hcju/isabelleps huggingface dataset:
enat_unfold: fixes stop :: "'c \<Rightarrow> bool"   and "next" :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "enat_unfold stop next a = (if stop a then 0 else eSuc (enat_unfold stop next (next a)))"
inject_5: fixes x3 :: "ty"   and x2 :: "trm"   and x1 :: "btrm"   and y3 :: "ty"   and y2 :: "trm"   and y1 :: "btrm" shows "(trm.Let x3 x2 x1 = trm.Let y3 y2 y1) = (x3 = y3 \<and> x2 = y2 \<and> x1 = y1)"
closed_segment_linear_image: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "linear f" shows "closed_segment (f a) (f b) = f ` closed_segment a b"
comp_ide_self: fixes a :: "'c" assumes "Cop.ide a" shows "a \<cdot>\<^sup>o\<^sup>p a = a"
f0_1: shows "f\<^sub>0 \<in> \<R>\<^sup>2"
bad_def: shows "bad \<equiv> SOME bad. \<exists>invK agrK sesK symK priSK. inj agrK \<and> inj sesK \<and> (\<forall>X X'. priSK X \<noteq> invK (priSK X')) \<and> (\<forall>X n. priSK X \<noteq> symK n) \<and> (\<forall>X n. invK (priSK X) \<noteq> symK n) \<and> (\<forall>a. invK (sesK a) = sesK a) \<and> (\<forall>n. invK (symK n) = symK n) \<and> (\<forall>m n. m \<in> bad \<longrightarrow> symK n = symK m \<longrightarrow> n \<in> bad)"
mem_val_mod_state: fixes asi :: "word8"   and a :: "word36"   and s :: "'b sparc_state"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "mem_val_alt asi a s = mem_val_alt asi a (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>)"
fsi: fixes i :: "nat" assumes "i < length gs" shows "gs ! i \<in> carrier_vec n"
assms_2: shows "\<Theta>,set Hs \<turnstile> B"
unit_circle_fix_preserve_is_poincare_line: fixes M :: "moebius"   and H :: "circline" assumes "unit_circle_fix M"   and "is_poincare_line H" shows "is_poincare_line (moebius_circline M H)"
T_def: shows "T = Pascal_Property.inter (line C' P) (line R B)"
SBHReadBuffered: fixes sb :: "'c memref list"   and a :: "nat"   and v :: "nat"   and volatile :: "bool"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "buffered_val sb a = Some v" shows "(Read volatile a t # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>(t \<mapsto> v), sb @ [Read\<^sub>s\<^sub>b volatile a t v], m, \<D>, \<O>, \<R>, \<S>)"
one_assn_def: shows "emp \<equiv> Abs_assn one_assn_raw"
distinct_5: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm" shows "PUnit \<noteq> PFn x41 x42 x43"
complex_sgn_def: fixes x :: "complex" shows "sgn x = x /\<^sub>R cmod x"
select_convs_2: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and more :: "'f" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
inverse_negative_imp_negative: fixes a :: "'a" assumes "inverse a < (0::'a)"   and "a \<noteq> (0::'a)" shows "a < (0::'a)"
Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> (Skip # cs, css, Normal s) \<rightarrow> (cs, css, Normal s)"
smooth_on_sum: fixes F :: "'c set"   and n :: "enat"   and S :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<And>i. \<lbrakk>i \<in> F; finite F\<rbrakk> \<Longrightarrow> n-smooth_on S (f i)"   and "open S" shows "n-smooth_on S (\<lambda>x. \<Sum>i\<in>F. f i x)"
is_local_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and I :: "'d set set"   and x :: "'d"   and U :: "'d set" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "stalk.is_local is_open \<FF> \<rho> add_str mult_str zero_str one_str I x U \<equiv> local_ring (stalk.carrier_stalk is_open \<FF> \<rho> x) (stalk.add_stalk \<FF> \<rho> add_str I) (stalk.mult_stalk \<FF> \<rho> mult_str I) (stalk.zero_stalk \<FF> \<rho> zero_str I U) (stalk.one_stalk \<FF> \<rho> one_str I U)"
0: shows "bprv (exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>))"
obj_fields_marked_good_ref_locs_def: shows "obj_fields_marked_good_ref_locs = mark_loop_mo_locs \<union> {mark_loop_mark_field_done}"
vec_setI: fixes v :: "'a vec"   and i :: "nat"   and a :: "'a" assumes "v $ i = a"   and "i < dim_vec v" shows "a \<in> set\<^sub>v v"
presWlsAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'sort, 'k, 'l, 'm, 'h, 'n) Recursion.model_scheme" shows "presWlsAbs hA MOD \<equiv> \<forall>us s A. wlsAbs (us, s) A \<longrightarrow> gWlsAbs MOD (us, s) (hA A)"
dm_downset: shows "dm \<circ> \<eta> = \<down>"
f4: shows "Deliver (r, Append r e1) \<notin> set (history (oid (i, Store e1 mo2 i)))"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and f :: "('fsym, 'tp) efsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "GE.int \<xi> (Fn f Tl) = GE_intF f (map (GE.int \<xi>) Tl)"
setep: shows "set p \<subseteq> insert (Upair u v) F"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 (\<mu> \<star> \<nu>) \<pi>"
rupture_carrier_as_pmod_image: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "UP.rupture_surj K p ` (\<lambda>q. UP.pmod q p) ` carrier (K [X]\<^bsub>K [X]\<^esub>) = carrier (Rupt\<^bsub>K [X]\<^esub> K p)"
mat_def: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list" shows "mat nr nc m = (length m = nc \<and> Ball (set m) (vec nr))"
min_dfa_accept: fixes M :: "nat bdd list \<times> bool list"   and v :: "nat"   and w :: "bool list list" assumes "wf_dfa M v"   and "list_all (is_alph v) w" shows "dfa_accepts (min_dfa M) w = dfa_accepts M w"
CFG_SSA_Transformed_code_axioms: shows "CFG_SSA_Transformed_code \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs (u_g g s1_uses) (p_g g s1_phis) var"
simple_path_def: fixes g :: "real \<Rightarrow> 'a" shows "simple_path g = (path g \<and> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0))"
subsetD: fixes A :: "'f set"   and B :: "'f set"   and c :: "'f" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
phi_nonzero: fixes n :: "nat"   and c :: "'a list"   and b :: "bool list"   and i :: "'a list" shows "0 \<le> \<phi> n (c, b, i)"
length_fold_remove1_le: fixes ys :: "'b list"   and xs :: "'b list" shows "length (fold remove1 ys xs) \<le> length xs"
sat_scale_rat: fixes v :: "nat \<Rightarrow> rat"   and c :: "constraint"   and r :: "rat" assumes "v \<Turnstile>\<^sub>c c" shows "v \<Turnstile>\<^sub>c (r *R c)"
Nil_mem_listset: fixes list :: "'c set list" shows "([] \<in> listset list) = (list = [])"
simps_3: fixes a :: "'a"   and b :: "'a" shows "(Num a \<le> Num b) = (a \<le> b)"
T_segerberg: fixes p :: "'a"   and x :: "'a" shows "d p \<cdot>  |x\<^sup>\<star>] (p \<leadsto> x \<leadsto> p) =  |x\<^sup>\<star>] d p"
Least_Min: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "finite {a. P a}"   and "\<exists>a. P a" shows "ord.Least less_eq P = linorder.Min less_eq {a. P a}"
good_item_simps_71: fixes rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rhoa"   and "good Y"   and "freshEnv ys y rhoa" shows "rhoa &[Y / y]_ys = rhoa [y \<leftarrow> Y]_ys"
intensional_simps_10: fixes y :: "'b \<Rightarrow> bool" shows "#True \<longrightarrow> y \<equiv> y"
lst: shows "lst_conv (t, xs @ snd ss2) = lst_conv ss2"
assms: shows "Diag (a \<^bold>\<otimes> b)"
nat_of_integer_code: fixes k :: "integer" shows "nat_of_integer k = (if k \<le> 0 then 0 else let (l, j) = divmod_integer k 2; l' = nat_of_integer l; l'' = l' + l' in if j = 0 then l'' else l'' + 1)"
carrier_one_not_zero: shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
invc_paint_Black: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "invc2 t" shows "invc (paint Black t)"
word_of_nat_eq_iff: fixes m :: "nat"   and n :: "nat" shows "(word_of_nat m = word_of_nat n) = (take_bit LENGTH('b) m = take_bit LENGTH('b) n)"
map_def: fixes f :: "'a" shows "\<Delta>.map f = (if arr f then J_C.MkArr (constant_functor.map J (\<cdot>) (local.dom f)) (constant_functor.map J (\<cdot>) (cod f)) (constant_transformation.map J (\<cdot>) f) else J_C.null)"
assms_2: shows "set (labels \<sigma>) \<subseteq> {\<alpha>}"
isOp_castOp: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "wlsInp (fst3 (castOp s X)) (snd3 (castOp s X)) \<and> wlsBinp (fst3 (castOp s X)) (trd3 (castOp s X)) \<and> stOf (fst3 (castOp s X)) = s \<and> Op (fst3 (castOp s X)) (snd3 (castOp s X)) (trd3 (castOp s X)) = X"
