hcju/isabelleps huggingface dataset:
mult_col_div_row_def: fixes a :: "'b"   and k :: "nat"   and A :: "'b mat" shows "mult_col_div_row a k A = multrow k (inverse a) (multcol k a A)"
update_conv': fixes k :: "'a"   and v :: "'b"   and al :: "('a \<times> 'b) list" shows "map_of (update k v al) = map_of al(k \<mapsto> v)"
rbt_del_rbt_sorted: fixes lt :: "('a, 'c) rbt"   and x :: "'a" assumes "rbt_sorted lt" shows "rbt_sorted (rbt_del x lt)"
set_filter_eq_P_forall: fixes X :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>X. P x" shows "Set.filter P X = X"
inputChainEq: fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi" assumes "inputChain xvec M P = inputChain yvec N Q"   and "xvec \<sharp>* yvec"   and "distinct xvec"   and "distinct yvec" obtains p :: "(name \<times> name) list" where "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)" and "distinctPerm p" and "yvec = p \<bullet> xvec" and "N = p \<bullet> M" and "Q = p \<bullet> P"
TE_wtFsym_def: shows "TE_wtFsym \<equiv> ProblemIkTpart.TE_wtFsym wtFsym resOf"
defs3: shows "f \<equiv> \<lambda>S. \<Union>x\<in>set S. case x of receive\<langle>x\<rangle> \<Rightarrow> fv x | \<langle>assign: x \<doteq> t'\<rangle> \<Rightarrow> fv t' | \<langle>check: x \<doteq> t'\<rangle> \<Rightarrow> {} | insert\<langle>t,t'\<rangle> \<Rightarrow> fv t \<union> fv t' | _ \<Rightarrow> {}"
intros_3: fixes p :: "'a pfm"   and q :: "'a pfm" assumes "\<turnstile>\<^sub>! p"   and "\<turnstile>\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q)" shows "\<turnstile>\<^sub>! q"
matrix_mul_assoc: fixes A :: "(('a, 'd) vec, 'c) vec"   and B :: "(('a, 'e) vec, 'd) vec"   and C :: "(('a, 'b) vec, 'e) vec" shows "A ** (B ** C) = A ** B ** C"
eq_onp_def: fixes R :: "'d \<Rightarrow> bool" shows "eq_onp R = (\<lambda>x y. R x \<and> x = y)"
fds_deriv_eq_imp_eq: fixes f :: "'b fds"   and g :: "'b fds" assumes "fds_deriv f = fds_deriv g"   and "fds_nth f (Suc 0) = fds_nth g (Suc 0)" shows "f = g"
DE_lemma: fixes I :: "('sf, 'sc, 'sz) interp"   and ab :: "(real, 'sz) vec"   and bb :: "(real, 'sz) vec"   and sol :: "real \<Rightarrow> (real, 'sz) vec"   and t :: "real" shows "repd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) vid1 (dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t))) = mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)"
simulation_behaviour: fixes s\<^sub>2 :: "'state2"   and b\<^sub>2 :: "'state2 behaviour"   and i :: "'index"   and s\<^sub>1 :: "'state1" assumes "L2.sem_behaves s\<^sub>2 b\<^sub>2"   and "\<not> is_wrong b\<^sub>2"   and "match i s\<^sub>1 s\<^sub>2" shows "\<exists>b\<^sub>1 i'. L1.sem_behaves s\<^sub>1 b\<^sub>1 \<and> rel_behaviour (match i') b\<^sub>1 b\<^sub>2"
pairs_iff_elts: fixes x :: "V"   and y :: "V"   and z :: "V" shows "((x, y) \<in> pairs z) = (\<langle>x, y\<rangle> \<in> elts z)"
take_append: fixes n :: "nat"   and xs :: "'e list"   and ys :: "'e list" shows "take n (xs @ ys) = take n xs @ take (n - length xs) ys"
preservation: shows "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a CWB)"
le_err_refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b err" assumes "\<forall>x. x \<sqsubseteq>\<^bsub>r\<^esub> x" shows "e \<sqsubseteq>\<^bsub>le r\<^esub> e"
all_total_imp_totalize: fixes F :: "'b program" assumes "all_total F" shows "totalize F = F"
complement_rep_number: assumes "\<And>bl. bl \<in># \<B> \<Longrightarrow> incomplete_block bl" shows "constant_rep_design \<V> \<B>\<^sup>C (\<b> - \<Lambda> * (\<v> - 1) div (\<k> - 1))"
last_ConsR: fixes xs :: "'g list"   and x :: "'g" assumes "xs \<noteq> []" shows "last (x # xs) = last xs"
