hcju/isabelleps huggingface dataset:
G_ide: fixes D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "left_adjoint_functor D C F"   and "partial_magma.ide C x" shows "left_adjoint_functor.G D C F x = left_adjoint_functor.Go D C F x"
hyps_3: shows "degree (prod f S) \<le> sum (degree \<circ> f) S"
nth_append1: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "n < length xs" shows "(xs @ ys) ! n = xs ! n"
conjugate_exponent_real_5: fixes p :: "real" assumes "1 < p" shows "p - p / conjugate_exponent p = 1"
cnf_BigAnd: fixes ls :: "'b formula list" shows "cnf \<^bold>\<And>ls = \<Union> (cnf ` set ls)"
ode_has_derivative: fixes x :: "'c" assumes "isFDERIV DIM('c) [0..<DIM('c)] ode_e (list_of_eucl x)"   and "x \<in> Csafe" shows "(ode has_derivative blinfun_apply (ode_d1 x)) (at x)"
min_eqR: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" shows "min a b = b"
if_bounded: fixes a :: "'a"   and i :: "'a"   and b :: "'a"   and v :: "real" shows "(if a \<le> i \<and> i \<le> b then v else 0) = v * indicat_real {a..b} i"
executableI_words_fin: fixes p :: "'a"   and execute :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and enabled :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and initial :: "'a \<Rightarrow> bool"   and w :: "'b list" assumes "p \<in> transition_system_initial.nodes execute enabled initial"   and "transition_system.path execute enabled w p" shows "set w \<subseteq> transition_system_initial.executable execute enabled initial"
endofunctor_axioms: shows "endofunctor (\<cdot>) (\<lambda>f. local.Prod (cod \<iota>, f))"
simps: fixes p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "int \<times> int" shows "dist_code p\<^sub>0 p\<^sub>1 = (fst p\<^sub>0 - fst p\<^sub>1)\<^sup>2 + (snd p\<^sub>0 - snd p\<^sub>1)\<^sup>2"
nfv_simps_10: fixes I :: "\<I>"   and r :: "formula Regex.regex" shows "nfv (MatchF I r) = nfv_regex r"
\<Delta>''_wf_residual_network: assumes "\<And>x. \<not> edge \<Delta> x x" shows "\<Delta>''.wf_residual_network"
acyclic_subcls1: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "wf_prog wf_md P" shows "acyclic (subcls1 P)"
AnalBinds_delete_lookup: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>" shows "(AnalBinds (delete x \<Gamma>)\<cdot>ae) x = \<bottom>"
strict_order_N_def: fixes a :: "'b N"   and b :: "'b N" shows "a < b \<equiv> a \<le> b \<and> \<not> b \<le> a"
disc_rfr_constant_time_pos: fixes r :: "real"   and n :: "nat" assumes "- 1 < r" shows "\<exists>c>0. \<forall>w\<in>space M. disc_rfr_proc r n w = c"
H: shows "(xs @ [y], {}) \<in> seq_comp_failures P Q"
hyps_2: fixes y :: "'a" assumes "y \<in> set ys" shows "compare cmp y__ y \<noteq> Greater"
mono_Retr_3: shows "Bisim.mono ZOretrT"
sigma_algebra_iff: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "sigma_algebra \<Omega> M = (algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> \<Union> (range A) \<in> M))"
simps: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec" shows "mat_leqb_eqc A b c = (let lst = matrix_to_lpolies (two_block_non_interfering A A\<^sup>T) in map (\<lambda>i. LEQ (lst ! i) (b $ i)) [0..<dim_vec b] @ map (\<lambda>i. EQ (lst ! i) ((b @\<^sub>v c) $ i)) [dim_vec b..<dim_vec (b @\<^sub>v c)])"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> (('e \<times> 'f list) \<times> 'c) spmf) \<Rightarrow> ('c \<Rightarrow> 'g \<Rightarrow> (('h \<times> 'f list) \<times> 'c) spmf) \<Rightarrow> 'a"   and rest_scheme :: "('c, 'f, 'd, 'g, 'e, 'h, 'b) rest_scheme" shows "P (case rest_scheme of Rest x xa xb \<Rightarrow> f x xa xb) = (\<forall>x1 x2 x3. rest_scheme = Rest x1 x2 x3 \<longrightarrow> P (f x1 x2 x3))"
that: fixes a :: "nat \<Rightarrow> real"   and b :: "nat \<Rightarrow> real"   and n :: "nat" assumes "l2norm {- pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))) < e" shows "thesis"
