hcju/isabelleps huggingface dataset:
split: fixes P :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and optionT :: "('f, 'e) optionT" shows "P (case optionT of OptionT x \<Rightarrow> f x) = (\<forall>x. optionT = OptionT x \<longrightarrow> P (f x))"
sets_into_space: fixes x :: "'c set"   and M :: "'c measure" assumes "x \<in> null_sets M" shows "x \<subseteq> space M"
Crypt_imp_OR1: fixes A :: "agent"   and evs :: "event list"   and NA :: "msg"   and B :: "agent" assumes "A \<notin> bad"   and "evs \<in> otway"   and "Crypt (shrK A) \<lbrace>NA, Agent A, Agent B\<rbrace> \<in> parts (knows Spy evs)" shows "Says A B \<lbrace>NA, Agent A, Agent B, Crypt (shrK A) \<lbrace>NA, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"
configuration_step: fixes x :: "nat"   and q' :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and q :: "'a ltl_prop_equiv_quotient" assumes "x \<in> \<UU>.configuration q' n"   and "q = \<up>af\<^sub>G\<^sub>\<UU> q' (w n)" shows "x \<in> \<UU>.configuration q (Suc n)"
map_subseqs: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map (map f) (subseqs xs) = subseqs (map f xs)"
simps_3: fixes v :: "'c"   and va :: "'c list"   and val :: "'d"   and k :: "nat" shows "make_insert (v # va) val (Suc k) = Insert (Some ((v # va) ! min k (length (v # va) - 1))) val"
qPsubstAll_simps_1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and xs :: "'varSort"   and x :: "'var" shows "qVar xs x #[[rho]] = (case rho xs x of None \<Rightarrow> qVar xs x | Some X \<Rightarrow> X)"
case_1: fixes f1 :: "int \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> int \<Rightarrow> MIR.num \<Rightarrow> 'a"   and f4 :: "MIR.num \<Rightarrow> 'a"   and f5 :: "MIR.num \<Rightarrow> MIR.num \<Rightarrow> 'a"   and f6 :: "MIR.num \<Rightarrow> MIR.num \<Rightarrow> 'a"   and f7 :: "int \<Rightarrow> MIR.num \<Rightarrow> 'a"   and f8 :: "MIR.num \<Rightarrow> 'a"   and f9 :: "int \<Rightarrow> MIR.num \<Rightarrow> MIR.num \<Rightarrow> 'a"   and x1 :: "int" shows "(case C x1 of C x \<Rightarrow> f1 x | Bound x \<Rightarrow> f2 x | CN x xa xb \<Rightarrow> f3 x xa xb | Neg x \<Rightarrow> f4 x | Add x xa \<Rightarrow> f5 x xa | Sub x xa \<Rightarrow> f6 x xa | Mul x xa \<Rightarrow> f7 x xa | Floor x \<Rightarrow> f8 x | CF x xa xb \<Rightarrow> f9 x xa xb) = f1 x1"
assms_2: shows "lose t = [True] \<longrightarrow> ack t = [x] \<and> st_out t = x"
cases: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and rest :: "('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme" assumes "WT_rest \<I>_adv \<I>_usr I rest" obtains "\<And>s x y es s'. \<lbrakk>((y, es), s') \<in> set_spmf (rfunc_adv rest s x); x \<in> outs_\<I> \<I>_adv; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_adv x \<and> I s'" and "\<And>s x y es s'. \<lbrakk>((y, es), s') \<in> set_spmf (rfunc_usr rest s x); x \<in> outs_\<I> \<I>_usr; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_usr x \<and> I s'" and "I (rinit rest)"
ASSUMPTION_D: fixes A :: "bool" assumes "ASSUMPTION A" shows "A"
n_preserve: fixes x :: "'a"   and y :: "'a" shows "n x \<cdot> n y \<cdot> n x = n y \<cdot> n x"
Interleaves_swap: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "xs \<cong> {ys, zs, P} = xs \<cong> {zs, ys, \<lambda>w ws. \<not> P w ws}"
cdle: shows "\<exists>j. k + n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> j < k"