match_all_preserve_1: fixes cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and v0 :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and e :: "exp0"   and p :: "pat"   and env' :: "(char list \<times> v) list" assumes "cupcake_match_result cenv v0 pes err_v = Rval (e, p, env')"   and "cupcake_c_ns cenv"   and "is_cupcake_value v0"   and "cupcake_clauses pes" shows "list_all (is_cupcake_value \<circ> snd) env'"
has_derivative_Uniq: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "\<exists>\<^sub>\<le>\<^sub>1 F. (f has_derivative F) (at x)"
norm_cauchy_schwarz: fixes x :: "'a"   and y :: "'a" shows "inner x y \<le> norm x * norm y"
node_proc_reachable: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and i :: "nat"   and \<zeta> :: "'e"   and R :: "nat set"   and ii :: "nat"   and T :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "(\<sigma>, NodeS i \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) (otherwith S {ii} (oarrivemsg I)) (other U {ii})"   and "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'" shows "(\<sigma>, \<zeta>) \<in> oreachable T (otherwith S {ii} (orecvmsg I)) (other U {ii})"
ideal_inverse: fixes a :: "'a"   and A :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "a \<in> A"   and "ideal A R (+) mult \<zero> \<one>" shows "- a \<in> A"
metric_bound_lemma: fixes x :: "complex"   and y :: "complex" shows "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
simps_2: fixes m :: "'a delete_message" shows "is_delete (Delete m) = True"
subprob_emeasure_le_1: fixes s :: "'a cfg"   and X :: "'a cfg stream set" shows "emeasure (MC.T s) X \<le> 1"
permutations_of_set_nonempty: fixes A :: "'b set" assumes "A \<noteq> {}" shows "permutations_of_set A = (\<Union>x\<in>A. (#) x ` permutations_of_set (A - {x}))"
separated_def: fixes g :: "graph"   and V :: "nat set" shows "separated g V \<equiv> separated\<^sub>2 g V \<and> separated\<^sub>3 g V"
simps_2: fixes "is" :: "nat list"   and f :: "'var \<Rightarrow> 'var"   and i :: "nat"   and s :: "'subst" shows "globalize_node is f (H i s) = H (isidx is) (subst_renameLCs f s)"
axioms_1: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "semiconnex_irreflexive A less" shows "semiconnex A less"
non_extensible_Borsuk_map: fixes s :: "'a set"   and c :: "'a set"   and a :: "'a" assumes "compact s"   and "c \<in> components (- s)"   and "bounded c"   and "a \<in> c" shows "\<nexists>g. continuous_on (s \<union> c) g \<and> g ` (s \<union> c) \<subseteq> sphere (0::'a) 1 \<and> (\<forall>x\<in>s. g x = (x - a) /\<^sub>R norm (x - a))"
ereal_divide_zero_left: fixes a :: "ereal" shows "0 / a = 0"
elims_3: fixes x :: "Monitor.args"   and xa :: "nat"   and xb :: "nat"   and xc :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat"   and xd :: "(nat \<times> 'a option list set \<times> 'a option list set) list" assumes "\<not> valid_mmuaux' x xa xb xc xd" obtains args :: "Monitor.args" and cur :: "nat" and dt :: "nat" and tp :: "nat" and tss :: "nat queue" and len :: "nat" and maskL :: "bool list" and maskR :: "bool list" and a1_map :: "('a option list, nat) mapping" and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping" and "done" :: "'a option list set list" and done_length :: "nat" and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list" where "x = args" and "xa = cur" and "xb = dt" and "xc = (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length)" and "xd = auxlist" and "\<not> (args_L args \<subseteq> args_R args \<and> maskL = join_mask (args_n args) (args_L args) \<and> maskR = join_mask (args_n args) (args_R args) \<and> len \<le> tp \<and> length (linearize tss) = len \<and> sorted (linearize tss) \<and> (\<forall>t\<in>set (linearize tss). t \<le> cur \<and> enat cur \<le> enat t + right (args_ivl args)) \<and> table (args_n args) (args_L args) (Mapping.keys a1_map) \<and> Mapping.keys a2_map = {tp - len..tp} \<and> (\<forall>xs\<in>Mapping.keys a1_map. case Mapping.lookup a1_map xs of Some tp' \<Rightarrow> tp' < tp) \<and> (\<forall>tp'\<in>Mapping.keys