assms_3: shows "\<forall>x\<in>L. b \<le> x"
intros_2: fixes A :: "'c set"   and a :: "'c" assumes "finite A" shows "finite (insert a A)"
is_discrete: fixes f :: "(bool, unit) concrete_category.arr" shows "partial_magma.ide (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) f = partial_magma.arr (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) f"
has_as_product_def: fixes J :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'a"   and a :: "'a" shows "\<pi>.Cop.has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J (\<cdot>\<^sup>o\<^sup>p) D a \<pi>"
int_of_uint64_plus: fixes x :: "uint64"   and y :: "uint64" shows "int_of_uint64 (x + y) = (int_of_uint64 x + int_of_uint64 y) mod 18446744073709551616"
small_refl_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(f \<in> \<omega>[F](f)) = (\<forall>\<^sub>F x in F. f x = (0::'c))"
partition_filter_conv: fixes f :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "partition f xs = (filter f xs, filter (Not \<circ> f) xs)"
eog: shows "exponential_order L c (\<lambda>x. - g x)"
conductor_gr_0: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "dcharacter n \<chi>" shows "0 < dcharacter.conductor n \<chi>"
that_3: fixes i :: "'a" assumes "i \<in> I" shows "open (U i)"
ANR_sphere: fixes a :: "'a"   and r :: "real" shows "ANR (sphere a r)"
theFT_Un_theNFT: fixes cl :: "('d, 'e, 'f) cmd list" shows "theFT cl \<union> theNFT cl = {..<length cl}"
fun_hD: shows "h \<in> D \<rightarrow> D"
bounded_clinear_mult_const: fixes g :: "'d \<Rightarrow> 'e"   and y :: "'e" assumes "bounded_clinear g" shows "bounded_clinear (\<lambda>x. g x * y)"
induct: fixes x :: "'a"   and p :: "Player"   and W :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> attractor_inductive p W"   and "\<And>v. v \<in> W \<Longrightarrow> P v"   and "\<And>v. \<lbrakk>v \<in> VV p; \<exists>w. v \<rightarrow> w \<and> w \<in> attractor_inductive p W \<and> P w\<rbrakk> \<Longrightarrow> P v"   and "\<And>v. \<lbrakk>v \<in> VV p**; \<not> deadend v; \<forall>w. v \<rightarrow> w \<longrightarrow> w \<in> attractor_inductive p W \<and> P w\<rbrakk> \<Longrightarrow> P v" shows "P x"
matrix_vector_mult_linear_continuous_on: fixes S :: "('a, 'b) Finite_Cartesian_Product.vec set"   and A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "continuous_on S ((*v) A)"
min_cons_hyp_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and e :: "nat" shows "min_cons_hyp \<psi> e \<equiv> if \<exists>i. consistent \<psi> i e \<down>= 0 then Some (LEAST i. consistent \<psi> i e \<down>= 0) else None"
set_sel: fixes a :: "'c option" assumes "a \<noteq> None" shows "the a \<in> set_option a"
IH: fixes stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and ST :: "ty list" assumes "exec_moves_d P t es h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"   and "n + max_varss es' \<le> length xs"   and "bsoks es n"   and "P,h \<turnstile> stk [:\<le>] ST"   and "conf_xcp' (compP2 P) h xcp" shows "\<exists>es'' xs''. P,es,h' \<turnstile> (es'', xs'') [\<leftrightarrow>] (stk', loc', pc', xcp') \<and> (if \<tau>moves2 (compP2 P) h stk es pc xcp then h' = h \<and> (if xcp' = None \<longrightarrow> pc < pc' then \<tau>reds1r else \<tau>reds1t) P t h (es', xs) (es'', xs'') else \<exists>ta' es'a xs'. \<tau>reds1r P t h (es', xs) (es'a, xs') \<and> True,P,t \<turnstile>1 \<langle>es'a,(h, xs')\<rangle> [-ta'\<rightarrow>] \<langle>es'',(h', xs'')\<rangle> \<and> ta_bisim wbisim1 (extTA2J1 P ta') ta \<and> \<not> \<tau>moves1 P h es'a \<and> (calls1 es' = None \<or> no_calls2 es pc \<or> es'a = es' \<and> xs' = xs))"
smap_alt: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c stream"   and s' :: "'b stream" shows "(smap f s = s') = (\<forall>n. f (s !! n) = s' !! n)"
sub_def: shows "sub \<equiv> {(f a, Get f) |a f. True}"
