hcju/isabelleps huggingface dataset:
entries_eqI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t1 :: "('a, 'b) RBT_Impl.rbt"   and t2 :: "('a, 'b) RBT_Impl.rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t1"   and "ord.rbt_sorted less t2"   and "ord.rbt_lookup less t1 = ord.rbt_lookup less t2" shows "RBT_Impl.entries t1 = RBT_Impl.entries t2"
induct: fixes P :: "('c, 'd) Digit \<Rightarrow> ('c, 'd) FingerTreeStruc \<Rightarrow> ('c, 'd) Node list \<Rightarrow> bool"   and a0 :: "('c, 'd) Digit"   and a1 :: "('c, 'd) FingerTreeStruc"   and a2 :: "('c, 'd) Node list" assumes "\<And>pr m. P pr m []"   and "\<And>pr m v va. P pr m (v # va)" shows "P a0 a1 a2"
bind_spmf_def: fixes x :: "'h spmf"   and f :: "'h \<Rightarrow> 'g spmf" shows "x \<bind> f = x \<bind> (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
less_add_diff: fixes k :: "nat"   and n :: "nat"   and m :: "nat" assumes "k < n" shows "m < n + m - k"
vdomain_vid_on: fixes A :: "V" shows "\<D>\<^sub>\<circ> (vid_on A) = A"
ccompatible3: fixes c :: "real"   and k :: "'d \<Rightarrow> nat"   and x :: "'d"   and X :: "'d set" assumes "c \<le> real (k x)"   and "c \<in> \<nat>"   and "x \<in> X" shows "ccompatible {region X I r |I r. valid_region X k I r} (LE x c)"
Collect_is_subset: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "{x \<in> A. P x} \<subseteq> A"
IH1: assumes "q \<in> subterms p1"   and "r \<in> stermsl q" shows "r \<in> ctermsl p1"
member_product: fixes rbt1 :: "'b set_rbt"   and rbt2 :: "'c set_rbt" assumes "ID ccompare \<noteq> None"   and "ID ccompare \<noteq> None" shows "RBT_Set2.member (RBT_Set2.product rbt1 rbt2) = (\<lambda>ab. ab \<in> Collect (RBT_Set2.member rbt1) \<times> Collect (RBT_Set2.member rbt2))"
homE_2: fixes CC :: "('d, 'e, 'f) category_scheme"   and A :: "'d"   and B :: "'d"   and f :: "'e" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>"   and "B \<in> Ob\<^bsub>CC\<^esub>"   and "f \<in> Hom\<^bsub>CC\<^esub> A B" shows "Cod\<^bsub>CC\<^esub> f = B"
list_rest_coinc: fixes s2 :: "'f list"   and s1 :: "'f list"   and r1 :: "'f list"   and r2 :: "'f list" assumes "length s2 \<le> length s1"   and "s1 @ r1 = s2 @ r2" shows "\<exists>r1p. r2 = r1p @ r1"
simps_15: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "sat \<sigma> V v i (Until \<phi> I \<psi>) = (\<exists>j\<ge>i. mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> sat \<sigma> V v j \<psi> \<and> (\<forall>k\<in>{i..<j}. sat \<sigma> V v k \<phi>))"
size_Diff_submset: fixes M :: "'a multiset"   and M' :: "'a multiset" assumes "M \<subseteq># M'" shows "size (M' - M) = size M' - size M"
sets_measure_of_eq: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" assumes "sigma_algebra \<Omega> M" shows "sets (measure_of \<Omega> M \<mu>) = M"
failures_traces: fixes xs :: "'e list"   and X :: "'e set"   and P :: "'e process" assumes "(xs, X) \<in> failures P" shows "xs \<in> traces P"
homotopic_paths_continuous_image: fixes s :: "'c set"   and f :: "real \<Rightarrow> 'c"   and g :: "real \<Rightarrow> 'c"   and h :: "'c \<Rightarrow> 'd"   and t :: "'d set" assumes "homotopic_paths s f g"   and "continuous_on s h"   and "h ` s \<subseteq> t" shows "homotopic_paths t (h \<circ> f) (h \<circ> g)"
rel_interior_injective_on_span_linear_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "bounded_linear f"   and "inj_on f (span S)" shows "rel_interior (f ` S) = f ` rel_interior S"
inf_llist_rec: fixes f :: "nat \<Rightarrow> 'c" shows "inf_llist f = LCons (f 0) (inf_llist (\<lambda>n. f (Suc n)))"
