hcju/isabelleps huggingface dataset:
hid_is_CSP1: fixes P :: "('c, 'd) action"   and cs :: "'c set" shows "hid P cs is CSP1 healthy"
assms_5: shows "\<not> (\<exists>(a', b', c')\<in>set b. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
pred_def: shows "pred = (\<Lambda> x. pred_Arity x)"
get_put: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f" assumes "wb_lens x" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
cases: fixes a1 :: "trm"   and a2 :: "trm" assumes "a1 \<longrightarrow>\<^sub>c a2" obtains M :: "trm" and a :: "coname" and N :: "trm" and x :: "name" where "a1 = Cut <a__>.Ma__ xa__.Na__" and "a2 = Ma__{a__:=xa__.Na__}" and "\<not> fic Ma__ a__" and "a__ \<sharp> Na__" and "xa__ \<sharp> Ma__"   | N :: "trm" and x :: "name" and a :: "coname" and M :: "trm" where "a1 = Cut <a>.M x.N" and "a2 = N{x:=<a>.M}" and "\<not> fin N x" and "a \<sharp> N" and "x \<sharp> M"
retract_of_path_connected: fixes T :: "'a set"   and S :: "'a set" assumes "path_connected T"   and "S retract_of T" shows "path_connected S"
prod_mset_singleton: fixes x :: "'a" shows "\<Prod>\<^sub># {#x#} = x"
2: shows "signof p * det A = (\<Sum>q | q permutes {0..<n}. signof p * signof q * (\<Prod>i = 0..<n. A $$ (i, q i)))"
allocate_SomeD: fixes h' :: "'heap"   and a :: "'addr"   and h :: "'heap"   and hT :: "htype" assumes "(h', a) \<in> allocate h hT"   and "is_htype P hT" shows "typeof_addr h' a = \<lfloor>hT\<rfloor>"
quotient_web_sel_1: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" shows "edge (quotient_web \<Gamma> f) x y = (edge \<Gamma> x y \<and> x \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<and> y \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f))"
qs: shows "qs \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y])"
vsv_vimage_vsingleton: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> A" shows "A `\<^sub>\<circ> ZFC_in_HOL.set {a} = ZFC_in_HOL.set {A\<lparr>a\<rparr>}"
Abs_supports_1: fixes x :: "'a"   and as :: "atom set" shows "(supp x - as) supports ([as]set. x)"
computable_complement_2: fixes A :: "nat set" assumes "computable (- A)" shows "computable A"
assms_1: shows "\<lparr>es\<rparr> \<leadsto> \<lparr>es'\<rparr>"
H1: shows "I \<subseteq> carrier (G\<lparr>carrier := J\<rparr>)"
WHILE_refine: fixes x :: "'c"   and x' :: "'d"   and R :: "('c \<times> 'd) set"   and b :: "'c \<Rightarrow> bool"   and b' :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and f' :: "'d \<Rightarrow> 'd nres" assumes "(x, x') \<in> R"   and "\<And>x x'. (x, x') \<in> R \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; b x; b' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "WHILE b f x \<le> \<Down> R (WHILE b' f' x')"
eq_divide_eq_numeral_1: fixes w :: "num"   and b :: "'b"   and c :: "'b" shows "(numeral w = b / c) = (if c \<noteq> (0::'b) then numeral w * c = b else numeral w = (0::'b))"
key_hyp: shows "satisfies_properties p qs (get_subsets (reduce_system p (qs, base_case_info))) (get_signs (reduce_system p (qs, base_case_info))) (get_matrix (reduce_system p (qs, base_case_info)))"
h7b: assumes "Av < 0"   and "0 < Bv"   and "Bv\<^sup>2 * Cv < Av\<^sup>2" shows "Av < - Bv * sqrt Cv"
simps_1: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "funpower f 0 x = x"
vector_complement_closed: fixes x :: "'a" assumes "x \<sqinter> top = x" shows "- x \<sqinter> top = - x"
cases: fixes a1 :: "'d + 'd"   and a2 :: "'d + 'd" assumes "ignore_Inl a1 a2" obtains x :: "'d" and y :: "'d" where "a1 = Inl x__" and "a2 = Inl y"   | x :: "'d" where "a1 = Inr x" and "a2 = Inr x"
