hcju/isabelleps huggingface dataset:
eq2: shows "lt (monom_mult ((1::'b) / punit.lc (rep_list p)) (l - punit.lt (rep_list p)) p) = (l - punit.lt (rep_list p)) \<oplus> lt p"
assms_3: shows "d \<turnstile> ev \<mapsto> d'"
fund_sol_def: shows "fund_sol = (THE z. is_arg_min (\<lambda>x. pell_valuation (case x of (x, y) \<Rightarrow> (int x, int y))) nontriv_solution z)"
ab_def_3: fixes n :: "nat" shows "B n = snd (AB n)"
2: shows "is_tangle_diagram (x * xs) \<and> is_tangle_diagram (basic y) \<longrightarrow> codomain_block x = domain_wall xs"
Ri_subset_\<Gamma>: fixes N :: "'a literal multiset set" shows "sr.Ri N \<subseteq> gr.ord_\<Gamma>"
dom_augment_rels_rels_eq: fixes \<R>' :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option"   and S :: "nat set"   and R :: "nat set" assumes "dom \<R>' = dom \<R>" shows "dom (augment_rels S R \<R>') = dom (augment_rels S R \<R>)"
assms_4: shows "scc \<inter> dom (discovered s) \<subseteq> (tree_edges s)\<^sup>* `` {v}"
intro: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and test_num' :: "int \<Rightarrow> 'a \<Rightarrow> bool"   and filter_plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"   and filter_less' :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a" assumes "Val_abs1 num' plus' \<gamma> test_num' filter_plus' filter_less'" shows "Abs_Int1 num' plus' test_num' filter_plus' filter_less' \<gamma>"
tab_simps_2: shows "src Tfg.\<rho>\<sigma>.tab = src (tab\<^sub>0 g \<star> Tfg.p\<^sub>0)"
wf_disc_sel_common_primitive_1: shows "wf_disc_sel (is_Src_Ports, src_ports_sel) Src_Ports"
ap_spmf_homo: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "ap_spmf (return_spmf f) (return_spmf x) = return_spmf (f x)"
A: fixes z :: "complex"   and n :: "nat" assumes "0 < Re z" shows "((\<lambda>t. complex_of_real t powr (z - 1) * complex_of_real ((1 - t) ^ n)) has_integral fact n / pochhammer z (n + 1)) {0..1}"
intro: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "rational_preference carrier relation" shows "real_vector_rpr carrier relation"
neq_LeafD: fixes t :: "('c \<times> 'd) tree" assumes "t \<noteq> \<langle>\<rangle>" shows "\<exists>l x c r. t = \<langle>l, (x, c), r\<rangle>"
simps_2: shows "oneof_closed [] = err_expecting ''one of [/>, >]'' []"
lnth_LNil: fixes n :: "nat" shows "lnth LNil n = undefined n"
S4aRNext_enabled: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $(S4 rmhist p \<and> ires!p = #NotAResult) \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> $Enabled (<RNext rmCh mm ires p>_(m p))"
inverse_eq_divide: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse a = divide one a"
smap_ub: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "sleq x (smap A)"
bi_unique_alt_def: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
contr: assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Fault f" shows "False"
in_set_butlast_appendI: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<in> set (butlast xs) \<or> x \<in> set (butlast ys)" shows "x \<in> set (butlast (xs @ ys))"
exist: shows "{0..} \<subseteq> existence_ivl0 x"
implmt_methd: fixes G :: "prog"   and C :: "qtname"   and I :: "qtname"   and im :: "qtname \<times> mhead"   and sig :: "sig" assumes "wf_prog G"   and "G\<turnstile>C\<leadsto>I"   and "is_iface G I"   and "im \<in> DeclConcepts.imethds G I sig" shows "? cm:methd G C sig: \<not> is_static cm \<and> \<not> is_static im \<and> G\<turnstile>resTy cm\<preceq>resTy im \<and> accmodi im = Public \<and> accmodi cm = Public"
Pred\<^sub>\<alpha>_def: shows "Pred\<^sub>\<alpha> \<equiv> (id ---> id ---> abs_Tree\<^sub>\<alpha>) tPred"