VSigmaD2: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> VSigma A B" shows "b \<in>\<^sub>\<circ> B a"
wf_syscls_is_class_Throwable: fixes P :: "'a prog" assumes "wf_syscls P" shows "is_class P Throwable"
pre_splitFace_edges_f_in_g2: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and f :: "face"   and vs :: "nat list"   and x :: "nat \<times> nat" assumes "pre_splitFace g ram1 ram2 f vs"   and "x \<in> \<E> f" shows "x \<in> \<E> g"
simps_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and at :: "('a, 'b) atm" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.glitOfL prot protFw (Neg at) = {ProblemIkTpartG.grdLit x |x. x \<in> varsA at \<and> prot (Signature.tpOfV x)}"
tail_in_verts: fixes G :: "('c, 'd) pre_digraph"   and e :: "'d" assumes "wf_digraph G"   and "e \<in> arcs G" shows "tail G e \<in> verts G"
tARR_def: fixes x2 :: "ty"   and x1 :: "ty" shows "x2 \<rightarrow> x1 \<equiv> Abs_ty (tARR_Rep (Rep_ty x2) (Rep_ty x1))"
eventually_at_right_field: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d" shows "eventually P (at_right x) = (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
simps_5: fixes ch :: "'c"   and c1 :: "('a, 'b, 'c) cmd"   and c2 :: "('a, 'b, 'c) cmd" shows "proper (Ch ch c1 c2) = (proper c1 \<and> proper c2)"
unit_table: fixes V :: "nat set"   and n :: "nat" assumes "V = {}" shows "table n V (unit_table n)"
prv_eqv_imp_trans: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (eqv \<phi> \<chi>) (imp (eqv \<chi> \<psi>) (eqv \<phi> \<psi>)))"
takeWhile_eq_filter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "\<And>x. x \<in> set (dropWhile P xs) \<Longrightarrow> \<not> P x" shows "takeWhile P xs = filter P xs"
that: fixes x :: "real \<times> real" assumes "(x, y) \<in> set (segments_of_aform X)" shows "thesis"
rev_nth: fixes n :: "nat"   and xs :: "'g list" assumes "n < length xs" shows "rev xs ! n = xs ! (length xs - Suc n)"
after_def: fixes f :: "'a \<Rightarrow> 'b" shows "f$ \<equiv> %s. f (second s)"
simps_2: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and j :: "nat" shows "sig_crit bs ss (Inr j) = is_pred_syz ss (term_of_pair (0::'a, j))"
hyperplane_eq_empty: fixes a :: "'a"   and b :: "real" shows "({x. a \<bullet> x = b} = {}) = (a = (0::'a) \<and> b \<noteq> 0)"
FrameStatEqSym: fixes F :: "'b frame"   and G :: "'b frame" assumes "F \<simeq>\<^sub>F G" shows "G \<simeq>\<^sub>F F"
sum_constant: fixes y :: "'a"   and A :: "'b set" shows "(\<Sum>x\<in>A. y) = of_nat (card A) * y"
mem_UnitsI: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "monoid M composition unit"   and "monoid.invertible M composition unit u"   and "u \<in> M" shows "u \<in> monoid.Units M composition unit"
linear_span_pOp_closed: fixes A :: "'b set"   and H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "a \<in> linear_span R M A H"   and "b \<in> linear_span R M A H" shows "a \<plusminus> b \<in> linear_span R M A H"
f_shrink_last_take: fixes n :: "nat"   and xs :: "'b list"   and k :: "nat" shows "xs \<down> n \<div>\<^sub>l k = xs \<div>\<^sub>l k \<down> (n div k)"
rel_spmf_return_spmf1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and p :: "'b spmf" shows "rel_spmf R (return_spmf x) p = (lossless_spmf p \<and> (\<forall>a\<in>set_spmf p. R x a))"
Ds_mono': fixes es :: "'a exp list"   and A :: "'a set option"   and A' :: "'a set option" assumes "\<D>s es A"   and "A \<sqsubseteq> A'" shows "\<D>s es A'"
