hcju/isabelleps huggingface dataset:
pole_abs: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0" shows "pole (proj2_line_abs v) = proj2_abs (M *v v)"
gcd_0_left: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd zero a = normalize a"
IV: assumes "enabled cfg1 msg"   and "\<not> enabled cfg2 msg" shows "stepReachable cfg1 msg cfg2"
abs_conv_abscissa_power_le: fixes f :: "'a fds"   and n :: "nat" shows "abs_conv_abscissa (f ^ n) \<le> abs_conv_abscissa f"
is_rbt_rbt_sorted: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) rbt" assumes "ord.is_rbt less t" shows "ord.rbt_sorted less t"
61311: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - y = - (- - y \<sqinter> x)"
inv_gbijec_gbijec: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "gbij\<^bsub>F,G\<^esub> f" shows "gbij\<^bsub>G,F\<^esub> (f\<inverse>\<^bsub>carrier G,carrier F\<^esub>)"
l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs \<equiv> l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs \<and> l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf"
set_map0_1: fixes f1 :: "'l \<Rightarrow> 'n"   and f2 :: "'m \<Rightarrow> 'o" shows "F1set1 \<circ> F1map f1 f2 = (`) f1 \<circ> F1set1"
atLeastatMost_empty_iff2: fixes a :: "'a"   and b :: "'a" shows "({} = ordered_powerprod_lin.atLeastAtMost a b) = (\<not> a \<preceq> b)"
hseqI': fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "D.hseq \<nu> \<mu>"
bres_def: fixes x :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "prod.bres x z = Sup_prod {y. less_eq_prod (times_prod' x y) z}"
OUT_g_a: shows "d_OUT g a < weight \<Gamma> a"
liminf_SUP_INF: fixes f :: "nat \<Rightarrow> 'a" shows "liminf f = (SUP n. Inf (f ` {n..}))"
Max_in_lits: fixes C :: "'b multiset" assumes "C \<noteq> {#}" shows "Max_mset C \<in># C"
replacefacesAt_eq: fixes ns :: "nat list"   and oldf :: "face"   and newfs :: "face list"   and F :: "face list list" shows "replacefacesAt ns oldf newfs F \<equiv> replacefacesAt2 ns oldf newfs F"
intros_1: fixes c :: "'c \<Rightarrow> 'd" shows "ce_star c [] c"
powf: fixes z :: "complex" assumes "z \<in> ball \<xi> r" shows "(\<lambda>n. (deriv ^^ n) f \<xi> / fact n * (z - \<xi>) ^ n) sums f z"
erl2: shows "l\<^sub>2 \<equiv>\<^sub>E rl2"
nsq: shows "\<not> ps (S t j) (occurs_on (t ! j)) \<noteq> None"
listset_map_imageE: fixes ys :: "'b list"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c set list" assumes "ys \<in> listset (map ((`) f) xs)" obtains ys' :: "'c list" where "ys' \<in> listset xs" and "ys = map f ys'"
Union_atLeastAtMost_real_of_nat: fixes a :: "nat"   and b :: "nat" assumes "a < b" shows "(\<Union>n\<in>{a..<b}. {real n..real (n + 1)}) = {real a..real b}"
map_to_sorted_list_by_tsl: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Rk :: "('g \<times> 'f) set"   and Rv :: "('h \<times> 'i) set"   and Rs :: "('g \<times> 'f) set \<Rightarrow> ('h \<times> 'i) set \<Rightarrow> ('j \<times> ('f \<Rightarrow> 'i option)) set"   and tsl :: "'j \<Rightarrow> ('g \<times> 'h) list" assumes "MINOR_PRIO_TAG (- 11)"   and "SIDE_GEN_ALGO (is_map_to_sorted_list R Rk Rv Rs tsl)" shows "(\<lambda>s. RETURN (tsl s), map_to_sorted_list R) \<in> \<langle>Rk, Rv\<rangle>Rs \<rightarrow> \<langle>\<langle>Rk \<times>\<^sub>r Rv\<rangle>list_rel\<rangle>nres_rel"