lens_put_irr2: fixes X :: "'b \<Longrightarrow> 'c"   and Y :: "'d \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and u :: "'b" assumes "X \<bowtie> Y" shows "get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>"
LIMSEQ_INF: fixes X :: "nat \<Rightarrow> 'a" assumes "decseq X" shows "X \<longlonglongrightarrow> Inf (range X)"
subprob_space_kernel: fixes K :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and a :: "'c" assumes "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "a \<in> space M" shows "subprob_space (K a)"
of_nat_eq_signed_scast: fixes x :: "nat"   and y :: "'b signed word" shows "(word_of_nat x = y) = (word_of_nat x = SCAST('b signed \<rightarrow> 'b) y)"
ide_cod: fixes f :: "'a1" assumes "A1.arr f" shows "A1.ide (A1.cod f)"
theI: fixes P :: "'g \<Rightarrow> bool"   and a :: "'g" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "P (THE x. P x)"
ntrs_mon_increasing_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> ntrs fg" shows "mon_s fg s \<subseteq> mon_s fg s' \<and> mon_c fg c = mon_c fg c'"
a: shows "Cut <a>.M x.N \<longrightarrow>\<^sub>c R"
continuous_map_Euclidean_space_add: fixes X :: "'a topology"   and n :: "nat"   and f :: "'a \<Rightarrow> nat \<Rightarrow> real"   and g :: "'a \<Rightarrow> nat \<Rightarrow> real" assumes "continuous_map X (Euclidean_space n) f"   and "continuous_map X (Euclidean_space n) g" shows "continuous_map X (Euclidean_space n) (\<lambda>x i. f x i + g x i)"
inj_on_convol_ident: fixes f :: "'a \<Rightarrow> 'b"   and X :: "'a set" shows "inj_on (\<lambda>x. (x, f x)) X"
rep_zero_iff: fixes x :: "'a" shows "(rep_nat x = 0) = (x = (0::'a))"
lv_relI: fixes x :: "real list" assumes "length x = DIM('b)" shows "(x, eucl_of_list x) \<in> lv_rel"
fib_mult_eq_sum_nat: fixes n :: "nat" shows "fib (Suc n) * fib n = (\<Sum>k\<le>n. fib k * fib k)"
star_inductl_var_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y = y" shows "x\<^sup>\<star> \<cdot> y \<le> y"
deadlocked'_implies_deadlock: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "deadlocked' s" shows "deadlock s"
simple_product_run: fixes \<delta>\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and \<delta>\<^sub>2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>1 :: "'a"   and q\<^sub>2 :: "'b"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "run \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w i = (run \<delta>\<^sub>1 q\<^sub>1 w i, run \<delta>\<^sub>2 q\<^sub>2 w i)"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_erase m lm (l, r) = (\<exists>rn. l = Oc \<up> Suc m \<and> tl r = Bk # <lm> @ Bk \<up> rn)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat list" assumes "\<And>i l. P i 0 l"   and "\<And>i j l. (i \<le> j \<Longrightarrow> P i j ([j] @ l)) \<Longrightarrow> P i (Suc j) l" shows "P a0 a1 a2"
sorted_pivot_positions: fixes A :: "'b mat" shows "strict_sorted (map fst (pivot_positions A))"
goodEnv_def: fixes rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option" shows "goodEnv rho \<equiv> (\<forall>ys. liftAll good (rho ys)) \<and> (\<forall>ys. |{y. rho ys y \<noteq> None}| <o |UNIV|)"
eq': shows "S \<union> V = S \<union> V \<inter> T"
M_M: shows "(\<rightarrow>\<^sub>M) M = (\<rightarrow>\<^sub>M) N"
convex_on_subset: fixes t :: "'b set"   and f :: "'b \<Rightarrow> real"   and S :: "'b set" assumes "convex_on t f"   and "S \<subseteq> t" shows "convex_on S f"
cut_Rep_preal: fixes x :: "preal" shows "Dedekind_Real.cut (Rep_preal x)"
count_space_uminus: shows "count_space UNIV = distr (count_space UNIV) (count_space UNIV) uminus"