bounded_lattice_top_ow_def: fixes U :: "'c set"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ls :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and top :: "'c" shows "bounded_lattice_top_ow U inf le ls sup top \<equiv> lattice_ow U inf le ls sup \<and> order_top_ow U le ls top"
of_bool_eq_iff: fixes one :: "'k"   and zero :: "'k"   and p :: "bool"   and q :: "bool" assumes "class.zero_neq_one one zero" shows "(zero_neq_one.of_bool one zero p = zero_neq_one.of_bool one zero q) = (p = q)"
UN_space_closed: fixes S :: "'b measure set" shows "\<Union> (sets ` S) \<subseteq> Pow (\<Union> (space ` S))"
unity_periodic_arithmetic: fixes k :: "nat" shows "periodic_arithmetic (\<lambda>x. unity_root k (int x)) k"
sim: shows "(S ===> eq_on A ===> rel_spmf (rel_prod (=) S)) r1 r2"
proj_tuple_join_mask_restrict: fixes as :: "'b option list"   and n :: "nat"   and X :: "nat set" assumes "length as = n" shows "proj_tuple (join_mask n X) as = restrict X as"
length_replicate: fixes n :: "nat"   and x :: "'d" shows "length (replicate n x) = n"
invertible_interchange_rows: fixes a :: "'b"   and b :: "'b" shows "invertible (interchange_rows (mat (1::'a)) a b)"
tau_step: fixes P :: "'state"   and P' :: "'state"   and P'' :: "'state" assumes "P \<rightarrow> \<langle>\<tau>,P'\<rangle>"   and "P' \<Rightarrow> P''" shows "P \<Rightarrow> P''"
simps_3: fixes f :: "'d \<Rightarrow> 'c"   and p :: "('b, 'd) fm" shows "sub f (\<^bold>\<not> p) = (\<^bold>\<not> sub f p)"
integral_indicator: fixes M :: "'a measure"   and A :: "'a set" shows "integral\<^sup>L M (rind A) = Sigma_Algebra.measure M (A \<inter> space M)"
solves_ode_cong: fixes T :: "real set"   and x :: "real \<Rightarrow> 'b"   and y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "real set"   and X :: "'b set"   and Y :: "'b set" assumes "\<And>t. t \<in> T \<Longrightarrow> x t = y t"   and "\<And>t. t \<in> T \<Longrightarrow> f t (x t) = g t (x t)"   and "T = S"   and "X = Y" shows "(x solves_ode f) T X = (y solves_ode g) S Y"
rep_eq: fixes x :: "'c set"   and xa :: "('c, 'd) fmap" shows "fmlookup (fmrestrict_set x xa) = map_restrict_set x (fmlookup xa)"
list_def: fixes h :: "'b \<Rightarrow> 'b ref"   and p :: "'b ref" shows "list h p = (SOME as. List h p as)"
rel_distinct_2: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y2 :: "bool"   and x11 :: "'b"   and x12 :: "bool" shows "\<not> rel_message R1 R2 <\<bottom>, outM y2> <x11, inM x12>"
sstar_fixpoint: fixes y :: "'a" assumes "has_least_prefixpoint (Sf y)" shows "sstar y = \<mu> Sf y"
prod_id: fixes p :: "nat \<Rightarrow> nat" assumes "p permutes {0..<Suc n}" shows "signof p * (\<Prod>i = 0..<Suc n. A $$ (p i, i)) = A $$ (p n, n) * signof p * (\<Prod>i = 0..<n. A $$ (p i, i))"
T_A_eq_lem: shows "sum t_A {0..<length rs} = int (T (s_A 0) (drop 0 rs) (drop 0 as))"
WT': shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2, \<I>1' \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C map_converter id id (map_sum f1 f2) (map_sum g1 g2) 1\<^sub>C \<surd>"
\<epsilon>_def: shows "FG.\<epsilon> \<equiv> transformation_by_components.map (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) FG.FG.map FG.\<epsilon>o"
inject_4: fixes x4 :: "nat"   and y4 :: "nat" shows "(\<not>unicast x4 = \<not>unicast y4) = (x4 = y4)"
ext_inject: fixes verts :: "'b set"   and arcs :: "'c set"   and tail :: "'c \<Rightarrow> 'b"   and head :: "'c \<Rightarrow> 'b"   and more :: "'d"   and verts' :: "'b set"   and arcs' :: "'c set"   and tail' :: "'c \<Rightarrow> 'b"   and head' :: "'c \<Rightarrow> 'b"   and more' :: "'d" shows "(\<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = \<lparr>verts = verts', arcs = arcs', tail = tail', head = head', \<dots> = more'\<rparr>) = (verts = verts' \<and> arcs = arcs' \<and> tail = tail' \<and> head = head' \<and> more = more')"