bequiv_up_to_trans: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and b'' :: "bexp" assumes "P \<Turnstile> b <\<sim>> b'"   and "P \<Turnstile> b' <\<sim>> b''" shows "P \<Turnstile> b <\<sim>> b''"
prod_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "HF'.prod f g \<equiv> HF'.tuple (local.comp f (pr1 (local.dom f) (local.dom g))) (local.comp g (pr0 (local.dom f) (local.dom g)))"
round_down_uminus_eq: fixes p :: "int"   and x :: "real" shows "round_down p (- x) = - round_up p x"
sort_eqv_imp_normalize_sort_eq: fixes cs :: "(String.literal \<times> String.literal) set"   and s1 :: "String.literal set"   and s2 :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_ex cs s1"   and "sort_ex cs s2"   and "sort_eqv cs s1 s2" shows "normalize_sort cs s1 = normalize_sort cs s2"
natUnion_ex: fixes x :: "'a"   and f :: "nat \<Rightarrow> 'a set" assumes "x \<in> natUnion f" shows "\<exists>i. x \<in> f i"
cong: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a measure"   and h :: "'b \<Rightarrow> 'a measure" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sup_measure.F g A = sup_measure.F h B"
residue_primroot_prime_lift_iff: fixes p :: "nat"   and g :: "nat" assumes "prime p"   and "odd p"   and "residue_primroot p g" shows "(\<forall>k>0. residue_primroot (p ^ k) g) = [g ^ (p - 1) \<noteq> 1] (mod p\<^sup>2)"
setdist_closure_2: fixes T :: "'b set"   and S :: "'b set" shows "setdist T (closure S) = setdist T S"
HLD_iff: fixes s :: "'b set"   and \<omega> :: "'b stream" shows "HLD s \<omega> = (shd \<omega> \<in> s)"
c_fst_at_0: shows "c_fst 0 = 0"
asubg_pOp_closed: fixes A :: "('e, 'f) aGroup_scheme"   and H :: "'e set"   and x :: "'e"   and y :: "'e" assumes "aGroup A"   and "A +> H"   and "x \<in> H"   and "y \<in> H" shows "x \<plusminus>\<^bsub>A\<^esub> y \<in> H"
finite_supp_implies_hereditarily_fs_Conj\<^sub>\<alpha>: fixes xset :: "('e, 'f, 'g) formula set['e]" assumes "finite (supp xset)" shows "hereditarily_fs (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
if2: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e1 :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and e2 :: "exp0"   and e3 :: "exp0" assumes "evaluate ck env s1 e1 (s2, Rval v1) \<and> do_if v1 e2 e3 = None" shows "evaluate ck env s1 (exp0.If e1 e2 e3) (s2, Rerr (Rabort Rtype_error))"
progress: fixes xp :: "nat option"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list" assumes "xp = None"   and "frs \<noteq> []" shows "\<exists>\<sigma>'. P \<turnstile> (xp, h, frs) -jvm\<rightarrow>\<^sub>1 \<sigma>'"
store_sub2_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and addr :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (store_sub2 instr curr_win rd 10 addr s1)) \<and> t2 = snd (fst (store_sub2 instr curr_win rd 10 addr s2))" shows "low_equal t1 t2"
finite_t1_space_imp_discrete_topology: fixes X :: "'b topology"   and U :: "'b set" assumes "topspace X = U"   and "finite U"   and "t1_space X" shows "X = discrete_topology U"
effect_raiseE: fixes x :: "String.literal"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "effect (raise x) h h' r" obtains "False"
m_rcancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> \<zero>\<^bsub>K\<^esub>"   and "a \<in> carrier K"   and "b \<in> carrier K"   and "c \<in> carrier K" shows "(b \<otimes>\<^bsub>K\<^esub> a = c \<otimes>\<^bsub>K\<^esub> a) = (b = c)"