ide_composite: shows "ide (\<pi> \<star> \<rho>)"
noStuck_def': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Stuck} = (\<not> \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Stuck)"
r_neg: fixes x :: "'a" assumes "x \<in> carrier G" shows "x \<oplus> \<ominus> x = \<zero>"
F: fixes i :: "'c" assumes "i \<in> I" shows "F i \<in> borel_measurable M"
lengthAux2: fixes xvec :: "name list"   and yvec :: "name list"   and y :: "name"   and zvec :: "name list" assumes "length xvec = length (yvec @ y # zvec)" obtains xvec1 :: "name list" and x :: "name" and xvec2 :: "name list" where "xvec = xvec1 @ x # xvec2" and "length xvec1 = length yvec" and "length xvec2 = length zvec"
INT_lower: fixes a :: "'b"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "a \<in> A" shows "\<Inter> (B ` A) \<subseteq> B a"
group_partitions: shows "partition_on (\<V> \<times> I) {G \<times> I |G. G \<in> \<G>}"
real_norm_def: fixes r :: "real" shows "norm r = \<bar>r\<bar>"
construct_basis: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B :: "'e set"   and b :: "'e"   and f :: "'e \<Rightarrow> 'f" assumes "vector_space_pair s1 s2"   and "\<not> module.dependent s1 B"   and "b \<in> B" shows "vector_space_pair.construct s1 s2 B f b = f b"
smulextp_code: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "('a, nat) alist"   and ys :: "('a, nat) alist" shows "smulextp f (Bag xs) (Bag ys) = fst (mul_ext f (ass_list_to_single_list (alist.impl_of xs)) (ass_list_to_single_list (alist.impl_of ys)))"
q_Imp_def: fixes t :: "hf"   and u :: "hf" shows "q_Imp t u \<equiv> q_Disj (q_Neg t) u"
Suc_2: shows "swapidseq (Suc n) p"
Nil_subset_T: fixes P :: "'a process" shows "{[]} \<subseteq> T P"
map_sel_1: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d stream" shows "shd (smap f a) = f (shd a)"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end5 n (l, r) = (inv_end5_loop n (l, r) \<or> inv_end5_exit n (l, r))"
slice_kind_Call_in_slice: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "sourcenode a \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "slice_kind S a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>cspp (targetnode a) (HRB_slice S) fs"
AutoGroup_def: fixes G :: "('c, 'd) monoid_scheme" shows "AutoGroup G = BijGroup (carrier G)\<lparr>carrier := auto G\<rparr>"
union: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "semilattice_set f"   and "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "semilattice_set.F f (A \<union> B) = f (semilattice_set.F f A) (semilattice_set.F f B)"
HPT_target_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and T :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd" assumes "TS \<in> HPT ST"   and "T \<in> TS" shows "target T \<in> StepConf (HA ST) (Conf ST) TS"
degree_cf_pos_poly: fixes p :: "int poly" shows "degree (cf_pos_poly p) = degree p"
nth_arg_def: fixes T :: "Unary_PCF.type"   and i :: "nat" shows "T !- i \<equiv> dest_fun T ! i"
HFail_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HFail s s' p"   and "HInv6 s"   and "Inv2c s" shows "HInv6 s'"
word_mod_less_divisor: fixes n :: "'b word"   and m :: "'b word" assumes "0 < n" shows "m mod n < n"
basis_basis_reduction_add_rows_loop_leave: fixes i :: "nat"   and fs :: "int vec list" assumes "LLL_bound_invariant False True i fs"   and "\<mu>_small_row i fs 0"   and "\<mu>_bound_row_inner fs i 0"   and "i < m" shows "LLL_bound_invariant True False i fs"