permute_def: fixes p :: "perm"   and a :: "'a" shows "p \<bullet> a = Abs (p \<bullet> Rep a)"
FG_proj_mult_right: fixes x :: "('a, 'g) aezfun"   and y :: "('a, 'g) aezfun" assumes "x \<in> ActingGroup.group_ring" shows "FG_proj (y * x) = FG_proj y * x"
Arzela_Ascoli: fixes S :: "'a set"   and \<F> :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "real" assumes "compact S"   and "\<And>n x. x \<in> S \<Longrightarrow> norm (\<F> n x) \<le> M"   and "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>n y. y \<in> S \<and> norm (x - y) < d \<longrightarrow> norm (\<F> n x - \<F> n y) < e" obtains g :: "'a \<Rightarrow> 'b" and k :: "nat \<Rightarrow> nat" where "continuous_on S g" and "strict_mono k" and "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. N \<le> n \<and> x \<in> S \<longrightarrow> norm (\<F> (k n) x - g x) < e"
do_store_action_prj_simps_4: fixes fl :: "bool"   and w :: "('a, 'c, 'd) mem_store_action"   and s :: "('a, 'b, 'c, 'd) Model.local_state" shows "(fl = fA (do_store_action w s)) = (fl = fA s \<and> w \<noteq> mw_fA (\<not> fA s) \<or> w = mw_fA fl)"
as_raw_\<alpha>_def: fixes s :: "'c array \<times> nat" shows "as_raw_\<alpha> s \<equiv> take (snd s) (list_of_array (fst s))"
f4: shows "rng_to_rng_of_frac \<one> = \<one>\<^bsub>rec_rng_of_frac\<^esub>"
simps_2: fixes r :: "'b memref"   and rs :: "'b memref list" shows "distinct_read_tmps (r # rs) = (case r of Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> t \<notin> read_tmps rs \<and> distinct_read_tmps rs | _ \<Rightarrow> distinct_read_tmps rs)"
map_cons_eq: fixes m :: "nat"   and xs :: "nat list" shows "map ((\<lambda>a. rec_exec a (m # xs)) \<circ> recf.id (Suc (length xs))) [Suc 0..<Suc (length xs)] = map (\<lambda>i. xs ! (i - 1)) [Suc 0..<Suc (length xs)]"
weakly_preserves_objects: fixes a :: "'c" assumes "C.obj a" shows "D.isomorphic (G.map\<^sub>0 a) (G a)"
assms: shows "s' = snd (fst (nop_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
infinite_iff_alw_ev: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "infinite {m. P (sdrop m \<omega>)} = alw (ev P) \<omega>"
notallz: shows "\<exists>v\<in>W \<union> X. c v \<noteq> (0::'a)"
WTDynCast: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and D :: "char list"   and C :: "char list" assumes "P,E \<turnstile> e :: Class D"   and "is_class P C"   and "P \<turnstile> Path D to C unique \<or> (\<forall>Cs. \<not> P \<turnstile> Path D to C via Cs )" shows "P,E \<turnstile> Cast C e :: Class C"
gsi: shows "fs.gs.gso (i - 1) \<in> Rn"
apartment_galleryD_chamber: fixes Xa :: "'b set set"   and \<A> :: "'b set set set"   and X :: "'b set set"   and xs :: "'b set list"   and x :: "'b set" assumes "ChamberComplexWithApartmentSystem Xa \<A>"   and "X \<in> \<A>"   and "SimplicialComplex.maxsimpchain X xs"   and "x \<in> set xs" shows "SimplicialComplex.maxsimp X x"
spectrum_def: fixes A :: "'b mat" shows "spectrum A = Collect (eigenvalue A)"
abs_convergent_prod_imp_ev_nonzero: fixes f :: "nat \<Rightarrow> 'e" assumes "abs_convergent_prod f" shows "\<forall>\<^sub>F n in sequentially. f n \<noteq> (0::'e)"
bin_exhaust: fixes bin :: "int" obtains x :: "int" and b :: "bool" where "bin = of_bool b + 2 * x"
preserves_reflects_arr: fixes f :: "'a" shows "S.arr (\<ll> f) = S.arr f"
trg_src: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "trg (src \<mu>) = src \<mu>"
star_inductl: fixes x :: "'a"   and y :: "'a" assumes "x \<odot> y \<le> y" shows "x\<^sup>\<star> \<odot> y \<le> y"
select_convs_13: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getCll \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getCll"
elims: fixes x :: "'a"   and xa :: "'a tree"   and y :: "'a tree" assumes "local.insert x xa = y" obtains x :: "'a" and t :: "'a tree" where "xa__ = x" and "xa = t" and "y = (case ins (size t) 0 x t of Same \<Rightarrow> t | Bal t' \<Rightarrow> t')"