simps_4: fixes v :: "'a formula" shows "is_cnf (\<^bold>\<not> v) = is_disj (\<^bold>\<not> v)"
cos_def: shows "cos = (\<lambda>x. \<Sum>n. cos_coeff n *\<^sub>R x ^ n)"
b0: shows "(fps_exp a - 1) $ 0 = (0::'a)"
antisym: fixes x :: "'c list"   and y :: "'c list" assumes "prefix x y"   and "prefix y x" shows "x = y"
ifci_def: fixes v :: "nat"   and t :: "nat"   and e :: "nat"   and bdd :: "'a bddi_scheme" shows "ifci v t e bdd \<equiv> if t = e then return (t, bdd) else pointermap_getmki (v, t, e) (dpmi bdd) \<bind> (\<lambda>(p, u). return (Suc (Suc p), dpmi_update (const u) bdd))"
wf_digraphI_app_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "aG.digraph_isomorphism hom" shows "wf_digraph (app_iso hom (app_iso hom G))"
product_comm: fixes A :: "'b fset set"   and B :: "'b fset set" shows "A \<otimes> B = B \<otimes> A"
Comp_Id_Hom: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'a"   and f :: "'b"   and A :: "'a" assumes "concrete_category Obj Hom Id Comp"   and "B \<in> Obj"   and "f \<in> Hom A B" shows "Comp B B A (Id B) f = f"
Seq_Ord_D: fixes s :: "hf"   and k :: "hf"   and l :: "hf" assumes "Seq s k"   and "l \<^bold>\<in> k"   and "Ord k" shows "Seq s l"
foldl_append: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'e"   and xs :: "'f list"   and ys :: "'f list" shows "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
isCont_polynom: fixes a :: "'a"   and c :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "isCont (\<lambda>w. \<Sum>i\<le>n. c i * w ^ i) a"
conjugate_exponent_ennreal_1_2_PInf_2: shows "conjugate_exponent \<infinity> = 1"
map_upd_submap: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'b option"   and y :: "'b"   and m' :: "'a \<Rightarrow> 'b option" assumes "x \<notin> dom m" shows "(m(x \<mapsto> y) \<subseteq>\<^sub>m m') = (m' x = Some y \<and> m \<subseteq>\<^sub>m m')"
t_heapify_bound: fixes i :: "nat"   and xs :: "'b list" assumes "i \<le> length xs" shows "t_heapify i xs \<le> 5 * i + 1"
sigma_eq_Dynkin: fixes M :: "'c set set"   and \<Omega> :: "'c set" assumes "M \<subseteq> Pow \<Omega>"   and "Int_stable M" shows "sigma_sets \<Omega> M = Dynkin \<Omega> M"
meet_ac_2: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" shows "x \<sqinter>\<^bsub>L\<^esub> y = y \<sqinter>\<^bsub>L\<^esub> x"
intros_2: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and es :: "e list"   and t1s :: "t list"   and t2s :: "t list"   and e :: "e"   and t3s :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> es : t1s _> t2s"   and "\<S>\<bullet>\<C> \<turnstile> [e] : t2s _> t3s" shows "\<S>\<bullet>\<C> \<turnstile> es @ [e] : t1s _> t3s"
folds_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and S :: "'c set" shows "folds f z S \<equiv> SOME x. fold_graph f z S x"
del_bd: fixes t :: "('a, 'b) rbt"   and x :: "'a" assumes "bd_inv t"   and "cl_inv t" shows "bd_inv (del x t) \<and> (if cl t = R then black_depth (del x t) = black_depth t else black_depth (del x t) = black_depth t - 1)"
is_abelian_group: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "abelian_group R"
eval_fds_deriv: fixes f :: "'a fds"   and s :: "'a" assumes "conv_abscissa f < ereal (s \<bullet> (1::'a))" shows "eval_fds (fds_deriv f) s = deriv (eval_fds f) s"
mult_commute_abs: fixes c :: "'b" shows "(\<lambda>x. x * c) = (*) c"
cos_tan: fixes x :: "real" assumes "\<bar>x\<bar> < pi / 2" shows "cos x = 1 / sqrt (1 + (tan x)\<^sup>2)"