simps: fixes \<A>\<^sub>1 :: "'grp \<Rightarrow> ((bool list \<times> bool list) \<times> 'b, 'grp, bool list) gpv"   and \<A>\<^sub>2 :: "'grp \<times> bool list \<Rightarrow> 'b \<Rightarrow> (bool, 'grp, bool list) gpv" shows "game (\<A>\<^sub>1, \<A>\<^sub>2) = TRY lift_spmf key_gen \<bind> (\<lambda>(pk, sk). \<A>\<^sub>1 pk \<bind> (\<lambda>((m\<^sub>0, m\<^sub>1), \<sigma>). assert_gpv (valid_plains m\<^sub>0 m\<^sub>1) \<bind> (\<lambda>_. lift_spmf coin_spmf \<bind> (\<lambda>b. encrypt pk (if b then m\<^sub>0 else m\<^sub>1) \<bind> (\<lambda>cipher. \<A>\<^sub>2 cipher \<sigma> \<bind> (\<lambda>b'. Done (b' = b))))))) ELSE lift_spmf coin_spmf"
smods_nil_eq: fixes p :: "real poly"   and q :: "real poly" shows "(smods p q = []) = (p = 0)"
span_image: fixes A :: "(('a, 'c) vec, 'b) vec"   and S :: "('a, 'c) vec set" shows "vec.span ((*v) A ` S) = (*v) A ` vec.span S"
srev_bottom_iff: fixes xs :: "[:'a:]" shows "(srev\<cdot>xs = \<bottom>) = (xs = \<bottom>)"
inf_cvec_def: shows "\<infinity>\<^sub>v = (1, 0)"
load_tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'a"   and \<O> :: "'b" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>load_tmps is. i < t"
setBit_no: fixes bin :: "num"   and n :: "nat" shows "setBit (numeral bin) n = word_of_int (bin_sc n True (numeral bin))"
simps_5: fixes P :: "interface"   and D :: "sdiagram"   and Q :: "interface" shows "com_cell (Loop_sdia P D Q) = Loop (com_sdia D)"
even_plus1: fixes x :: "nat" assumes "ya_odd x" shows "ya_even (x + 1)"
Abs_1: shows "(\<lambda>x:T. t) = (\<lambda>x':T'. t')"
fpxs_nth_mult: fixes f :: "'b fpxs"   and g :: "'b fpxs"   and r :: "rat" shows "fpxs_nth (f * g) r = (\<Sum>(y, z)\<in>{(y, z). y \<in> fpxs_supp f \<and> z \<in> fpxs_supp g \<and> r = y + z}. fpxs_nth f y * fpxs_nth g z)"
prod_nonneg: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<forall>a\<in>A. (0::'b) \<le> f a" shows "(0::'b) \<le> prod f A"
subst_support_def: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'c) Term.term" shows "(\<theta> supports \<tau>) = (\<tau> = \<theta> \<circ>\<^sub>s \<tau>)"
num_params_tm_union_le: fixes t1 :: "taylor_model"   and X :: "nat"   and t2 :: "taylor_model"   and prec :: "nat"   and I :: "float interval list"   and a :: "float interval list" assumes "num_params (tm_poly t1) \<le> X"   and "num_params (tm_poly t2) \<le> X" shows "num_params (tm_poly (tm_union prec I a t1 t2)) \<le> X"
compose_axiom_def: shows "compose_axiom \<equiv> \<langle> Game hgid1 ;; Game hgid2 \<rangle> pusall \<leftrightarrow> \<langle> Game hgid1 \<rangle> \<langle> Game hgid2 \<rangle> pusall"
add_zero_left: fixes x :: "'a" assumes "x \<in> V" shows "(0::'a) + x = x"
subtermeq_imp_subtermtypeeq: fixes t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "s \<sqsubseteq> t" shows "\<Gamma> s \<sqsubseteq> \<Gamma> t"
OrL_inv: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F \<^bold>\<or> G, \<Gamma> \<Rightarrow> \<Delta>" shows "F, \<Gamma> \<Rightarrow> \<Delta> \<and> G, \<Gamma> \<Rightarrow> \<Delta>"
suffix_tl_subset: fixes xs :: "'a list"   and ys :: "'a list" assumes "suffix xs ys" shows "set (tl xs) \<subseteq> set (tl ys)"
rep_eq: fixes x :: "Vars.var set"   and xa :: "CoCalls" shows "Rep_CoCall (xa G|` x) = Set.filter (\<lambda>(xa, y). xa \<in> x \<and> y \<in> x) (Rep_CoCall xa)"
odlist_all_def: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c odlist" shows "odlist_all P xs \<equiv> list_all P (toList xs)"
span_finiteE: fixes B :: "'c set"   and p :: "'c" assumes "finite B"   and "p \<in> ideal B" obtains q :: "'c \<Rightarrow> 'c" where "p = (\<Sum>b\<in>B. q b * b)"