lead_coeff_i_def': fixes x :: "'i list" shows "lead_coeff_i ff_ops x = coeff_i ff_ops x (degree_i x)"
invertible: fixes u :: "'a" assumes "u \<in> I" shows "maxI.additive.sub.invertible u"
pull_up_configuration_rank: fixes x :: "nat"   and q :: "'b"   and n :: "nat"   and y :: "nat" assumes "x \<in> configuration q n"   and "y \<in> configuration q n" shows "rank x n = rank y n"
greaterThanAtMost_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and k :: "'b" assumes "class.order less_eq less"   and "less_eq l k" shows "ord.greaterThanAtMost less_eq less k l = {}"
dvd_div_neg: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.idom_divide times one plus zero minus uminus divide"   and "dvd.dvd times b a" shows "divide a (uminus b) = uminus (divide a b)"
Pi_split_insert_domain: fixes x :: "'e \<Rightarrow> 'f"   and i :: "'e"   and I :: "'e set"   and X :: "'e \<Rightarrow> 'f set" shows "(x \<in> Pi (insert i I) X) = (x \<in> Pi I X \<and> x i \<in> X i)"
better_E_Abs: fixes v2 :: "trm"   and x :: "vrs"   and t12 :: "trm"   and T11 :: "ty" assumes "val v2" shows "(\<lambda>x:T11. t12) \<cdot> v2 \<longmapsto> t12[x \<mapsto> v2]"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom (f \<otimes> \<I>) = local.dom f \<otimes> \<I>"
inv_letter_def: fixes l :: "accent \<times> 'b" shows "inv_letter l \<equiv> case l of (a, x) \<Rightarrow> (case a of Acute \<Rightarrow> Grave | Grave \<Rightarrow> Acute | Macron \<Rightarrow> Macron, x)"
enumerator_bound2: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat"   and x :: "nat" assumes "ls \<in> set (enumerator m nmax)"   and "1 < nmax"   and "x \<in> set (butlast ls)" shows "x < nmax - Suc 0"
fmlookup_map: fixes f :: "'c \<Rightarrow> 'a"   and m :: "('b, 'c) fmap"   and x :: "'b" shows "fmlookup (fmmap f m) x = map_option f (fmlookup m x)"
acyclic_heap_def: fixes h :: "(_) heap" shows "local.a_acyclic_heap h = acyclic (parent_child_rel h)"
poly_minus_def: fixes f :: "('c monom \<times> 'd) list"   and g :: "('c monom \<times> 'd) list" shows "poly_minus f g = poly_add f (monom_mult_poly (1, - (1::'d)) g)"
A': shows "A' \<in> carrier_mat (dim_row A') (dim_row A')"
simps_3: fixes C :: "ty"   and D :: "ty"   and X :: "ntrm set" shows "NEGc (C AND D) X = AXIOMSc (C AND D) \<union> BINDINGc (C AND D) X \<union> ANDRIGHT (C AND D) (\<parallel><C>\<parallel>) (\<parallel><D>\<parallel>)"
no_in_one_ll: fixes pret :: "dag"   and levellista :: "ref list list"   and var :: "ref \<Rightarrow> nat"   and i :: "nat"   and j :: "nat"   and no :: "ref" assumes "wf_ll pret levellista var"   and "i < length levellista"   and "j < length levellista"   and "no \<in> set (levellista ! i)"   and "i \<noteq> j" shows "no \<notin> set (levellista ! j)"
img_is_subgroup: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h" shows "subgroup (h ` carrier G) H"
axioms_1: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c"   and B :: "'c"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and \<tau>s :: "'c list" assumes "lbvc A r f T B step \<tau>s" shows "lbv A r f T B"
pseudominimal_in_pow_singleton: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "order.pseudominimal_in (Pow A) {a}"
b1: fixes v :: "'b" shows "cblinfun_of_mat M *\<^sub>V v = basis_enum_of_vec (M *\<^sub>v vec_of_basis_enum v)"
seq_app_tail: fixes x :: "'b"   and s :: "nat \<Rightarrow> 'b" shows "tail (x ## s) = s"
cri_asym2: fixes x :: "'loc \<times> 't"   and y :: "'loc \<times> 't" shows "x<\<^sub>py \<longrightarrow> x \<noteq> y"
that: fixes n :: "'node" assumes "n \<in> set ns"   and "v \<in> defs g n" shows "thesis"