measurable_on_sf_limit_lemma1: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "\<And>a b. {x \<in> S. a \<le> f x \<and> f x < b} \<in> sets (lebesgue_on S)" obtains g :: "nat \<Rightarrow> 'c \<Rightarrow> real" where "\<And>n. g n \<in> borel_measurable (lebesgue_on S)" and "\<And>n. finite (range (g n))" and "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x"
depth_nmset_MSet: fixes x :: "'b nmultiset"   and M :: "'b nmultiset multiset" assumes "x \<in># M" shows "|x| < |MSet M|"
puree_alt: fixes \<Phi> :: "bool" shows "\<up> \<Phi> = (\<lambda>s. \<Phi> \<and> \<box> s)"
lmap_prepend: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c llist" shows "lmap f (prepend xs ys) = prepend (map f xs) (lmap f ys)"
normalize_bound_1_poly: fixes rai :: "int poly \<times> rat \<times> rat" shows "poly_real_alg_1 (normalize_bounds_1 rai) = poly_real_alg_1 rai"
lengthGtOneTwoDistinctElements: fixes l :: "'a list" assumes "uniq l"   and "1 < length l"   and "l \<noteq> []" shows "\<exists>a1 a2. a1 \<in> set l \<and> a2 \<in> set l \<and> a1 \<noteq> a2"
L3: shows "Domain r \<union> Range r \<in> ce_sets"
w: shows "wave (\<Gamma>\<lparr>weight := weight \<Gamma> - u\<rparr>) f"
TS_subset: fixes i :: "nat"   and j :: "nat"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat" assumes "i \<le> j" shows "TS M2 M1 \<Omega> V m i \<subseteq> TS M2 M1 \<Omega> V m j"
pbdom_def: fixes f :: "'a"   and g :: "'a" shows "ECP.pbdom f g \<equiv> local.dom (prj0 f g)"
Lin_def: shows "Lin \<equiv> {L. strict_linear_order L}"
c_top_welldef: fixes N :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "N \<bullet> (a \<cdot> b) = c_top G N (N \<bullet> a) (N \<bullet> b)"
intros_14: fixes \<C> :: "t_context"   and t :: "t" shows "\<C> \<turnstile> [Select] : [t, t, T_i32] _> [t]"
birkhoff_sum_1_2: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "A.birkhoff_sum f 1 x = f x"
less_eq_admS_def: fixes x :: "'b admS"   and y :: "'b admS" shows "x \<le> y \<equiv> unlr x \<subseteq> unlr y"
strict_trans1: fixes b :: "nat list"   and a :: "nat list"   and c :: "nat list" assumes "b \<le>\<^sub>v a"   and "c <\<^sub>v b" shows "c <\<^sub>v a"
empty_replicate: fixes n :: "nat"   and x :: "'b" shows "([] = replicate n x) = (n = 0)"
mk_dimvec_def: fixes ds :: "'a list" shows "mk_dimvec ds = (if length ds = CARD('b) then dim_lambda (\<lambda>d. ds ! enum_ind d) else 1)"
finite_wfvarsoccs\<^sub>s\<^sub>t\<^sub>p: fixes x :: "('d, 'c) strand_step" shows "finite (wfvarsoccs\<^sub>s\<^sub>t\<^sub>p x)"
intro: fixes get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>h shadow_root_ptr h' r ptr'. \<lbrakk>h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'; r \<in> get_shadow_root_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_delete_shadow_root_get_shadow_root get_shadow_root_locs"
less_integer_def: shows "(<) \<equiv> map_fun int_of_integer (map_fun int_of_integer id) (<)"
starfun_inverse2: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b star" shows "inverse ((*f* f) x) = (*f* (\<lambda>x. inverse (f x))) x"
neqE: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<noteq> y" obtains "x \<prec> y"   | "y \<prec> x"
ideD_1: fixes a :: "'c" assumes "C.ide a" shows "C.arr a"