a2_map. case Mapping.lookup a2_map tp' of Some m \<Rightarrow> table (args_n args) (args_R args) (Mapping.keys m) \<and> (\<forall>xs\<in>Mapping.keys m. case Mapping.lookup m xs of Some tstp \<Rightarrow> tstp_lt tstp (cur - (left (args_ivl args) - 1)) tp \<and> isl tstp = (0 < left (args_ivl args)))) \<and> length done = done_length \<and> length done + len = length auxlist \<and> rev done = map proj_thd (take (length done) auxlist) \<and> Ball (set (take (length done) auxlist)) (check_before (args_ivl args) dt) \<and> sorted (map fst auxlist) \<and> list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map (args_pos args) tp' a1_map) (\<lambda>ts' tp'. filter_a2_map (args_ivl args) ts' tp' a2_map)) (drop (length done) auxlist) (zip (linearize tss) [tp - len..<tp]))"
power_up: fixes x :: "real"   and n :: "nat"   and p :: "nat" assumes "0 \<le> x" shows "x ^ n \<le> power_up p x n"
closed_real_atLeast: fixes a :: "real" shows "closed {a..}"
count_space_IntVal_prod: shows "stock_measure INTEG \<Otimes>\<^sub>M stock_measure INTEG = count_space (range IntVal \<times> range IntVal)"
coeff_mult_0: fixes f :: "'a poly"   and g :: "'a poly" shows "coeff (f * g) 0 = coeff f 0 * coeff g 0"
Pred_eqvt: fixes p :: "perm"   and \<phi> :: "'f" shows "p \<bullet> Pred \<phi> = Pred (p \<bullet> \<phi>)"
braun: shows "braun \<langle>t1, x, t2\<rangle>"
simps: fixes a :: "'node"   and n :: "'node"   and S :: "'node set" shows "(a \<in> obs_intra n S) = (\<exists>as n'. a = n' \<and> n -as\<rightarrow>\<^sub>\<iota>* n' \<and> (\<forall>nx\<in>set (sourcenodes as). nx \<notin> S) \<and> n' \<in> S)"
terminal_no_progress: fixes es :: "e list"   and s :: "s"   and vs :: "v list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list" assumes "const_list es \<or> es = [Trap]" shows "\<not> \<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
path_connected_space_iff_components_subset_singleton: fixes X :: "'b topology" shows "path_connected_space X = (\<exists>a. path_components_of X \<subseteq> {a})"
agree2: shows "Vagree \<nu> \<nu>' (FVDiff t2)"
integrable_r: fixes x :: "real"   and y :: "real" assumes "2 \<le> x" shows "r integrable_on {x..y}"
of_nat_div: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "class.field_char_0 divide inverse times one plus zero minus uminus" shows "semiring_1.of_nat one plus zero (m div n) = divide (minus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero (m mod n))) (semiring_1.of_nat one plus zero n)"
sublinear_wp_PC: fixes x :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "sublinear (wp x)"   and "sublinear (wp y)"   and "unitary P" shows "sublinear (wp (x \<^bsub>P\<^esub>\<oplus> y))"
chfindom_monofun2cont: fixes f :: "'a \<Rightarrow> 'b" assumes "monofun f" shows "cont f"
image_Int_subset: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" shows "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B"
wave_not_RF_IN_zero: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_IN f x = 0"
W_def: shows "W \<equiv> hmemrec (\<lambda>f z. if z = 0 then Zero else Q_Succ (f (pred z)))"
rbt_greater_prop: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and t :: "('a, 'b) rbt" shows "ord.rbt_greater less k t = (\<forall>x\<in>set (keys t). less k x)"
fref_compI': fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool"   and R1 :: "('a \<times> 'c) set"   and R2 :: "('b \<times> 'd) set"   and h :: "'e \<Rightarrow> 'f"   and Q :: "'e \<Rightarrow> bool"   and S1 :: "('c \<times> 'e) set"   and S2 :: "('d \<times> 'f) set" assumes "(f, g) \<in> [P]\<^sub>f R1 \<rightarrow> R2"   and "(g, h) \<in> [Q]\<^sub>f S1 \<rightarrow> S2" shows "(f, h) \<in> [\<lambda>x. Q x \<and> (\<forall>y. (y, x) \<in> S1 \<longrightarrow> P y)]\<^sub>f R1 O S1 \<rightarrow> R2 O S2"
max3_def: fixes it :: "nat idx_interval"   and b :: "nat"   and c :: "nat" shows "max3 it b c = max (high (idx_interval.int it)) (max b c)"