finite_total_preorder_on_iff: fixes A :: "'c set"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "finite_total_preorder_on A R = (total_preorder_on A R \<and> finite A)"
Kas_Issues_A: fixes A :: "agent"   and authK :: "nat"   and Peer :: "msg"   and Ta :: "msg"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "Kas Issues A with Crypt (shrK A) \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace> on evs"
S_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and ecalling :: "bool"   and ccalling :: "bool"   and rcalling :: "bool"   and cs :: "mClkState"   and rs :: "rpcState"   and hs1 :: "histState"   and hs2 :: "histState"   and p :: "PrIds" shows "S rmhist ecalling ccalling rcalling cs rs hs1 hs2 p = (Calling memCh p = #ecalling \<and> Calling crCh p = #ccalling \<and> (#ccalling \<longrightarrow> arg<crCh!p> = MClkRelayArg<arg<memCh!p>>) \<and> (\<not> #ccalling \<and> cst!p = #clkB \<longrightarrow> MVOKBARF<res<crCh!p>>) \<and> Calling rmCh p = #rcalling \<and> (#rcalling \<longrightarrow> arg<rmCh!p> = RPCRelayArg<arg<crCh!p>>) \<and> (\<not> #rcalling \<longrightarrow> ires!p = #NotAResult) \<and> (\<not> #rcalling \<and> rst!p = #rpcB \<longrightarrow> MVOKBA<res<rmCh!p>>) \<and> cst!p = #cs \<and> rst!p = #rs \<and> (rmhist!p = #hs1 \<or> rmhist!p = #hs2) \<and> MVNROKBA<ires!p>)"
phi_def: shows "\<tau>s \<equiv> map (\<lambda>pc. if c ! pc = \<bottom> then wtl (take pc ins) c 0 s\<^sub>0 else c ! pc) [0..<length ins]"
pq_def: shows "create_insert s n \<sigma> new_id = Inr (Insert (InsertMessage p new_id q \<sigma>))"
G1_def: shows "G1 \<equiv> fps_shift 1 (fps_exp (- Ln 2) - 1)"
set_category_axioms: shows "set_category (\<cdot>\<^sub>S) (cardSuc (cmax |UNIV| natLeq))"
nth_equalityI: fixes xs :: "'a list"   and ys :: "'a list" assumes "List.length xs = List.length ys"   and "\<And>i. i < List.length xs \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
trg_cod: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "trg\<^sub>C (C.cod \<mu>) = trg\<^sub>C \<mu>"
circuit_in_sub_cong: fixes \<E> :: "'a set"   and \<E>' :: "'a set"   and C :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>' \<subseteq> \<E>" shows "indep_system.circuit_in (\<E>.indep_in \<E>) \<E>' C = \<E>.circuit_in \<E>' C"
eval\<kappa>: fixes x :: "\<kappa>" shows "eval\<kappa> x \<in> UNIV"
is_faithful: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and f :: "'d"   and f' :: "'d" assumes "faithful_functor A B F"   and "partial_magma.arr A f \<and> partial_magma.arr A f' \<and> partial_magma.dom A f = partial_magma.dom A f' \<and> partial_magma.cod A f = partial_magma.cod A f'"   and "F f = F f'" shows "f = f'"
carrier_mat_triv: fixes m :: "'b mat" shows "m \<in> carrier_mat (dim_row m) (dim_col m)"
approx_unique_complex: fixes r :: "complex star"   and s :: "complex star"   and x :: "complex star" assumes "r \<in> SComplex"   and "s \<in> SComplex"   and "r \<approx> x"   and "s \<approx> x" shows "r = s"
proj_def: fixes n :: "'c"   and S :: "('c strand_label \<times> 'd) list" shows "proj n S \<equiv> filter (\<lambda>s. is_LabelN n s \<or> is_LabelS s) S"
path_connectedin_topspace: fixes X :: "'a topology" shows "path_connectedin X (topspace X) = path_connected_space X"
wellorder_Least_lemma_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
inner_eq_zero_iff: fixes x :: "'a" shows "(inner_field x x = (0::'a)) = (x = (0::'a))"
ii: shows "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'"
deg_nonzero_represents: fixes p :: "int poly" assumes "Polynomial.degree p \<noteq> 0" shows "\<exists>x. p represents x"