flush_step_sb_subset: fixes m :: "nat \<Rightarrow> nat"   and sb :: "'a memref list"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)"   and m' :: "nat \<Rightarrow> nat"   and sb' :: "'a memref list"   and \<O>' :: "nat set \<times> (nat \<Rightarrow> bool option) \<times> (nat \<Rightarrow> bool option)" assumes "(m, sb, \<O>) \<rightarrow>\<^sub>f (m', sb', \<O>')" shows "set sb' \<subseteq> set sb"
inject_3: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and y41 :: "('a, 'b, 'c) Stmt"   and y42 :: "('a, 'b, 'c) Stmt" shows "(x41 ;; x42 = y41 ;; y42) = (x41 = y41 \<and> x42 = y42)"
lift_Def_node: fixes V :: "'f"   and Def :: "'g \<Rightarrow> 'f set"   and n :: "'g"   and E :: "'g"   and X :: "'g"   and H :: "'f set"   and L :: "'f set" assumes "V \<in> Def n" shows "(Node n, V) \<in> lift_Def_set Def E X H L"
Im_complex_div_eq_0: fixes a :: "complex"   and b :: "complex" shows "(Im (a / b) = 0) = (Im (a * cnj b) = 0)"
contains_key_hassoc_lookup_def: shows "contains_key_hassoc_lookup \<equiv> hassoc.contains_key"
reflp_on_converse_simp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "reflp_on P\<inverse>\<inverse> A = reflp_on P A"
mantissa_def: fixes f :: "float" shows "mantissa f = fst (SOME p. f = 0 \<and> fst p = 0 \<and> snd p = 0 \<or> f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> odd (fst p))"
ceiling_diff_floor_le_1: fixes x :: "'a" shows "\<lceil>x\<rceil> - \<lfloor>x\<rfloor> \<le> 1"
cancel_append_same1: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "cancel xs ys" shows "cancel (zs @ xs) (zs @ ys)"
runit_naturality: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "\<mu> \<cdot> \<r>[local.dom \<mu>] = \<r>[cod \<mu>] \<cdot> (\<mu> \<star> src \<mu>)"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat"   and y :: "'a" assumes "\<sigma> x xa xb = y" obtains i :: "nat" and j :: "nat" where "x = 0" and "xa = ia__" and "xb = ja__" and "y = (0::'a)"   | l :: "nat" and i :: "nat" and j :: "nat" where "x = Suc l" and "xa = i" and "xb = j" and "y = (d (Suc l) * \<sigma> l i j + \<mu>' i l * \<mu>' j l) / d l"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_4: fixes l :: "'e strand_label"   and t :: "('f, 'g) Term.term"   and s :: "('f, 'g) Term.term"   and A :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, insert\<langle>t,s\<rangle>) # A) = (l, insert\<langle>t,s\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
acc_err_def: fixes p :: "nat"   and x :: "'a \<times> real"   and e :: "real" shows "acc_err p x e \<equiv> (fst x, truncate_up p (snd x + e))"
ennreal_divide_numeral: fixes x :: "real"   and b :: "num" assumes "0 \<le> x" shows "ennreal x / numeral b = ennreal (x / numeral b)"
encode_0_iff: fixes base :: "nat"   and e :: "nat"   and n :: "nat" assumes "2 \<le> base" shows "(encode base e n = 0) = (n = 0)"
joinable_def: fixes G :: "graph"   and p :: "nat list"   and q :: "nat list" shows "joinable G p q = (is_path G p \<and> is_path G q \<and> last p = hd q)"
lcm_mult_unit1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "lcm (b * a) c = lcm b c"
ReachableTrans: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and cfg2 :: "('a, 'c, 'b) configuration"   and cfg3 :: "('a, 'c, 'b) configuration" assumes "asynchronousSystem.reachable trans sends cfg1 cfg2"   and "asynchronousSystem.reachable trans sends cfg2 cfg3" shows "asynchronousSystem.reachable trans sends cfg1 cfg3"