ffi_bad: fixes i :: "nat" shows "bad (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) (worst_chain (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) (\<lambda>ys xs. length xs < length ys) i)"
simps_13: fixes C :: "qtname"   and sig :: "sig" shows "assignsE (Methd C sig) = {}"
agg_matrix_bot: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<forall>i j. f (i, j) = bot" shows "f = mbot"
n_nmod_e: shows "\<not> gr.INTERP M N \<TTurnstile> E"
trms\<^sub>s\<^sub>s\<^sub>t_unlabel_suffix_subset_1: fixes B :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list"   and A :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t B \<subseteq> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A @ B)"
action'': shows "a'' \<in> actions (justifying_exec (J (Suc n)))"
empty_eqvt: fixes p :: "perm" shows "p \<bullet> {} = {}"
ordsetmap_smap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and a :: "'b"   and b :: "'b" assumes "ComplexLikePoset less_eq less P"   and "a \<in> P"   and "b \<in> P"   and "less_eq a b" shows "order.greater_eq (ordering.poset_simplex_map less_eq less P b) (ordering.poset_simplex_map less_eq less P a)"
length_subst_atm_mset_list: fixes AAs :: "'a multiset list"   and \<eta> :: "'s" shows "length (AAs \<cdot>aml \<eta>) = length AAs"
M1_RelM_rcosetGM1g: fixes g :: "'a" assumes "g \<in> carrier G" shows "(M1, M1 #> g) \<in> RelM"
fun_eq_onD: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and a :: "'c" assumes "fun_eq_on f g A"   and "a \<in> A" shows "f a = g a"
ord_pos: fixes x :: "'a" assumes "x \<in> carrier G" shows "0 < ord x"
PfP_fresh_iff: fixes a :: "atom"   and y :: "tm" shows "a \<sharp> PfP y = a \<sharp> y"
Pow_def: fixes A :: "'b set" shows "Pow A = {B. B \<subseteq> A}"
smult_sum: fixes a :: "'b"   and f :: "'d \<Rightarrow> ('b, 'c) sq_matrix"   and I :: "'d set" shows "(\<Sum>i\<in>I. a *\<^sub>S f i) = a *\<^sub>S sum f I"
word_bw_assocs_1: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "(x AND y) AND z = x AND y AND z"
pps_to_list_nth_leI: fixes j :: "nat"   and i :: "nat"   and S :: "'t set" assumes "j \<le> i"   and "i < card S" shows "pps_to_list S ! i \<preceq>\<^sub>t pps_to_list S ! j"
rsd_winners_manipulation: fixes Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and agents' :: "'agent list" assumes "total_preorder_on alts Ri'"   and "i \<in> agents"   and "set agents' = agents"   and "finite agents"   and "finite alts"   and "alts \<noteq> {}" shows "\<forall>x\<in>rsd_winners (R(i := Ri')) alts agents'. \<forall>y\<in>rsd_winners R alts agents'. R i x y"
assms_2: fixes p :: "'a poly" assumes "\<And>a. poly p a \<noteq> (0::'a)" shows "Q p"
IH_2: fixes w :: "nat \<Rightarrow> 'a set" assumes "w \<Turnstile> \<psi>" shows "\<exists>j. \<G>\<^sub>F\<^sub>G \<psi> w \<Turnstile>\<^sub>P af\<^sub>G \<psi> (prefix j w)"
abs_eq: fixes x :: "'a set" assumes "eq_onp closed_csubspace x x" shows "Proj (Abs_clinear_space x) = CBlinfun (projection x)"
n2_type: fixes t1 :: "'a bro"   and h :: "nat"   and t2 :: "'a bro"   and a :: "'a" shows "(t1 \<in> Bp h \<and> t2 \<in> T h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h)) \<and> (t1 \<in> T h \<and> t2 \<in> Bp h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h))"
arctan_divide_mono: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "x \<le> y" shows "arctan y / y \<le> arctan x / x"
