "x ! i -pc\<rightarrow> x ! Suc i" shows "(snd (x ! i), snd (x ! Suc i)) \<in> guar"
a_known_ptrs_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and h :: "(_) heap" shows "l_known_ptrs\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_known_ptrs known_ptr h = (\<forall>ptr\<in>fset (object_ptr_kinds h). known_ptr ptr)"
ordinal_times_assoc: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" shows "x * y * z = x * (y * z)"
closed_translation_iff_1: fixes f :: "'c \<Rightarrow> 'b"   and y :: "'b"   and S :: "'c set" shows "closed ((\<lambda>x. f x + y) ` S) = closed (f ` S)"
div_mult1_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (times a b) c = plus (times a (divide b c)) (divide (times a (modulo b c)) c)"
simps_15: fixes x2 :: "'a" shows "set_option (Some x2) = {x2}"
s_App1: fixes \<pi> :: "Syntax.term list"   and e\<^sub>1 :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e\<^sub>1' :: "Syntax.term"   and e\<^sub>2 :: "Syntax.term" assumes "\<lless>\<pi>, e\<^sub>1\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>1'\<ggreater>" shows "\<lless>\<pi>, App e\<^sub>1 e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', App e\<^sub>1' e\<^sub>2\<ggreater>"
cHeap_simp: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "cHeap \<Gamma> e\<cdot>a = pathsCard (paths (Theap \<Gamma> e\<cdot>a))"
bal_slim_node: fixes k :: "nat"   and a :: "'b"   and h :: "nat" shows "bal (slim_node k a h)"
axioms_2: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "currying A1 A2 B" shows "category A2"
pre: shows "wf_matchF_aux \<sigma> V n R I r aux ne 0"
is_inf_minimum: fixes x :: "'c"   and y :: "'c" shows "is_inf x y (minimum x y)"
elementary_union_interval: fixes p :: "'a set set"   and a :: "'a"   and b :: "'a" assumes "p division_of \<Union> p" obtains q :: "'a set set" where "q division_of cbox a b \<union> \<Union> p"
inc: fixes s :: "'a"   and v :: "'b"   and s' :: "'a"   and M :: "('a \<times> ('a, 'b) Op) set"   and vs :: "'b \<Rightarrow> nat" assumes "(s, Inc v s') \<in> M" shows "((s, vs), s', \<lambda>x. if x = v then Suc (vs v) else vs x) \<in> step M"
return\<^sub>C_def: fixes upd :: "('d list \<Rightarrow> 'd list) \<Rightarrow> 'c control_state_scheme \<Rightarrow> 'c control_state_scheme"   and rhs :: "'c control_state_scheme \<Rightarrow> 'd" shows "return\<^sub>C upd rhs = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some ((), \<sigma>) else (assign_local upd rhs ;- set_return_status) \<sigma>)"
matrix_vector_mul_assoc: fixes A :: "(('b, 'd) vec, 'c) vec"   and B :: "(('b, 'e) vec, 'd) vec"   and x :: "('b, 'e) vec" shows "A *v (B *v x) = A ** B *v x"
odd_even_add: 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 a :: "'b"   and b :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a b)"
scaleR_scaleR: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "real"   and b :: "real"   and x :: "'e" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (scaleR b x) = scaleR (a * b) x"
prime_nonzero: fixes p :: "nat" assumes "prime p" shows "p \<noteq> 0"
Fow_def: fixes A :: "'b set" shows "Fow A = {x. \<langle>x\<rangle>\<^sub>f \<subseteq> A}"
ParTL_MtransC: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>*t s'" shows "(c1 | c2, s) \<rightarrow>*c (c2, s')"
unlock_lock_SomeD: fixes l :: "('b \<times> nat) option"   and t' :: "'b"   and n :: "nat" assumes "unlock_lock l = \<lfloor>(t', n)\<rfloor>" shows "l = \<lfloor>(t', Suc n)\<rfloor>"
hd_drop_conv_nth: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "hd (drop n xs) = xs ! n"
normalizability_I: fixes a :: "'c"   and b :: "'c"   and A :: "('c \<times> 'c) set" assumes "(a, b) \<in> A\<^sup>*"   and "b \<in> NF A" shows "(a, b) \<in> A\<^sup>!"