coprime_pderiv_imp_squarefree: fixes p :: "'a poly" assumes "algebraic_semidom_class.coprime p (pderiv p)" shows "squarefree p"
OP_def: fixes x :: "'g" shows "OP x \<equiv> x"
N: shows "N \<subseteq> timpl_closure_set M TI"
increasing_conv_subseq_fun: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a = acc_point s"   and "\<exists>s'. s' = convergent_subseq s" shows "is_increasing (convergent_subseq_fun s a)"
ipurge_tr_aux_foldr_eq: fixes P :: "'e \<Rightarrow> bool"   and I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and xs :: "'e list" shows "filter P (ipurge_tr_aux I D U xs) = ipurge_tr_aux_foldr I D P U xs"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Pos of Pos \<Rightarrow> f1 | Neg \<Rightarrow> f2) = f1"
Cauchy_uniformly_convergent: fixes X :: "'c set"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" assumes "uniformly_Cauchy_on X f" shows "uniformly_convergent_on X f"
a: shows "find (\<lambda>x. vec_to_iarray A !! x \<noteq> (0::'a)) [mod_type_class.to_nat i..<IArray.length (vec_to_iarray A)] = Some a"
rec_not_def: shows "rec_not = CN rec_minus [constn 1, recf.Id 1 0]"
has_integral: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c"   and a :: "'b"   and b :: "'b" shows "(f has_integral y) (cbox a b) = (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - y) < e))"
Abs1_eq_2: fixes a :: "'b"   and x :: "'a"   and y :: "'a" shows "([{atom a}]res. x = [{atom a}]res. y) = (x = y)"
comp_in_homI: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and c :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.in_hom f a b"   and "S.VV.in_hom g b c" shows "S.VV.in_hom (S.VV.comp g f) a c"
tendsto_diff_ereal: fixes x :: "ereal"   and y :: "ereal"   and f :: "'a \<Rightarrow> ereal"   and F :: "'a filter"   and g :: "'a \<Rightarrow> ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "\<bar>y\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x - g x) \<longlongrightarrow> x - y) F"
assms_4: shows "separated_by {f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>} B C"
simps_5: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "fol fm"   and q :: "fol fm" shows "semantics e f g (Iff p q) = (semantics e f g p = semantics e f g q)"
Cons_3: shows "combine Ks (u # Us) = \<zero>"
assms: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "\<exists>y. y \<in>\<^sub>\<circ> B \<and> x \<subseteq>\<^sub>\<circ> y"
partn_lst_two_swap: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and x :: "V"   and y :: "V"   and n :: "nat" assumes "Partitions.partn_lst r B [x, y] n" shows "Partitions.partn_lst r B [y, x] n"
assms_1: fixes x :: "'a" shows "P x \<longrightarrow> P' x"
isometryD: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "isometry U" shows "U* o\<^sub>C\<^sub>L U = id_cblinfun"
iffs: fixes om :: "'d \<Rightarrow> 'f"   and am :: "'e \<Rightarrow> 'g"   and more :: "'h"   and om' :: "'d \<Rightarrow> 'f"   and am' :: "'e \<Rightarrow> 'g"   and more' :: "'h" shows "(\<lparr>om = om, am = am, \<dots> = more\<rparr> = \<lparr>om = om', am = am', \<dots> = more'\<rparr>) = (om = om' \<and> am = am' \<and> more = more')"
assms_4: shows "D1 \<in> carrier_mat nr2 n2"
fresh_ssubst_dbtm: fixes i :: "'b"   and V :: "name set"   and t :: "dbtm" assumes "atom i \<sharp> p \<bullet> V"   and "V \<subseteq> Vs" shows "atom i \<sharp> ssubst (vquot_dbtm V t) V F"