kpower_def: fixes f :: "'b \<Rightarrow> 'b set"   and n :: "nat" shows "kpower f n = ((;) f ^^ n) skip"
witness_length: fixes X :: "nat set"   and n :: "nat" shows "length (witness X n) = n"
coboundedI2: fixes c :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd" assumes "c \<preceq> b" shows "c \<preceq> a \<oplus> b"
assms_6: shows "fs' = fs[i := fs ! (i - 1), i - 1 := fs ! i]"
wfP_subset: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP r"   and "p \<le> r" shows "wfP p"
uint_divmod_def: fixes x :: "uint"   and y :: "uint" shows "uint_divmod x y = (if y = 0 then (undefined (div) x 0, undefined (mod) x 0) else (x div y, x mod y))"
exhaust_sel: fixes sum :: "'e + 'f" obtains "sum = Inl (projl sum)"   | "sum = Inr (projr sum)"
simps_1: fixes x1 :: "nat"   and y1 :: "nat" shows "((_ x1 _) = (_ y1 _)) = (x1 = y1)"
vimage_vsingleton: fixes r :: "V"   and a :: "V" shows "r `\<^sub>\<circ> set {a} = set {b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
joinablep_refl: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "reflp P" shows "joinablep P x x"
PQ: fixes x :: "'a" assumes "x \<in> lset ys" shows "P x = Q x"
0: assumes "Ide a" shows "Can (a\<^bold>\<down>) \<and> a\<^bold>\<down> \<in> Hom a \<^bold>\<lfloor>a\<^bold>\<rfloor>"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sup_monoid.sum g A = sup_monoid.sum h B"
get_alloc: fixes x :: "'b"   and h :: "heap"   and x' :: "'b" shows "Ref.get (snd (Ref.alloc x h)) (fst (Ref.alloc x' h)) = x"
in_nodes_lt_n: fixes G :: "nat graph_rec"   and x :: "nat" assumes "dom_sl G"   and "x \<in> set (g_V G)" shows "x < length (cfg_doms.nodes G)"
elements_mat_h_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "elements_mat_h A = range (\<lambda>(i, j). A $h i $h j)"
index_eq_app_right_lower: shows "card (UNIV // eq_app_right language) \<le> card (dfa.states M)"
divides_degree: fixes p :: "'b poly"   and q :: "'b poly" assumes "p dvd q" shows "degree p \<le> degree q \<or> q = 0"
functor_ToTC: shows "functor VxVxV.comp (\<cdot>) H.ToTC"
simps_4: fixes v :: "'b formula" shows "is_cnf (\<^bold>\<not> v) = is_disj (\<^bold>\<not> v)"
labeled_sat_ineq_dbproj_sem_equiv: fixes i :: "'a"   and F' :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "('a \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and X :: "'var list"   and M :: "('fun, 'var) Term.term set"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<forall>(j, p)\<in>(\<lambda>(t, s). (i, t, s)) ` set F' \<union> set D. \<forall>(k, q)\<in>(\<lambda>(t, s). (i, t, s)) ` set F' \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"   and "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (map snd D) \<inter> set X = {}" shows "\<lbrakk>M; map (\<lambda>G. \<forall>X\<langle>\<or>\<noteq>: (F @ G)\<rangle>\<^sub>s\<^sub>t) (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' (map snd D))\<rbrakk>\<^sub>d \<I> = \<lbrakk>M; map (\<lambda>G. \<forall>X\<langle>\<or>\<noteq>: (F @ G)\<rangle>\<^sub>s\<^sub>t) (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' (map snd (dbproj i D)))\<rbrakk>\<^sub>d \<I>"
pmf_False_conv_True: fixes p :: "bool pmf" shows "pmf p False = 1 - pmf p True"
reach_l_x: shows "s\<turnstile> l reachable_from x"
group_axioms: shows "Group_Theory.group stfx.carrier_direct_lim stfx.add_rel (stfx.class_of V (zero_str\<^sub>Y V))"
collinear_subset: fixes T :: "'b set"   and S :: "'b set" assumes "collinear T"   and "S \<subseteq> T" shows "collinear S"
bit_push_bit_iff: fixes m :: "nat"   and a :: "'a"   and n :: "nat" shows "bit (push_bit m a) n = (m \<le> n \<and> (2::'a) ^ n \<noteq> (0::'a) \<and> bit a (n - m))"