ldv'': fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<cdot> (z \<sqinter> \<top> \<cdot> x) = y \<cdot> z \<sqinter> \<top> \<cdot> x"
arr_iff_in_hom: fixes f :: "'a" shows "hom'.arr f = hom'.in_hom f (hom'.dom f) (hom'.cod f)"
simple_ownership_distinct: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'p"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'p memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'p"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'p memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "\<O>\<^sub>i \<inter> \<O>\<^sub>j = {}"
is_monoid: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
has_sgnx_eq_rhs: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and F :: "real filter"   and y :: "real" assumes "(f has_sgnx x) F"   and "x = y" shows "(f has_sgnx y) F"
Solutions_def: shows "Solutions = {(x, y). a \<bullet> x = b \<bullet> y \<and> length x = m \<and> length y = n}"
suffix_append: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "suffix xs (ys @ zs) = (suffix xs zs \<or> (\<exists>xs'. xs = xs' @ zs \<and> suffix xs' ys))"
G: shows "UNlift01 cl dl P F II1 = UNlift01 cl dl P F II2"
r_match_alt: fixes L :: "'d set list"   and l :: "'d list" shows "r_match L l = (length L = length l \<and> (\<forall>i<length l. l ! i \<in> L ! i))"
simps_6: fixes vs :: "'b list"   and bs :: "'b list"   and c :: "poly"   and a :: "tm" shows "Itm vs bs (tm.Mul c a) = \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> * Itm vs bs a"
comp_ide_self: fixes a :: "'a" assumes "ide a" shows "a \<cdot> a = a"
a_in_carr_fgmodule: fixes a :: "'d"   and A :: "'d set"   and z :: "'d"   and i :: "'d \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd" assumes "a \<in> A" shows "a \<in> carrier (fgmodule R A z i f s)"
leq: fixes s :: "real" assumes "s \<in> {0..t}" shows "1 * frechet I (f1 fid2 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol s))) (snd (mk_v I (OVar vid1) (sol 0, b) (sol s))) \<le> 1 * frechet I (f1 fid1 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol s))) (snd (mk_v I (OVar vid1) (sol 0, b) (sol s)))"
cases: fixes a :: "'b list" assumes "tail_side a" obtains uu :: "'b" and xs :: "'b list" where "a = uu # xs"
simps_2: fixes env :: "v sem_env"   and uw :: "char list"   and e :: "exp0" shows "is_cupcake_value (Closure env uw e) = (is_cupcake_exp e \<and> is_cupcake_all_env env)"
exec_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,s\<rangle> \<Rightarrow> t" obtains f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | s :: "'e" and s' :: "('e, 'f) xstate" where "sb__ = Normal sc__" and "\<Gamma>\<turnstile> \<langle>c1,Normal sc__\<rangle> \<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> \<Rightarrow> t"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
IH': fixes j :: "'sz" assumes "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT (x2a j)}. FVT (\<sigma> i))" shows "sterm_sem (adjointFO I \<sigma> \<nu>) (x2a j) = sterm_sem (adjointFO I \<sigma> \<omega>) (x2a j)"
ind: fixes as :: "nat"   and lm :: "nat list"   and n :: "nat" shows "abc_steps_l (as, lm) ap (m + n) = abc_steps_l (abc_steps_l (as, lm) ap m) ap n"
MaxR_f_def: fixes d :: "'d" shows "MaxR_f d = (\<lambda>x acc. if x \<in> Field (Pd d) then Some (case acc of None \<Rightarrow> x | Some y \<Rightarrow> maxR d x y) else acc)"
p: shows "p = fst (bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k))"
same_sgn_sgn_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "sgn b = sgn a" shows "sgn (plus a b) = sgn a"
zero_le_power2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a 2)"
uniform_limitI: fixes S :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter" assumes "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e" shows "uniform_limit S f l F"