min_product_empty_singleton_2: fixes A :: "'b fset set" shows "{{||}} \<otimes>\<^sub>m A = min_set A"
state_var_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "state_var s1 = state_var s2"
fcard_funion_disjoint: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<inter>| B = {||}" shows "fcard (A |\<union>| B) = fcard A + fcard B"
assms_1: shows "der \<in> borel_measurable (G matur)"
linorder_lattice_sup_least: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> x" shows "sup y z \<le> x"
dgrad_p_set_closed_uminus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "p \<in> dgrad_p_set d m" shows "- p \<in> dgrad_p_set d m"
dist_0_norm: fixes x :: "'d" shows "dist (0::'d) x = norm x"
fresh_distinct_fv: fixes S :: "'c set"   and l :: "'b" assumes "atom ` S \<sharp>* l" shows "S \<inter> fv l = {}"
in_segment_2: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x \<in> open_segment a b) = (a \<noteq> b \<and> (\<exists>u>0. u < 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b))"
simps_14: fixes va :: "formula list" shows "is_constraint (Neg (Ands va)) = False"
Gen_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and P :: "(symbol \<times> character list) list set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc P = {uu_. \<exists>x p. uu_ = x \<and> p \<in> P \<and> LocalLexing.pvalid \<NN> \<TT> \<RR> \<SS> Doc p x}"
sqrt_iteration_lower_bound: fixes x :: "float"   and prec :: "nat"   and n :: "nat" assumes "0 < real_of_float x" shows "0 < real_of_float (sqrt_iteration prec n x)"
L2_set_triangle_ineq: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real"   and A :: "'a set" shows "L2_set (\<lambda>i. f i + g i) A \<le> L2_set f A + L2_set g A"
simps_3: fixes uz :: "dynamic list" shows "inl OpNeg uz = None"
verit_negate_coefficient_3: fixes a :: "'c"   and b :: "'c" assumes "a = b" shows "- a = - b"
hyps_3: shows "rev s2 \<in> (rev ` A)\<star>"
Quotient_rel_rep: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "Quotient R Abs Rep T" shows "R (Rep a) (Rep b) = (a = b)"
eq_elapsed_def: fixes u :: "'c \<Rightarrow> 'a"   and u' :: "'c \<Rightarrow> 'a" shows "eq_elapsed u u' \<equiv> (0::'a) < elapsed u u' \<longrightarrow> (\<forall>c\<in>\<X>. u' c - u c = elapsed u u')"
map_to_set_def: fixes m :: "'f \<Rightarrow> 'g option" shows "map_to_set m = {(k, v). m k = Some v}"
ask_nondet_def: fixes ask :: "('c \<Rightarrow> 'm) \<Rightarrow> 'm"   and f :: "'c \<Rightarrow> ('a, 'm) nondetT" shows "ask_nondet ask f = NondetT (ask (\<lambda>r. run_nondet (f r)))"
d: fixes \<phi>' :: "'a"   and \<psi>' :: "'a" shows "K \<^bold>* \<phi>' \<subseteq> (K \<^bold>* (\<phi>' .\<or>. \<psi>')) \<oplus> \<phi>'"
units_power_order_eq_one: fixes a :: "int" assumes "finite (Units R)"   and "a \<in> Units R" shows "a [^] card (Units R) = \<one>"
nth_default_nth: fixes n :: "nat"   and xs :: "'f list"   and dflt :: "'f" assumes "n < length xs" shows "nth_default dflt xs n = xs ! n"
scaleR_cancel_left: fixes a :: "real"   and x :: "'b"   and y :: "'b" shows "(a *\<^sub>R x = a *\<^sub>R y) = (x = y \<or> a = 0)"
msetext_huet_map: fixes A :: "'b set"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "finite A"   and "ys \<in> lists A"   and "xs \<in> lists A"   and "\<forall>x\<in>A. \<not> gt (f x) (f x)"   and "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt (f z) (f y) \<longrightarrow> gt (f y) (f x) \<longrightarrow> gt (f z) (f x)"   and "\<forall>y\<in>A. \<forall>x\<in>A. gt y x \<longrightarrow> gt (f y) (f x)"   and "msetext_huet gt ys xs" shows "msetext_huet gt (map f ys) (map f xs)"