opp_power_def: fixes x :: "'a"   and n :: "nat" shows "power.power (1::'a) (\<odot>) x n = x\<^bsup>n\<^esup>"
induct: fixes x1 :: "Var \<Rightarrow> L"   and x2 :: "Expr"   and x3 :: "L"   and P :: "(Var \<Rightarrow> L) \<Rightarrow> Expr \<Rightarrow> L \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> HS_E"   and "\<And>G x. P G (varE x) (G x)"   and "\<And>G c. P G (valE c) bottom"   and "\<And>G e1 p1 e2 p2 p f. \<lbrakk>(G, e1, p1) \<in> HS_E; P G e1 p1; (G, e2, p2) \<in> HS_E; P G e2 p2; p = LUB p1 p2\<rbrakk> \<Longrightarrow> P G (opE f e1 e2) p"   and "\<And>G e p q. \<lbrakk>(G, e, p) \<in> HS_E; P G e p; LEQ p q\<rbrakk> \<Longrightarrow> P G e q" shows "P x1 x2 x3"
ChamberComplexRetraction_axioms: shows "ChamberComplexRetraction X f"
card_Diff1_less_iff: fixes A :: "'d set"   and x :: "'d" shows "(card (A - {x}) < card A) = (finite A \<and> x \<in> A)"
sup_continuous_inf: fixes P :: "'c \<Rightarrow> 'd"   and Q :: "'c \<Rightarrow> 'd" assumes "sup_continuous P"   and "sup_continuous Q" shows "sup_continuous (\<lambda>x. inf (P x) (Q x))"
expectation_prob_atLeast_1: fixes N :: "nat pmf" assumes "(\<lambda>i. measure_pmf.prob N {i..}) abs_summable_on {1..}" shows "measure_pmf.expectation N real = (\<Sum>\<^sub>ai\<in>{1..}. measure_pmf.prob N {i..})"
branch_prob_def: fixes \<Y> :: "nat set \<times> nat set \<times> (nat \<Rightarrow> pdf_type) \<times> ((nat \<Rightarrow> val) \<Rightarrow> ennreal)"   and \<rho> :: "nat \<Rightarrow> val" shows "branch_prob \<Y> \<rho> = emeasure (dens_ctxt_measure \<Y> \<rho>) (space (dens_ctxt_measure \<Y> \<rho>))"
non_empty: fixes U :: "'c set"   and V :: "'c set" assumes "Subspace.subspace U V" shows "U \<noteq> {}"
intro: fixes M :: "('a, 'b) message set"   and m :: "('a, 'b) insert_message"   and a :: "'a extended \<Rightarrow> 'a sort_key extended" assumes "consistent (M \<union> {Insert m})"   and "Insert m \<notin> M"   and "a_conditions (insert_messages (M \<union> {Insert m})) a" shows "integrate_insert_commute_insert_axioms M a m"
rel_pmf_conj_2: fixes Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "bool"   and x :: "'b pmf"   and y :: "'c pmf" shows "rel_pmf (\<lambda>x y. Q x y \<and> P) x y = (P \<and> rel_pmf Q x y)"
explode_inverse: fixes x :: "literal" shows "Abs_literal (explode x) = x"
nslimI: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L" shows "nslim X = L"
forball_contra: fixes A :: "'c set"   and P :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'d"   and Q :: "'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> bool" assumes "\<forall>y\<in>A. P x y \<longrightarrow> \<not> Q y"   and "\<forall>y\<in>A. Q y \<or> R y" shows "\<forall>y\<in>A. \<not> P x y \<or> R y"
max_subgraph_def: fixes G :: "('c, 'd) pre_digraph"   and P :: "('c, 'd) pre_digraph \<Rightarrow> bool"   and H :: "('c, 'd) pre_digraph" shows "pre_digraph.max_subgraph G P H \<equiv> subgraph H G \<and> P H \<and> (\<forall>H'. H' \<noteq> H \<and> subgraph H H' \<longrightarrow> \<not> (subgraph H' G \<and> P H'))"
compactin_absolute: fixes X :: "'a topology"   and S :: "'a set" shows "compactin (subtopology X S) S = compactin X S"
closure_approachable_le: fixes x :: "'a"   and S :: "'a set" shows "(x \<in> closure S) = (\<forall>e>0. \<exists>y\<in>S. dist y x \<le> e)"
word_bw_same_1: fixes x :: "'b word" shows "x AND x = x"
nth_least_ext_strict_mono: fixes k :: "nat"   and l :: "nat"   and s :: "nat set" assumes "k < l" shows "nth_least_ext s k < nth_least_ext s l"