S4: shows "c_fold (c_unfold k (c_snd u)) = c_snd u"
dg_prod_is_arrD_3: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" shows "vsv a"
continuous_within_compose2: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "continuous (at x within s) f"   and "continuous (at (f x) within f ` s) g" shows "continuous (at x within s) (\<lambda>x. g (f x))"
index_mat_addrow_2: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and ad :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_addrow_gen ad mul a i l A $$ (i, j) = ad (mul a (A $$ (l, j))) (A $$ (i, j))"
21: shows "irreflexive ((w \<sqinter> - prim_EP w v e)\<^sup>+ * e * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star>)"
size_def: fixes m :: "('c, 'd) mapping" shows "Mapping.size m = (if finite (Mapping.keys m) then card (Mapping.keys m) else 0)"
not_less: fixes y :: "'d"   and x :: "'d" shows "(\<not> y < x) = (x \<le> y)"
interval_lebesgue_integral_mult_right: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and c :: "'a"   and f :: "real \<Rightarrow> 'a" shows "LINT x=a..b|M. (c * f x) = c * interval_lebesgue_integral M a b f"
assms_4: shows "lincomb_list f ss = 0\<^sub>v n"
ir_Skip_Skip: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "ir_hoare P SKIP SKIP P"
signed_take_bit_Suc: fixes n :: "nat"   and a :: "'a" shows "signed_take_bit (Suc n) a = a mod (2::'a) + (2::'a) * signed_take_bit n (a div (2::'a))"
ground_list: shows "is_ground_cls_list (subst_cls_list Cs \<sigma>)"
pre_transl: fixes C :: "Nielson_VCG.acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "Nielson_VCGi.pre (transl C) Q = Nielson_VCG.pre C Q"
\<beta>_numbound0: fixes p :: "fm"   and bs :: "real list" assumes "iszlfm p bs" shows "\<forall>b\<in>set (\<beta> p). numbound0 b"
P_def: shows "P = (\<lambda>n. \<forall>i>n. local.coeff p i = \<zero>)"
nextl_path_to: fixes q :: "hf" assumes "q \<in> accessible" shows "nextl (dfa.init M) (path_to q) = q"
assms_3: fixes \<chi> :: "'a ltl"   and j :: "nat" assumes "\<chi> \<in> \<G>" shows "suffix i w \<Turnstile> \<chi> = suffix (i + j) w \<Turnstile> \<chi>"
\<beta>t: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau>' t $ Fv x \<tau>') (subst_bv (Fv x \<tau>') t)"
image_add_atLeastLessThan': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "'a"   and i :: "'a"   and j :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "(\<lambda>n. plus n k) ` ord.atLeastLessThan less_eq less i j = ord.atLeastLessThan less_eq less (plus i k) (plus j k)"
simps_2: fixes I :: "interp"   and r :: "real" shows "term_sem I (Number r) = (\<lambda>\<omega>. r)"
ground_img_obtain_fun: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term"   and x :: "'a" assumes "ground (subst_range s)"   and "x \<in> subst_domain s" obtains f :: "'b" and T :: "('b, 'a) Term.term list" where "s x = Fun f T" and "Fun f T \<in> subst_range s" and "fv (Fun f T) = {}"
UnI1: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A" shows "c \<in> A \<union> B"
zero_comp_diff_simps_1: fixes a :: "'a"   and b :: "'a" shows "((0::'a) \<le> a - b) = (b \<le> a)"
inf_compl_bot_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (inf y (uminus x)) = bot"
trancl_into_trancl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "r\<^sup>+\<^sup>+ a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
sum_inverse_squares_expansion: fixes n :: "nat"   and N :: "nat" assumes "0 < n"   and "0 < N" shows "(\<Sum>k = 1..n. 1 / (real k)\<^sup>2) = pi\<^sup>2 / 6 - 1 / real n + 1 / (2 * (real n)\<^sup>2) - (\<Sum>i = 1..N. bernoulli (2 * i) / real (n ^ (2 * i + 1))) - EM_remainder (2 * N + 1) (\<lambda>x. - fact (2 * N + 2) / x ^ (2 * N + 3)) (int n)"
iteratei_def: fixes t :: "('e, 'f) trie" shows "iteratei t = set_iterator_image trie_reverse_key (Trie_Impl.iteratei (impl_of t))"