Cons_1: fixes x :: "'a"   and y :: "'a" assumes "set qs \<subseteq> {x, y}" shows "T\<^sub>p [x, y] qs (OPT2 qs [x, y]) \<le> length qs"
topological_basis_def: fixes "open" :: "'b set \<Rightarrow> bool"   and B :: "'b set set" assumes "class.topological_space open" shows "topological_space.topological_basis open B = ((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union> B' = x)))"
Y_ends: shows "\<forall>c\<in>Y. a \<noteq> c \<and> b \<noteq> c \<longrightarrow> [[a c b]]"
ty_dom_cons: fixes \<Gamma> :: "binding list"   and X :: "vrs"   and Q :: "ty"   and \<Delta> :: "binding list" shows "ty_dom (\<Gamma> @ [VarB X Q] @ \<Delta>) = ty_dom (\<Gamma> @ \<Delta>)"
opp_fold_A_min_gallery_from1_map: fixes F :: "'a set"   and Fs :: "'a set list" assumes "chamber F"   and "F \<in> fold_A \<turnstile> A"   and "min_gallery (C # Fs @ [F])" shows "min_gallery (D # opp_fold_A \<Turnstile> (Fs @ [F]))"
lubY_le_flubY: shows "(lub Y cl, f (lub Y cl)) \<in> r"
dyn_array_def: fixes xs :: "'a list"   and a :: "'a dynamic_array" shows "dyn_array xs a = (\<exists>\<^sub>Ap. dyn_array_raw p a * \<up> (xs = abs_array p) * \<up> (snd p \<le> length (fst p)))"
restrict_codomain: fixes Z :: "'b set set" assumes "codomain.ChamberSubcomplex Z"   and "order.greater_eq Z (f \<turnstile> X)" shows "ChamberComplexMorphism X Z f"
pred_def: fixes G :: "('c, 'd) graph"   and v :: "'c" shows "pred G v \<equiv> {(v', w). (v', w, v) \<in> edges G}"
1: fixes k :: "nat" assumes "0 < k"   and "\<not> coprime k n" shows "gauss_sum k = 0"
dom_cod: fixes f :: "'c" assumes "C.arr f" shows "C.dom (C.cod f) = C.cod f"
fds_nth_zero: shows "fds_nth 0 = (\<lambda>_. 0::'a)"
unwind_exit_def: fixes \<Delta>e :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'b \<Rightarrow> bool" shows "unwind_exit \<Delta>e \<equiv> \<forall>s vl s1 vl1. reachNT s \<and> reach s1 \<and> \<Delta>e s vl s1 vl1 \<longrightarrow> vl \<noteq> [] \<and> exit s (hd vl)"
standard_classes_def: shows "standard_classes = [ObjectC, SXcptC Throwable, SXcptC NullPointer, SXcptC OutOfMemory, SXcptC ClassCast, SXcptC NegArrSize, SXcptC IndOutBound, SXcptC ArrStore]"
Methd_no_jump: fixes G :: "prog"   and s0 :: "state"   and D :: "qtname"   and sig :: "sig"   and v :: "val"   and s1 :: "state"   and j :: "jump" assumes "G\<turnstile>s0 \<midarrow>Methd D sig-\<succ>v\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)" shows "fst s1 \<noteq> Some (Jump j)"
substrT_set: fixes a :: "char list"   and T :: "POPLmarkRecord.type"   and fs :: "(char list \<times> POPLmarkRecord.type) list"   and k :: "nat"   and U :: "POPLmarkRecord.type" assumes "(a, T) \<in> set fs" shows "(a, T[k \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau>) \<in> set (fs[k \<mapsto>\<^sub>\<tau> U]\<^sub>r\<^sub>\<tau>)"
matrix_rational_approximation: fixes e :: "real"   and A :: "((real, 'a) vec, 'b) vec" assumes "0 < e" obtains B :: "((real, 'a) vec, 'b) vec" where "\<And>i j. B $ i $ j \<in> \<rat>" and "onorm ((*v) (A - B)) < e"
gfp_upperbound: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and X :: "'d"   and f :: "'d \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq X (f X)" shows "less_eq X (complete_lattice.gfp Sup less_eq f)"
scaleRat_leq2: fixes a :: "'a"   and b :: "'a"   and k :: "rat" assumes "a \<le> b"   and "k < 0" shows "k *R b \<le> k *R a"
owner_seskey_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "(Owner n, SesKey SK) \<in> s" shows "(\<exists>S. fst SK = Some S \<and> Crypt (Auth_ShaKey n) (PriKey S) \<in> used s) \<or> fst SK = None"