dupe_monic_i_int_def: fixes ops :: "'a arith_ops_record"   and D :: "int poly"   and H :: "int poly"   and S :: "int poly"   and T :: "int poly" shows "dupe_monic_i_int ops D H S T = (let d = of_int_poly_i ops D; h = of_int_poly_i ops H; s = of_int_poly_i ops S; t = of_int_poly_i ops T in (\<lambda>U. case dupe_monic_i ops d h s t (of_int_poly_i ops U) of (D', H') \<Rightarrow> (to_int_poly_i ops D', to_int_poly_i ops H')))"
conc_trans: fixes C :: "'a nres"   and B :: "'b nres"   and A :: "'b nres" assumes "C \<le> \<Down> R B"   and "B \<le> \<Down> Id A" shows "C \<le> \<Down> R (\<Down> Id A)"
i_set0_mult_mod_0_div: fixes I :: "nat set"   and k :: "nat"   and d :: "nat" assumes "I \<in> i_set0_mult k"   and "k mod d = 0" shows "I \<oslash> d \<in> i_set0_mult (k div d)"
inf_raw: fixes g :: "nat \<Rightarrow> V" shows "range g \<in> range elts"
fmlas_UNIV: shows "i.R fmlas = UNIV"
search_diverg: fixes b :: "nat"   and i :: "nat" assumes "s b \<down>= i"   and "search b \<up>" shows "\<nexists>y. \<phi> i y \<down> \<and> e_length b \<le> y"
extremum_unique: fixes a :: "'b::order_bot" shows "((a::'b::order_bot) \<le> \<bottom>) = (a = \<bottom>)"
b_2: shows "\<forall>y\<in>local.degenerate_lotteries. y \<succeq>[\<R>] b"
strongly_connectedE: fixes G :: "('c, 'd) pre_digraph" assumes "strongly_connected G" obtains "\<And>u v. u \<in> verts G \<and> v \<in> verts G \<Longrightarrow> u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
assms: shows "(a, b) = chophelper (to_list v p) []"
maximal_independent_subset: fixes V :: "'b set" obtains B :: "'b set" where "B \<subseteq> V" and "local.independent B" and "V \<subseteq> local.span B"
hext_upd_obj: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and S' :: "(char list list \<times> (char list \<Rightarrow> val option)) set" assumes "h a = \<lfloor>(C, S)\<rfloor>" shows "h \<unlhd> h(a \<mapsto> (C, S'))"
assms_2: shows "((\<lambda>x. g (eval c x)) expands_to G) bs'"
multiset_cases: fixes M :: "'a multiset" obtains "M = {#}"   | x :: "'a" and N :: "'a multiset" where "M = add_mset x N"
\<delta>\<^sub>p_def_1: shows "p\<^sub>1 = find_closest p\<^sub>0 \<delta> (p\<^sub>2 # ps)"
radiant_of_bounds_2: fixes p :: "(real, 2) vec" shows "radiant_of p < 2 * pi"
length_Suc_conv: fixes xs :: "'g list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
set_of_r_bal: fixes r :: "'a tree\<^sub>0"   and l :: "'a tree\<^sub>0"   and x :: "'a" assumes "height r = height l + 2" shows "set_of (r_bal\<^sub>0 x l r) = insert x (set_of l \<union> set_of r)"
of_int_power: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and z :: "int"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (z ^ n) = power.power one times (ring_1.of_int one plus zero minus z) n"
proj3_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "rt \<noteq> update rt dip (dsn, dsk, flg, hops, sip)" shows "\<pi>\<^sub>3 (the (update rt dip (dsn, dsk, flg, hops, sip) dip)) = dsk"
integrable_restrict_UNIV: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "S \<in> sets lebesgue" shows "integrable lebesgue (\<lambda>x. if x \<in> S then f x else (0::'d)) = integrable (lebesgue_on S) f"
snth_stranspose: fixes ws :: "'b stream list"   and k :: "nat" shows "stranspose ws !! k = map (\<lambda>w. w !! k) ws"
arr_mkArr: fixes A :: "'t set"   and B :: "'t set"   and F :: "'t \<Rightarrow> 't" shows "S'.arr (S'.mkArr A B F) = (A \<subseteq> S'.Univ \<and> |A| <o \<AA> \<and> B \<subseteq> S'.Univ \<and> |B| <o \<AA> \<and> F \<in> A \<rightarrow> B)"