weak_reduction_simulation_and_closures_1: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_simulation Rel Cal" shows "weak_reduction_simulation (Rel\<^sup>=) Cal"
divide_zero_left: fixes a :: "'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>" shows "\<zero> \<oslash> a = \<zero>"
pi: shows "pi = (atom i \<rightleftharpoons> atom i') + p"
star_right_induct_mult_iff: fixes y :: "'a"   and x :: "'a" shows "(y * x \<le> y) = (y * x\<^sup>\<star> \<le> y)"
polys_to_mat_def: fixes ts :: "'d list"   and ps :: "('d \<Rightarrow>\<^sub>0 'c) list" shows "polys_to_mat ts ps = mat_of_rows (length ts) (map (poly_to_row ts) ps)"
Plus_eq_empty_conv: fixes A :: "'c set"   and B :: "'d set" shows "(A <+> B = {}) = (A = {} \<and> B = {})"
of_int_le_numeral_power_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "int"   and x :: "num"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus a) (power.power one times (numeral.numeral one plus x) n) = (a \<le> numeral x ^ n)"
runit_char_3: fixes a :: "'a" assumes "ide a" shows "\<exists>!f. \<guillemotleft>f : a \<otimes> \<I> \<rightarrow> a\<guillemotright> \<and> f \<otimes> \<I> = (a \<otimes> \<iota>) \<cdot> \<a>[a, \<I>, \<I>]"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "I.wtE \<xi> \<equiv> \<forall>x. intT (tpOfV x) (\<xi> x)"
X_def: fixes x :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "X x i = [:- x i, 1::'b:]"
inline_sel: fixes c :: "('b, 'call, 'ret) gpv"   and s :: "'s" shows "the_gpv (local.inline c s) = map_spmf (\<lambda>result. case result of Inl xs \<Rightarrow> Pure xs | Inr (out, oracle, rpv) \<Rightarrow> IO out (\<lambda>input. oracle input \<bind> (\<lambda>(x, s'). local.inline (rpv x) s'))) (local.inline1 c s)"
v: shows "v = M.sumlist (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs])"
simps_2: fixes at :: "('a, 'c) atm" shows "gL (Neg at) = Neg (gA at)"
surj_Sup_pres_inj_on: fixes f :: "'c \<Rightarrow> 'b" assumes "top_pres ((`) f)"   and "Sup_pres f" shows "inj_on f (range (radj f \<circ> f))"
row_spaceI: fixes x :: "'c vec"   and v :: "'c vec"   and A :: "'c mat" assumes "x = v \<^sub>v* A" shows "x \<in> row_space A"
that: fixes g :: "'b \<Rightarrow> 'a" assumes "g = the_inv f"   and "g \<circ> f = id"   and "map_indets g \<circ> map_indets f = id" shows "thesis"
Tarskian_logic_def: fixes Cn :: "'b set \<Rightarrow> 'b set" shows "Tarskian_logic Cn \<equiv> (\<forall>A B. A \<subseteq> B \<longrightarrow> Cn A \<subseteq> Cn B) \<and> (\<forall>A. A \<subseteq> Cn A) \<and> (\<forall>A. Cn (Cn A) \<subseteq> Cn A)"
splitFace_face_face_op: fixes g :: "graph"   and u :: "nat"   and v :: "nat"   and f :: "face"   and vs :: "nat list"   and f\<^sub>1 :: "face"   and f\<^sub>2 :: "face"   and g' :: "graph" assumes "minGraphProps g"   and "pre_splitFace g u v f vs"   and "(f\<^sub>1, f\<^sub>2, g') = splitFace g u v f vs" shows "face_face_op g'"
inorder_inv_delete: fixes t :: "'t"   and a :: "'a" assumes "inv t \<and> sorted1 (inorder t)" shows "inv (delete a t)"
bisim1Sync5: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and v :: "'addr val"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (insync\<^bsub>V\<^esub> (a) Val v, xs) \<leftrightarrow> ([xs ! V, v], xs, 4 + length (compE2 e1) + length (compE2 e2), None)"
psubsetI: fixes A :: "'b set"   and B :: "'b set" assumes "(A::'b set) \<subseteq> (B::'b set)"   and "(A::'b set) \<noteq> (B::'b set)" shows "(A::'b set) \<subset> (B::'b set)"