sym_eq: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "pseudo_ordered_set A less_eq"   and "x \<in> A"   and "y \<in> A" shows "sympartp less_eq x y = (x = y)"
openin_Int: fixes U :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "openin U S"   and "openin U T" shows "openin U (S \<inter> T)"
lookup_a2_map: fixes tp' :: "nat"   and m :: "(event_data option list, nat + nat) mapping" assumes "Mapping.lookup a2_map tp' = Some m" shows "table n R (Mapping.keys m) \<and> (\<forall>xs\<in>Mapping.keys m. case Mapping.lookup m xs of Some tstp \<Rightarrow> tstp_lt tstp (cur - (left I - 1)) tp \<and> isl tstp = (0 < left I))"
wffp_cases2: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl" obtains n :: "nat" and st :: "'St" where "0 < n__" and "sl = map Idle (replicate n__ st__)"   | n :: "nat" and st :: "'St" and st1 :: "'St" and u :: "'U" and c :: "'C" and sl1 :: "('St, 'U, 'C) state list" where "sl = map Idle (replicate n st) @ [State st1 u c] @ sl1" and "sl1 \<noteq> [] \<Longrightarrow> wffp sl1 \<and> hd sl1 \<in> \<delta> (State st1 u c)"
arrow_of_spans_of_maps_axioms: shows "arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>0 h) (tab\<^sub>1 h) (tab\<^sub>0 h) (tab\<^sub>1 h) h.chine"
Ints_def: fixes one :: "'b"   and times :: "'b \<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" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.Ints one plus zero minus = range (ring_1.of_int one plus zero minus)"
vec_nth_cases: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> UNIV" obtains x :: "('c, 'b) Finite_Cartesian_Product.vec" where "y = ($h) x"
Dag_root_not_in_subdag_r: fixes r :: "ref \<Rightarrow> ref"   and p :: "ref"   and l :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "Dag (r p) l r t" shows "p \<notin> set_of t"
inj_iff: fixes f :: "'d \<Rightarrow> 'e" shows "inj f = (inv f \<circ> f = id)"
lprefix_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lprefix xs (lappend xs ys)"
ge_game_refl: fixes x :: "game" shows "ge_game (x, x)"
distinct_disc_2: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and pr_operation' :: "pr_operation'" assumes "Network c s t"   and "is_NONSAT_PUSH' pr_operation'" shows "\<not> is_RELABEL' pr_operation'"
simps_1: fixes ok :: "nat"   and f :: "'b list"   and f' :: "'b list"   and r :: "'b list"   and r' :: "'b list" shows "rem_steps (Rev ok f f' r r') = 2 * length f + ok + 2"
aux4_comm: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> - y + x \<cdot> y = x"
modified_view_valid: shows "isViewOn \<lparr>V = V\<^bsub>\<V>'\<^esub> \<union> N\<^bsub>\<V>'\<^esub>, N = {}, C = C\<^bsub>\<V>'\<^esub>\<rparr> E\<^bsub>SES\<^esub>"
minus_eq: fixes x :: "'a"   and y :: "'a" shows "x - y = abs_nat (rep_nat x - rep_nat y)"
M_def: shows "M \<equiv> case B of ereal m \<Rightarrow> (m + 1) / 2 | _ \<Rightarrow> 2"
from_digits_altdef: fixes n :: "nat"   and d :: "nat \<Rightarrow> nat" shows "from_digits n d = (\<Sum>i<n. d i * base ^ i)"
norm_lt: fixes x :: "'c"   and y :: "'d" shows "(norm x < norm y) = (x \<bullet> x < y \<bullet> y)"
pr_yields_binary_product: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "has_as_binary_product a b (pr1 a b) (pr0 a b)"
ccINF_inf_distrib: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A" shows "inf (Inf (f ` A)) (Inf (g ` A)) = Inf ((\<lambda>a. inf (f a) (g a)) ` A)"
rcons_def: fixes t :: "('a, 'b) FingerTreeStruc"   and a :: "'a \<times> 'b" shows "FingerTreeStruc.rcons t a = FingerTreeStruc.nrcons t (Tip (fst a) (snd a))"
left_interface_def: shows "left_interface \<equiv> corec_converter (\<lambda>conv a. map_gpv (map_prod id Inr) id (left_gpv (run_converter conv a)))"
residue_of_prod': fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" shows "(a \<otimes> b) k = a k * b k mod p ^ k"
dbl_dec_simps_2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one zero = uminus one"
apply_updates_def: fixes u :: "(nat \<times> vname aexp) list"   and old :: "vname \<Rightarrow> value option" shows "apply_updates u old = fold (\<lambda>h r. r(fst h $:= aval (snd h) old)) u"