Abs_formula_eqvt: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and p :: "perm" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "p \<bullet> Abs_formula t\<^sub>\<alpha> = Abs_formula (p \<bullet> t\<^sub>\<alpha>)"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "\<exists>x. least L x (carrier L)" shows "weak_partial_order_bottom_axioms L"
elims: fixes x :: "'b option"   and xa :: "'b option"   and y :: "'b option" assumes "x \<squnion> xa = y" obtains xb :: "'b" and ya :: "'b" where "x = Some xb" and "xa = Some ya__" and "y = Some (xb \<squnion> ya__)"   | ya :: "'b option" where "x = None" and "xa = ya" and "y = ya"   | v :: "'b" where "x = Some v" and "xa = None" and "y = Some v"
binop_v11I: fixes A :: "V"   and f :: "V" assumes "nop_v11 A (2\<^sub>\<nat>) f" shows "binop_v11 A f"
distinct_removeAll: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs" shows "distinct (removeAll x xs)"
pmf_map_inj: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c pmf"   and x :: "'c" assumes "inj_on f (set_pmf M)"   and "x \<in> set_pmf M" shows "pmf (map_pmf f M) (f x) = pmf M x"
start_state_def: fixes P :: "jvm_prog" shows "start_state P = (None, start_heap P, [([], [], Start, start_m, 0, No_ics)], [Start \<mapsto> (Map.empty, Done)])"
disjoint_support': fixes p :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "permutation p" shows "(set (support p a) \<inter> set (support p b) = {}) = (a \<notin> set (support p b))"
affine_bounded_eq_lowdim: fixes S :: "'a set" assumes "affine S" shows "bounded S = (aff_dim S \<le> 0)"
Acc_to_mojmir_accept: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "accepting_pair\<^sub>R (\<delta>\<^sub>\<UU> \<Sigma>) (\<iota>\<^sub>\<UU> \<phi>) (Acc\<^sub>\<UU> \<Sigma> \<pi> (G \<psi>)) w"   and "the (\<pi> (G \<psi>)) < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G\<^sub>\<UU> ((Abs \<circ> Unf\<^sub>G) \<psi>)" shows "mojmir_def.accept \<up>af\<^sub>G\<^sub>\<UU> ((Abs \<circ> Unf\<^sub>G) \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"
free_tv_of_substitutions_extend_to_schemes: fixes n :: "nat"   and sch :: "type_scheme"   and S :: "nat \<Rightarrow> typ" assumes "n \<in> free_tv sch" shows "free_tv (S n) \<subseteq> free_tv ($ S sch)"
similar_mat_witD2_2: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "Q * P = 1\<^sub>m n"
Image_closed_trancl: fixes r :: "('a \<times> 'a) set"   and X :: "'a set" assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
rec_twice_def: shows "rec_twice = Cn 1 rec_mult [recf.id 1 0, constn 2]"
hfd: fixes x :: "complex" assumes "x \<in> S" shows "((\<lambda>z. exp (g z) / f z) has_field_derivative 0) (at x)"
rejects_empty: fixes \<F> :: "nat set set"   and M :: "nat set" shows "rejects \<F> {} M = (Fpow M \<inter> \<F> = {})"
subterms_finite: fixes t :: "('a, 'b) Term.term" shows "finite (subterms t)"
y_max_Re_Im: fixes z :: "complex" assumes "z ^ n = x" shows "Re z < Re y \<or> Re z = Re y \<and> Im z \<le> Im y"
WhileTrue: fixes b :: "state \<Rightarrow> bool"   and s :: "state"   and c :: "com"   and t :: "state"   and u :: "state" assumes "b s"   and "s -c\<rightarrow> t"   and "t -WHILE b DO c\<rightarrow> u" shows "s -WHILE b DO c\<rightarrow> u"
sep_conj_sep_true: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and h :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "P h" shows "sep_algebra.sep_conj plus sep_disj P (\<lambda>s. True) h"
ps'_def: shows "ps' = filter (\<lambda>x. dist (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case (l, snd (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) of (x, y) \<Rightarrow> (real_of_int x, y)) < dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps"
state_and_event_determine_next: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and c'' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "distributed_system.next channel trans send recv c ev c''" shows "c' = c''"