\<T>e_ide_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b"   and b :: "'a" assumes "yoneda_lemma_fixed_e C S \<phi> F a e"   and "partial_magma.ide (dual_category.comp C) b" shows "transformation_by_components.map (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) (yoneda_lemma.\<T>o C S \<phi> F a e) b = set_category_given_img.mkArr S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set C \<phi> (b, a)) (set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F b)) (\<lambda>x. set_category_given_img.Fun S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F (hom_functor.\<psi> C \<phi> (b, a) x)) e)"
leadsETo_Un: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and C :: "'d set"   and B :: "'d set" assumes "F \<in> A leadsTo[CC] C"   and "F \<in> B leadsTo[CC] C" shows "F \<in> (A \<union> B) leadsTo[CC] C"
distinct_selsort_inl: fixes X :: "'b \<times> (real \<times> real) pdevs" shows "distinct (ccw.selsort 0 (inl (snd X)))"
assms_1: shows "set (coeffs p) \<subseteq> \<real>"
b_def: shows "b = (\<Sum>xa\<in>set (Gram_Schmidt xs). (x \<bullet> xa / (xa \<bullet> xa)) *\<^sub>R xa)"
assoc: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "semigroup f" shows "f (f a b) c = f a (f b c)"
left_implies_right: assumes "is_finished x" shows "\<exists>\<alpha>. x = finished_item \<alpha>"
I: shows "I = (\<lambda>l s. 0 \<le> s a)"
wset_thread_okD: fixes ws :: "'a \<Rightarrow> 'b wait_set_status option"   and ts :: "'a \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option"   and t :: "'a" assumes "wset_thread_ok ws ts"   and "ts t = None" shows "ws t = None"
continuous_map_componentwise_UNIV: fixes X :: "'d topology"   and Y :: "'e \<Rightarrow> 'f topology"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "continuous_map X (product_topology Y UNIV) f = (\<forall>k. continuous_map X (Y k) (\<lambda>x. f x k))"
prems_1: fixes p :: "'b"   and ls :: "'a list"   and eq :: "'a list"   and gs :: "'a list" assumes "(p__, xs, ls__, eq__, gs__) = (p, [], ls, eq, gs)" shows "P"
xor_left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b XOR a XOR c = a XOR b XOR c"
positions_in_row_nth: fixes s :: "('b \<times> bool list) stream"   and i :: "nat" assumes "\<exists>!p. snd (s !! p) ! i" shows "snd (s !! the_elem (positions_in_row s i)) ! i"
pnormalize_Nil: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.semiring_0 plus zero times" shows "semiring_0.pnormalize zero [] = []"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "Adj.C.seq g f" shows "G' (g \<cdot>\<^sub>C f) = G' g \<cdot>\<^sub>D G' f"
perm_int_def: fixes pi :: "('a \<times> 'a) list"   and i :: "int" shows "pi \<bullet> i = i"
pure_intuitionistic: fixes P :: "'a \<Rightarrow> bool" assumes "pure P" shows "intuitionistic P"
no_inter: shows "outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) xs \<inter> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>t) = {}"
simps_1: fixes xc :: "'a" shows "polychain_of xc [] = []"
fg_inv: shows "map_graph_fn G (f \<circ> g) = G"
Hermite_of_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "Hermite_of A ass res bezout = (let A' = echelon_form_of A bezout in Hermite_of_upt_row_i A' (nrows A) ass res)"
adjust_in_span: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us + w \<in> local.span (insert w (set us))"
pm_of_idx_pm_plus: fixes xs :: "'c list"   and f :: "nat \<Rightarrow>\<^sub>0 'd"   and g :: "nat \<Rightarrow>\<^sub>0 'd" shows "pm_of_idx_pm xs (f + g) = pm_of_idx_pm xs f + pm_of_idx_pm xs g"
le_fininf_singleton: fixes v :: "'action stream" shows "[shd v] \<preceq>\<^sub>F\<^sub>I v"
pos_minus_divideC_le_eq: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "complex"   and b :: "'b"   and a :: "'b" assumes "class.ordered_complex_vector scaleR scaleC plus zero minus uminus less_eq less"   and "0 < c" shows "less_eq (uminus (scaleC (inverse c) b)) a = less_eq (uminus b) (scaleC c a)"