add_set_single1Tr: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "\<forall>x y. x \<in> add_set f {a} n \<and> y \<in> add_set f {a} n \<longrightarrow> x = y"
WT_dem: fixes \<eta> :: "nat" shows "\<I>_uniform UNIV (insert None (Some ` vld \<eta>)), \<I>_full \<oplus>\<^sub>\<I> \<I>_uniform UNIV (insert None (Some ` (nlists UNIV \<eta> \<times> nlists UNIV \<eta>))) \<turnstile>\<^sub>C MAC.dem \<eta> \<surd>"
run_one_step_basic_store_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x201 :: "t"   and x202 :: "tp option"   and x203 :: "nat"   and x204 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Store x201 x202 x203 x204) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
strand_sem_mono_ik_2: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "M \<subseteq> M'"   and "\<lbrakk>M; S\<rbrakk>\<^sub>d \<theta>" shows "\<lbrakk>M'; S\<rbrakk>\<^sub>d \<theta>"
space_P: fixes J :: "nat set" assumes "finite J"   and "J \<subseteq> UNIV" shows "space (CI J) = space (Pi\<^sub>M J M)"
simps_4: fixes uw :: "'a" shows "(ext0.Val uw \<le> ext0.Top) = True"
inorder_empty: shows "Tree2.inorder RBT_Set.empty = []"
checker_r1_correctness: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real"   and \<delta> :: "real" shows "checker_r1 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> = (check_precond_r1 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> \<and> \<not> safe_distance_normal.collision_react a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> {0..})"
g: fixes x :: "'a" shows "prob_space (g x) \<and> sets (g x) = UNIV \<and> (AE y in g x. Sigma_Algebra.measure (g x) {y} \<noteq> 0)"
one_ge_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.ordered_semiring_1 one times plus zero less_eq" shows "less_eq zero one"
ringed_space_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" shows "ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
applicableLearnCharacterization: fixes stateA :: "State" shows "applicableLearn stateA = (getConflictFlag stateA = True \<and> \<not> getC stateA el getF stateA)"
cases: fixes x :: "'a \<times> 'b \<times> 'c \<times> 'd" obtains a :: "'a" and b :: "'b" and c :: "'c" and d :: "'d" where "x = (a, b, c, d)"
PredicatePairFunctions2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<forall>x1 x2. \<exists>y1 y2. P x1 x2 y1 y2" obtains f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" and f2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'e" where "\<forall>x1 x2. \<exists>y1 y2. f1 x1 x2 = y1 \<and> f2 x1 x2 = y2 \<and> P x1 x2 (f1 x1 x2) (f2 x1 x2)"
zprime_is_sum4sq: fixes p :: "int" assumes "prime (nat p)" shows "is_sum4sq_int p"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "(case option of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (if option = \<bottom> then f1 else f2 (the option))"
Sup_distr: fixes X :: "'a set"   and y :: "'a" shows "\<Squnion> X \<cdot> y = (\<Squnion>x\<in>X. x \<cdot> y)"
assms: fixes \<alpha> :: "act"   and Q' :: "ccs" assumes "Q \<longmapsto>\<alpha> \<prec> Q'" shows "\<exists>P'. P \<longmapsto>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
vdisjnt_iff: fixes A :: "V"   and B :: "V" shows "vdisjnt A B = (\<forall>x. \<not> (x \<in>\<^sub>\<circ> A \<and> x \<in>\<^sub>\<circ> B))"
elims: fixes x :: "('b \<times> 'a) list"   and xa :: "'b"   and y :: "'a option" assumes "lookup x xa = y" obtains xb :: "'b" where "x = []" and "xa = xb__" and "y = None"   | ya :: "'b" and v :: "'a" and ls :: "('b \<times> 'a) list" and xb :: "'b" where "x = (ya, v) # ls" and "xa = xb" and "y = (if xb = ya then Some v else lookup ls xb)"