simps_1: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b"   and s' :: "'b" shows "Sem (Basic f) s s' = (s' = f s)"
nz: shows "\<forall>\<^sub>F x in at_top. \<theta> x \<noteq> 0 \<or> x \<noteq> 0"
is_basis_def: fixes S :: "(real, 'a) vec set" shows "is_basis S \<equiv> independent S \<and> span S = UNIV"
Red_I_of_Inf_to_N: fixes \<iota> :: "'f inference"   and N :: "'f set" assumes "\<iota> \<in> Inf"   and "concl_of \<iota> \<in> N" shows "\<iota> \<in> Red_I N"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e llist" shows "lmap g (lmap f v) = lmap (g \<circ> f) v"
chain_mono: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and C :: "'c set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "pred_on.chain A P C" shows "pred_on.chain A Q C"
reduction_language_subset: shows "R.language \<subseteq> S.language"
dist_vec_def: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "dist x y = L2_set (\<lambda>i. dist (x $ i) (y $ i)) UNIV"
simps_6: fixes v :: "trm"   and va :: "trm" shows "get_and_list (LessEq v va) = [LessEq v va]"
cong_mat: fixes nr :: "nat"   and nr' :: "nat"   and nc :: "nat"   and nc' :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b"   and f' :: "nat \<times> nat \<Rightarrow> 'b" assumes "nr = nr'"   and "nc = nc'"   and "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> f (i, j) = f' (i, j)" shows "Matrix.mat nr nc f = Matrix.mat nr' nc' f'"
wp1WhileTrue': fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat" assumes "bval b s" shows "wp\<^sub>1 (WHILE b DO c) Q l s = wp\<^sub>1 c (wp\<^sub>1 (WHILE b DO c) Q) l s"
no_real_normal_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "simple_group G"   and "H \<lhd> G" shows "H = carrier G \<or> H = {\<one>\<^bsub>G\<^esub>}"
factorization_m_def: fixes m :: "int"   and f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "poly_mod.factorization_m m f cfs \<equiv> case cfs of (c, fs) \<Rightarrow> poly_mod.Mp m f = poly_mod.Mp m (Polynomial.smult c (\<Prod>\<^sub># fs)) \<and> (\<forall>f\<in>#fs. poly_mod.irreducible\<^sub>d_m m f \<and> monic (poly_mod.Mp m f))"
has_density_embed_measure': fixes f :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and f' :: "'d \<Rightarrow> 'c"   and M :: "'d measure"   and \<delta> :: "'d \<Rightarrow> ennreal" assumes "inj f"   and "\<And>x. x \<in> space N \<Longrightarrow> f' (f x) = x"   and "sets M = sets (embed_measure N f)" shows "has_density (distr M N f') N (\<delta> \<circ> f) = has_density M (embed_measure N f) \<delta>"
distinct_59: fixes x9 :: "('a, String.literal) phantom" shows "Integer \<noteq> Enum x9"
order_refl: fixes less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "'l" assumes "class.preorder less_eq less" shows "less_eq x x"
wf_design_iff: fixes bl :: "'a set" assumes "bl \<in># \<B> + mset_set \<G>" shows "design \<V> (\<B> + mset_set \<G>) = (bl \<subseteq> \<V> \<and> finite \<V> \<and> bl \<noteq> {})"
sorted_list_of_set_unique: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set"   and l :: "'g list" assumes "class.linorder less_eq less"   and "finite A" shows "(linorder.strict_sorted less l \<and> set l = A \<and> length l = card A) = (linorder.sorted_list_of_set less_eq A = l)"
factors_cong_unit: fixes G :: "('c, 'd) monoid_scheme"   and u :: "'c"   and a :: "'c"   and as :: "'c list" assumes "comm_monoid_cancel G"   and "u \<in> Units G"   and "a \<notin> Units G"   and "factors G as a"   and "set as \<subseteq> carrier G" shows "factors G (as[0 := as ! 0 \<otimes>\<^bsub>G\<^esub> u]) (a \<otimes>\<^bsub>G\<^esub> u)"