moebius_pt_comp_inv_left: fixes M :: "moebius"   and z :: "complex_homo" shows "moebius_pt (moebius_inv M) (moebius_pt M z) = z"
the_coext_prop_2: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' x f" shows "f = \<epsilon>o x \<cdot>\<^sub>C F (x\<epsilon>.the_coext x' f)"
mono_mprefix_D_process: fixes P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process"   and A :: "'a set" assumes "\<forall>x. P x \<sqsubseteq>\<^sub>D Q x" shows "Mprefix A P \<sqsubseteq>\<^sub>D Mprefix A Q"
simps: fixes i :: "nat" shows "local.der_stack i = (if local.has_left i then local.the_left i # local.der_stack (local.the_left i) else [])"
complex_diff_cnj: fixes z :: "complex" shows "z - cnj z = cor (2 * Im z) * \<i>"
sorted_rev_nth_mono: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and i :: "nat"   and j :: "nat" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (rev xs)"   and "i \<le> j"   and "j < length xs" shows "less_eq (xs ! j) (xs ! i)"
d: fixes x :: "real_alg" shows "real_alg_show_info x = convert_info (info_real_alg x)"
F2: shows "\<forall>x\<^sub>3 x\<^sub>1. (0::'a) \<le> x\<^sub>1 \<longrightarrow> \<bar>x\<^sub>3 * x\<^sub>1\<bar> = \<bar>x\<^sub>3\<bar> * x\<^sub>1"
signed_take_bit_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = or (take_bit n a) (times (zero_neq_one.of_bool one zero (bit a n)) (not (mask n)))"
app_vfieldD: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r" shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<union>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
trans_equiv_greater: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b = Equiv"   and "cmp b c = Greater" shows "cmp a c = Greater"
heap_is_wellformed_children_disc_nodes_different: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "set children \<inter> set disc_nodes = {}"
restrict_apply: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and x :: "'e" shows "restrict f A x = (if x \<in> A then f x else undefined)"
mset_factorsD_3: fixes F :: "'a multiset"   and p :: "'a" assumes "mset_factors F p" shows "\<Prod>\<^sub># F = p"
A2: shows "\<forall>SP TQ. SP \<in>S P \<and> TQ \<in>T Q \<longrightarrow> (\<lbrakk>SP\<rbrakk>, TQ) \<in> TRel"
Abs_nd_fun_inverse: fixes y :: "'b \<Rightarrow> 'b set" assumes "s2p {f. s2p UNIV f} y" shows "Rep_nd_fun (Abs_nd_fun y) = y"
totient_conv_totient_aux1: fixes ps :: "nat list"   and n :: "nat" assumes "set ps = prime_factors n"   and "distinct ps" shows "totient n = totient_aux1 n ps"
\<b>_one: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "valid_decomp X ps"   and "standard_decomp k ps" shows "\<b> ps (Suc 0) = (if ps\<^sub>+ = [] then 0 else Suc (Max (poly_deg ` fst ` set (ps\<^sub>+))))"
option_bindSomeE: fixes X :: "'d option"   and F :: "'d \<Rightarrow> 'c option"   and v :: "'c" assumes "X \<bind> F = Some v" obtains x :: "'d" where "X = Some x" and "F x = Some v"
insert_mono: fixes C :: "'g set"   and D :: "'g set"   and a :: "'g" assumes "C \<subseteq> D" shows "insert a C \<subseteq> insert a D"
conv_abscissa_mult_const_left: fixes c :: "'a"   and f :: "'a fds" shows "conv_abscissa (fds_const c * f) \<le> conv_abscissa f"
OutputR_def: fixes x2 :: "name"   and x1 :: "name" shows "x2[x1] \<equiv> Abs_freeRes (OutputR_Rep x2 x1)"
nb\<^sub>8_3: shows "disjnt (dom (map_of (map (\<lambda>v. (v, True)) (add_effects_of op')))) (dom (map_of (map (\<lambda>v. (v, False)) (delete_effects_of op'))))"
AE_lim_stream: fixes x :: "real \<times> 'a"   and P :: "(real \<times> 'a) stream \<Rightarrow> bool" assumes "x \<in> space S"   and "Measurable.pred (stream_space S) P" shows "almost_everywhere (K.lim_stream x) P = (AE y in K x. AE \<omega> in K.lim_stream y. P (y ## \<omega>))"