closure_subset: fixes S :: "'b set" shows "S \<subseteq> closure S"
Newman: fixes r :: "('b \<times> 'b) set" assumes "SN r"   and "WCR r" shows "CR r"
simply_connected_eq_homotopic_circlemaps1: fixes S :: "'a set"   and f :: "complex \<Rightarrow> 'a"   and g :: "complex \<Rightarrow> 'a" assumes "simply_connected S"   and "continuous_on (sphere 0 1) f"   and "f ` sphere 0 1 \<subseteq> S"   and "continuous_on (sphere 0 1) g"   and "g ` sphere 0 1 \<subseteq> S" shows "homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f g"
collect_clki_def: fixes I :: "'d \<Rightarrow> ('a, 'b) cconstraint" shows "PTA.collect_clki I = \<Union> {collect_clock_pairs (I x) |x. True}"
a1: shows "\<forall>k. prime (4 * k + 3) \<longrightarrow> P (4 * k + 3)"
dist_eq_on_real_2_vec: fixes a :: "(real, 2) vec"   and b :: "(real, 2) vec" shows "dist a b = sqrt ((a $ 1 - b $ 1)\<^sup>2 + (a $ 2 - b $ 2)\<^sup>2)"
dim_extract_matrix_2: fixes as :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "dim_col (extract_matrix as m n) = n"
r_lazyifzero_def: fixes n :: "nat"   and j\<^sub>1 :: "nat"   and j\<^sub>2 :: "nat" shows "r_lazyifzero n j\<^sub>1 j\<^sub>2 \<equiv> Cn (Suc (Suc n)) r_univ [Cn (Suc (Suc n)) r_ifz [recf.Id (Suc (Suc n)) 0, r_constn (Suc n) j\<^sub>1, r_constn (Suc n) j\<^sub>2], r_shift (r_list_encode n)]"
normalize_square_part: fixes x :: "'b" shows "normalize (square_part x) = square_part x"
heaps_eq_Dag_eq: fixes t :: "dag"   and l :: "ref \<Rightarrow> ref"   and l' :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and r' :: "ref \<Rightarrow> ref"   and p :: "ref" assumes "\<forall>x\<in>set_of t. l x = l' x \<and> r x = r' x" shows "Dag p l r t = Dag p l' r' t"
asubg_mOp_closed: fixes H :: "'a set"   and x :: "'a" assumes "M +> H"   and "x \<in> H" shows "-\<^sub>a x \<in> H"
simps: fixes a :: "'a list" shows "steps a = ((\<exists>x. a = [x]) \<or> (\<exists>x y xs. a = x # y # xs \<and> E x y \<and> steps (y # xs)))"
P1: shows "\<forall>x. ListMem x l \<longrightarrow> P x"
norm_minus_cancel: fixes x :: "'a" shows "norm (- x) = norm x"
isoE: fixes f :: "'a2" assumes "C2.iso f" obtains g :: "'a2" where "C2.inverse_arrows f g"
def_hashmap_size_uint64_def: shows "def_hashmap_size = (\<lambda>_. 16)"
inf_sup_aci_7: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.lattice inf less_eq less sup" shows "sup x (sup y z) = sup y (sup x z)"
Enc_def: shows "Enc \<equiv> map_fun Re (map_fun Re Ab) cEnc"
abs_eq: fixes xa :: "'b"   and x :: "'b" shows "Fract xa x = abs_fract (if x = (0::'b) then (0::'b, 1::'b) else (xa, x))"
ffilter_empty: fixes f :: "'b \<Rightarrow> bool" shows "ffilter f {||} = {||}"
min_int_poly_primitive: fixes x :: "'b" shows "Missing_Polynomial_Factorial.primitive (min_int_poly x)"
shiftl_def: fixes w :: "'b word"   and n :: "nat" shows "w << n = (shiftl1 ^^ n) w"
MGT\<^sub>t_def: fixes c :: "com" shows "MGT\<^sub>t c = (\<lambda>z s. z = s \<and> c \<down> s, c, \<lambda>z. exec z c)"
pairwiseI: fixes S :: "'a set"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; x \<noteq> y\<rbrakk> \<Longrightarrow> R x y" shows "pairwise R S"
complex_cone_inter_nonpos_Reals: fixes a :: "real"   and b :: "real" assumes "- pi < a"   and "a \<le> b"   and "b < pi" shows "complex_cone a b \<inter> \<real>\<^sub>\<le>\<^sub>0 = {0}"