type_copy_map_id0: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and M :: "'c \<Rightarrow> 'c" assumes "type_definition Rep Abs UNIV"   and "M = id" shows "Abs \<circ> M \<circ> Rep = id"
apply_times: fixes f :: "'a perm"   and g :: "'a perm" shows "(\<langle>$\<rangle>) (f * g) = (\<langle>$\<rangle>) f \<circ> (\<langle>$\<rangle>) g"
orbit_equality: fixes x :: "'a" assumes "x \<in> G" shows "orbit.Class x = {\<alpha> x |\<alpha>. \<alpha> \<in> translation ` H}"
relcomp_empty1: fixes R :: "('c \<times> 'b) set" shows "{} O R = {}"
iso_class_rep_def: fixes F :: "'b set" shows "iso_class_rep F \<equiv> SOME f. f \<in> F"
in_read_only_share_all_until_volatile_write': fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "read_only_unowned \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<forall>j<length ts. i \<noteq> j \<longrightarrow> (let (uu_, uu_, uu_, sb\<^sub>j, uu_, \<O>, uu_) = ts ! j in a \<notin> acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) \<O> \<and> a \<notin> all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j))"   and "a \<in> read_only (share sb \<S>)" shows "a \<in> read_only (share (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) (share_all_until_volatile_write ts \<S>))"
RPHIL_refines_RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m: fixes i :: "nat" assumes "0 < i" shows "RPHIL\<^sub>n\<^sub>o\<^sub>r\<^sub>m i 0 \<sqsubseteq> RPHIL i"
clop_wtrans: fixes x :: "'b" shows "cl_op (cl_op x) \<le> cl_op x"
fcompare_double_Eq1: fixes x :: "double"   and y :: "double"   and z :: "double"   and c :: "ccode" assumes "fcompare_double x y = ccode.Eq"   and "fcompare_double y z = c" shows "fcompare_double x z = c"
insert_Times_insert: fixes a :: "'c"   and A :: "'c set"   and b :: "'d"   and B :: "'d set" shows "insert a A \<times> insert b B = insert (a, b) (A \<times> insert b B \<union> insert a A \<times> B)"
flowsto_unionE: fixes X0 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and T :: "real set"   and CX :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and Y :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and Z :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" assumes "flowsto X0 T CX (Y \<union> Z)" obtains X1 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" and X2 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" where "X0 = X1 \<union> X2" and "flowsto X1 T CX Y" and "flowsto X2 T CX Z"
is_nextElem_last_hd: fixes vs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct vs"   and "is_nextElem vs x y"   and "x = last vs" shows "y = hd vs"
surj_imp_inj_inv: fixes f :: "'b \<Rightarrow> 'a" assumes "surj f" shows "inj (inv f)"
cobounded1: fixes a :: "'b word"   and b :: "'b word" shows "signed.min a b \<le>s a"
add_increasing2: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r"   and less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and c :: "'r"   and b :: "'r"   and a :: "'r" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero c"   and "less_eq b a" shows "less_eq b (plus a c)"
rbt_sorted_entries: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "linorder.sorted less_eq (map fst (RBT_Impl.entries t))"
homotopic_loopsI: fixes h :: "real \<times> real \<Rightarrow> 'b"   and s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "continuous_on ({0..1} \<times> {0..1}) h"   and "h ` ({0..1} \<times> {0..1}) \<subseteq> s"   and "\<And>x. x \<in> {0..1} \<Longrightarrow> h (0, x) = p x"   and "\<And>x. x \<in> {0..1} \<Longrightarrow> h (1, x) = q x"   and "\<And>x. x \<in> {0..1} \<Longrightarrow> pathfinish (h \<circ> Pair x) = pathstart (h \<circ> Pair x)" shows "homotopic_loops s p q"