char_one: shows "dc2c \<chi>\<^sub>0 \<one>\<^bsub>G\<^esub> = 1"
inject_1: fixes x3 :: "'a"   and y3 :: "'a" shows "(Atom x3 = Atom y3) = (x3 = y3)"
coprime_multiplicity: fixes p :: "nat"   and m :: "nat" assumes "prime p"   and "0 < m" shows "coprime p (m div p ^ multiplicity p m)"
transcendental_mono: fixes K :: "'a set"   and K' :: "'a set"   and x :: "'a" assumes "K \<subseteq> K'"   and "(transcendental over K') x" shows "(transcendental over K) x"
distinct_19: fixes x41 :: "typ"   and x42 :: "Core.term"   and x51 :: "Core.term"   and x52 :: "Core.term" shows "Abs x41 x42 \<noteq> x51 $ x52"
RBT_set_conv_keys: fixes t :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_set t = set (RBT_Set2.keys t)"
sum_bitset: fixes n :: "nat" shows "sum ((^) 2) (bitset n) = n"
compat2: fixes y :: "int"   and x :: "int"   and z :: "int" assumes "y < x"   and "z \<le> y" shows "z < x"
SCallInitReds: fixes P :: "(char list list \<times> char list exp) prog"   and es :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and vs :: "val list"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and v' :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and l\<^sub>2' :: "sfields"   and ef :: "char list exp"   and h\<^sub>3 :: "heap"   and l\<^sub>3 :: "sfields"   and sh\<^sub>3 :: "sheap"   and b\<^sub>3 :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>es,s\<^sub>0,b\<^sub>0\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),False\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "M \<noteq> clinit"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),False\<rangle> \<rightarrow>* \<langle>Val v',(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "length vs = length pns"   and "l\<^sub>2' = [pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3, sh\<^sub>3),b\<^sub>3\<rangle>"   and "final ef" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2, sh\<^sub>3),b\<^sub>3\<rangle>"
Cons_3: fixes j :: "nat" assumes "j < length (x # xs)"   and "j \<noteq> i" shows "(x # xs) ! j = (0::'a)"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>i a. (i = 0 \<Longrightarrow> P (Eq a)) \<Longrightarrow> P (Dvd i a)"   and "\<And>i a. (i = 0 \<Longrightarrow> P (NEq a)) \<Longrightarrow> P (NDvd i a)"   and "P T"   and "P F"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
nth_map: fixes n :: "nat"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "n < |xs|" shows "map f xs ! n = f (xs ! n)"
gamma_differentiable_2: shows "\<forall>x\<in>{0..1} - s. \<gamma> differentiable at x"
simps_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and ys :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # ys) = (Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
disj_zero_right: fixes x :: "'a" shows "x \<^bold>\<sqinter> \<one> = x"
assms_2: fixes r :: "complex"   and b :: "'a" shows "f (r *\<^sub>C b) = cnj r *\<^sub>C f b"
eq3: shows "adjoint (exexH_k k) * (adjoint (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) * P * tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) * exexH_k k = adjoint (exexH_k k) * adjoint (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) * P * (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K) * exexH_k k)"
sum_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x)" shows "less_eq zero (comm_monoid_add.sum plus zero f A)"
out_arcs_def: fixes G :: "('f, 'e) pre_digraph"   and v :: "'f" shows "out_arcs G v \<equiv> {e \<in> arcs G. tail G e = v}"
simps_6: fixes c :: "int"   and e :: "MIR.num" shows "plusinf (Le (CN 0 c e)) = F"
prems_3: shows "reads_consistent pending_write \<O> m (r # sb)"