get_M_Element_preserved8: fixes element_ptr :: "(_) element_ptr"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and setter :: "('j \<Rightarrow> 'j) \<Rightarrow> ('c, 'a, 'b, 'd, 'g option) RElement_scheme \<Rightarrow> ('c, 'a, 'b, 'd, 'g option) RElement_scheme"   and v :: "'j"   and h' :: "(_) heap"   and getter :: "(_) RObject_ext \<Rightarrow> 'k" assumes "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \<noteq> object_ptr"   and "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
lex_pm_except_max: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd"   and x :: "'c" assumes "lex_pm s t"   and "keys s \<union> keys t \<subseteq> {..x}" shows "lex_pm (except s {x}) (except t {x})"
exhaust: fixes y :: "face" obtains x1 :: "nat list" and x2 :: "facetype" where "y = Face x1 x2"
opt_mru_guard_imp_Quorum: fixes vs :: "process \<Rightarrow> (nat \<times> val) option"   and Q :: "process set"   and v :: "val" assumes "opt_mru_guard vs Q v" shows "Q \<in> Quorum"
sort_key_stable: fixes f :: "'g \<Rightarrow> 'h"   and k :: "'h"   and xs :: "'g list" shows "filter (\<lambda>y. f y = k) (sort_key f xs) = filter (\<lambda>y. f y = k) xs"
is_pseudonatural_equivalence: shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G F \<Phi>\<^sub>F \<sigma>.map\<^sub>0 \<sigma>.map\<^sub>1"
mask_mono: fixes sz' :: "nat"   and sz :: "nat" assumes "sz' \<le> sz" shows "mask sz' \<le> mask sz"
upower_inductl: fixes P :: "'b hrel"   and Q :: "('b, 'c) urel"   and R :: "('b, 'c) urel"   and n :: "nat" assumes "Q \<sqsubseteq> (P ;; Q) \<sqinter> R" shows "Q \<sqsubseteq> power.power II (;;) P n ;; R"
dij_def: fixes i :: "nat"   and j :: "nat" shows "local.dij i j = lcm (a ! i) (b ! j) div a ! i"
ennreal3_cases: fixes x :: "ennreal"   and xa :: "ennreal"   and xaa :: "ennreal" obtains r :: "real" and ra :: "real" and raa :: "real" where "0 \<le> r__" and "x = ennreal r__" and "0 \<le> ra__" and "xa = ennreal ra__" and "0 \<le> raa" and "xaa = ennreal raa"   | r :: "real" and ra :: "real" where "0 \<le> rb__" and "x = ennreal rb__" and "0 \<le> rab__" and "xa = ennreal rab__" and "xaa = \<top>"   | r :: "real" and ra :: "real" where "0 \<le> rc__" and "x = ennreal rc__" and "xa = \<top>" and "0 \<le> rac__" and "xaa = ennreal rac__"   | r :: "real" where "0 \<le> rd__" and "x = ennreal rd__" and "xa = \<top>" and "xaa = \<top>"   | r :: "real" and ra :: "real" where "x = \<top>" and "0 \<le> re__" and "xa = ennreal re__" and "0 \<le> ra" and "xaa = ennreal ra"   | r :: "real" where "x = \<top>" and "0 \<le> rf__" and "xa = ennreal rf__" and "xaa = \<top>"   | r :: "real" where "x = \<top>" and "xa = \<top>" and "0 \<le> r" and "xaa = ennreal r"   | "x = \<top>" and "xa = \<top>" and "xaa = \<top>"
up_to_less: fixes J :: "nat set"   and i :: "nat" assumes "finite J"   and "i \<in> J" shows "i < up_to J"
path_component_subset: fixes S :: "'a set"   and x :: "'a" shows "path_component_set S x \<subseteq> S"
eq_name_eq_pres': fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "nat \<Rightarrow> nat option"   and name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and name' :: "char list"   and pres' :: "(nat \<times> nat) list"   and effs' :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer' :: "nat" assumes "abs_ast_prob problem"   and "s \<in> ast_problem.valid_states problem"   and "is_standard_operator' (name, pres, effs, layer)"   and "(name, pres, effs, layer) \<in> set (ast_problem.ast\<delta> problem)"   and "ast_problem.lookup_operator problem ((fst \<circ> the \<circ> ast_problem.lookup_action problem \<circ> abs_ast_operator) (name, pres, effs, layer)) = Some (name', pres', effs', layer')" shows "pres = pres'"
assms_3: shows "snd ` set txs \<subseteq> var"
frange_fid_on: fixes A :: "V" shows "\<R>\<^sub>\<bullet> (fid_on A) = A"