Maclaurin_cos_expansion2: fixes x :: "real"   and n :: "nat" assumes "0 < x"   and "0 < n" shows "\<exists>t>0. t < x \<and> cos x = (\<Sum>m<n. cos_coeff m * x ^ m) + cos (t + 1 / 2 * real n * pi) / fact n * x ^ n"
i_take_drop_eq_map: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Up> m \<Down> n = map f [m..<m + n]"
finfun_All_All: fixes P :: "('a, bool) finfun" shows "finfun_All P = All (finfun_apply P)"
f_values: shows "\<forall>x\<in>{i. i < 2 ^ n}. f x = 0 \<or> f x = 1"
elementary_category_with_pullbacks_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "elementary_category_with_pullbacks C prj0 prj1 \<equiv> category C \<and> elementary_category_with_pullbacks_axioms C prj0 prj1"
elementary_category_with_pullbacks_axioms: shows "elementary_category_with_pullbacks (\<cdot>) prj0 prj1"
msetext_huet_irrefl: fixes xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> msetext_huet gt xs xs"
add_leq_two: shows "\<forall>c. |re c| + |cl c| \<le> 2"
lenext_hd_or_tl: fixes ys :: "'b list"   and xs :: "'b list"   and gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<lbrakk>length ys = length xs; gts (y # ys) (x # xs)\<rbrakk> \<Longrightarrow> gt y x \<or> gts ys xs"   and "lenext gts (y # ys) (x # xs)" shows "gt y x \<or> lenext gts ys xs"
Gmult_def: fixes g :: "'g"   and v :: "'v" shows "g *\<cdot> v \<equiv> ((1::'r) \<delta>\<delta> g) \<cdot> v"
word_arith_nat_defs_7: fixes a :: "'b word"   and b :: "'b word" shows "a mod b = word_of_nat (unat a mod unat b)"
in_cbox_complex_iff: fixes x :: "complex"   and a :: "complex"   and b :: "complex" shows "(x \<in> cbox a b) = (Re x \<in> {Re a..Re b} \<and> Im x \<in> {Im a..Im b})"
vcard_disjoint_sup: fixes x :: "V"   and y :: "V" assumes "x \<sqinter> y = 0" shows "vcard (x \<squnion> y) = vcard x \<oplus> vcard y"
pmf_map_inj: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b pmf"   and x :: "'b" assumes "inj_on f (set_pmf M)"   and "x \<in> set_pmf M" shows "pmf (map_pmf f M) (f x) = pmf M x"
eq_bottom_iff: fixes x :: "'d" shows "(x = \<bottom>) = (x \<sqsubseteq> \<bottom>)"
polyfun_linear_factor: fixes c :: "nat \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z ^ i) = (z - a) * (\<Sum>i<n. b i * z ^ i) + (\<Sum>i\<le>n. c i * a ^ i)"
prems_5: shows "elts (SUP \<xi>\<in>elts \<gamma>. \<xi>) \<subseteq> \<Union> (range X)"
valid_SDG_node_cases: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n = CFG_node (parent_node n) \<or> CFG_node (parent_node n) \<longrightarrow>\<^bsub>cd\<^esub> n"
prems_1: shows "Suc n \<le> mod_type_class.to_nat i"
M_subset_N: fixes k :: "nat" shows "M k \<subseteq> N"
bounded_linear_minus: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "bounded_linear (\<lambda>x. - f x)"
all_local_consts_listed: fixes r :: "'rule"   and p :: "('form, 'var) antecedent" assumes "r \<in> sset rules"   and "p \<in> set (antecedent r)" shows "lconsts (a_conc p) \<union> \<Union> (lconsts ` fset (a_hyps p)) \<subseteq> a_fresh p"
oexp_def: shows "rt.oexp = {(f, g). f \<in> rt.FINFUNC \<and> g \<in> rt.FINFUNC \<and> ((let m = wo_rel.max_fun_diff t f g in (f m, g m) \<in> r) \<or> f = g)}"
ih: fixes Q :: "('a literal list \<times> nat) list" shows "wstate_of_dstate (N, P, Q @ Q', n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, P, Q @ filter (Not \<circ> strictly_subsume [C] \<circ> fst) Q', n)"
MP: fixes p :: "('c, 'd) fm"   and q :: "('c, 'd) fm" assumes "\<turnstile> p \<^bold>\<longrightarrow> q"   and "\<turnstile> p" shows "\<turnstile> q"
getRel_def: fixes l :: "'b"   and G :: "('b, 'a) labeled_graph" shows "getRel l G = {(x, y). (l, x, y) \<in> edges G}"
lead_coeff_Q: shows "Polynomial.lead_coeff Q = int c"