a: assumes "R (p \<bullet> x) y"   and "R (q \<bullet> y) z" shows "R ((q + p) \<bullet> x) z"
nsD: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>ns v" shows "u \<le>s v"
span_base: fixes Sa :: "('a \<Rightarrow> real) set"   and a :: "'a \<Rightarrow> real" assumes "Sa \<subseteq> diff_fun_space"   and "a \<in> Sa" shows "a \<in> sub.diff_fun_space.m1.span Sa"
vec0I_def: fixes ze :: "'b"   and n :: "nat" shows "vec0I ze n = replicate n ze"
gena_rule_def: shows "gena_rule \<equiv> ([pusall], Exists xid1 pusall)"
nonzero_Basis: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and u :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "u \<noteq> zero"
distinct_clearjunk_id: fixes al :: "('c \<times> 'd) list" assumes "distinct (map fst al)" shows "clearjunk al = al"
disj_zero_left: fixes x :: "'c" shows "(0::'c) OR x = x"
not_eq_order_implies_strict: fixes a :: "'b list"   and b :: "'b list" assumes "a \<noteq> b"   and "prefix b a" shows "strict_prefix b a"
inorder_del: fixes t :: "'b tree23"   and x :: "'b" assumes "complete t"   and "Sorted_Less.sorted (inorder t)" shows "inorder (treeD (del x t)) = del_list x (inorder t)"
Range_empty_iff: fixes r :: "('b \<times> 'a) set" shows "(Range r = {}) = (r = {})"
ldropWhile_fixp_parallel_induct: fixes P :: "('c llist \<Rightarrow> 'c llist) \<Rightarrow> ('d llist \<Rightarrow> 'd llist) \<Rightarrow> bool"   and Pb :: "'c \<Rightarrow> bool"   and Pa :: "'d \<Rightarrow> bool" assumes "ccpo.admissible (prod_lub llist.lub_fun llist.lub_fun) (rel_prod llist.le_fun llist.le_fun) (\<lambda>x. P (fst x) (snd x))"   and "P (\<lambda>_. lSup {}) (\<lambda>_. lSup {})"   and "\<And>f g. P f g \<Longrightarrow> P (\<lambda>xs. case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if Pb x then f xs' else xs) (\<lambda>xs. case xs of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if Pa x then g xs' else xs)" shows "P (ldropWhile Pb) (ldropWhile Pa)"
triv_suf: fixes u :: "'b list"   and v :: "'b list" shows "u \<le>s v \<cdot> u"
matrix_vector_mult_def: fixes m :: "(('e, 'g) vec, 'f) vec"   and x :: "('e, 'g) vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $ i $ j * x $ j"
CastFail\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and h :: "heap"   and l :: "val list"   and sh :: "sheap"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l, sh)\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile>\<^sub>1 \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW ClassCast,(h, l, sh)\<rangle>"
map_key_def: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f" shows "Poly_Mapping.map_key f \<equiv> map_fun lookup Abs_poly_mapping (\<lambda>p. p \<circ> f)"
simps_1: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c"   and v0 :: "'c" assumes "ParityGame G" shows "\<not> lnull (ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0)"
cr_vint_def: fixes a :: "V"   and b :: "int" shows "cr_vint a b = (a = b\<^sub>\<int>)"
del_UnEdge_frame: fixes x :: "'a \<times> 'b \<times> 'a"   and g :: "('a, 'b, 'c) graph_scheme"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "x \<in> edges g"   and "x \<noteq> (v, e, v')"   and "x \<noteq> (v', e, v)" shows "x \<in> edges (del_unEdge v e v' g)"
simps: fixes V :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set"   and st :: "nat set"   and t :: "'a option list" shows "newQuery V Q (st, t) = (\<lambda>tab. projectTable V (semiJoin tab (st, t))) ` Q"