A71_Acc: shows "Acc level1 sA71 = {sA81, sA91, sA92}"
reachable_path_entry: fixes n :: "nat" assumes "reachable n" shows "\<exists>p. path_entry (g_E G) p n"
arith_sum2: fixes t :: "nat"   and k :: "nat"   and j :: "nat" shows "Suc (Suc (t + k + j)) = Suc (Suc (t + (k + j)))"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "finite_group G"   and "comm_group G" shows "finite_comm_group G"
vector_sub_project_orthogonal: fixes b :: "'b"   and x :: "'b" shows "inner b (x - scale (inner x b / inner b b) b) = (0::'a)"
Ccod: fixes f :: "'b" assumes "f \<in> mor" shows "cod f \<in> obj"
1: shows "M_mat_R (signs_smash signs1 signs2) (subsets_smash_R n1 subs1 subs2) $$ (i, j) = z_R (subsets_smash_R n1 subs1 subs2 ! i) (signs_smash signs1 signs2 ! j)"
skew_case: fixes t :: "('c \<times> nat) tree" shows "skew t = (case t of \<langle>\<rangle> \<Rightarrow> \<langle>\<rangle> | \<langle>\<langle>\<rangle>, a, t3\<rangle> \<Rightarrow> \<langle>\<langle>\<rangle>, a, t3\<rangle> | \<langle>\<langle>t1, (y, lvy), t2\<rangle>, (x, lvx), t3\<rangle> \<Rightarrow> if lvx = lvy then \<langle>t1, (y, lvx), \<langle>t2, (x, lvx), t3\<rangle>\<rangle> else t)"
extension_valid_maximal_conforming: fixes v' :: "'a" assumes "v' \<rightarrow> \<sigma> v0"   and "v' \<in> VV p \<Longrightarrow> \<sigma> v' = \<sigma> v0" shows "vmc_path G (LCons v' P) v' p \<sigma>"
reindex: fixes h :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'e \<Rightarrow> 'f" assumes "inj_on h A" shows "prod g (h ` A) = prod (g \<circ> h) A"
scaleR_mtx_vec_assoc: fixes c :: "real"   and A :: "'a sq_mtx"   and x :: "(real, 'a) vec" shows "(c *\<^sub>R A) *\<^sub>V x = c *\<^sub>R A *\<^sub>V x"
set_add_def': fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set"   and K :: "'c set" shows "H <+>\<^bsub>G\<^esub> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<oplus>\<^bsub>G\<^esub> k}"
birkhoff_sum_def: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b"   and n :: "nat"   and x :: "nat \<Rightarrow> real" shows "T.birkhoff_sum f n x = (\<Sum>i<n. f ((T ^^ i) x))"
nat_pow_pow: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and n :: "nat"   and m :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "(x [^]\<^bsub>G\<^esub> n) [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> (n * m)"
l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, 'ai option) prog) \<Rightarrow> ((_) heap, exception, 'ai option) prog" shows "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order \<equiv> first_in_tree_order = l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_first_in_tree_order to_tree_order"
remove_code_1: fixes x :: "'b"   and xs :: "'b list" shows "Set.remove x (set xs) = set (removeAll x xs)"
d_p_prod_eq_meet: fixes x :: "'a"   and y :: "'a" shows "d x \<parallel> d y = d x \<sqinter> d y"
\<Upsilon>_cong_aux: fixes U :: "(MIR.num \<times> int) list"   and x :: "real"   and bs :: "real list" assumes "\<forall>(t, n)\<in>set U. numbound0 t \<and> 0 < n" shows "(\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` set (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs U)) = (\<lambda>((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (set U \<times> set U)"
kruskal_injective_inv_3: fixes e :: "'b"   and f :: "'b" assumes "stone_algebra_class.comp_inf.arc e"   and "stone_algebra_class.comp_inf.forest f" shows "bounded_distrib_lattice_class.comp_inf.coreflexive (id (bounded_distrib_lattice_class.comp_inf.tc (top \<sqinter> e)) \<sqinter> bounded_distrib_lattice_class.comp_inf.tc (top \<sqinter> e) \<sqinter> (id f \<sqinter> f))"
servK_authentic: fixes authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "authK \<notin> range shrK"   and "evs \<in> kerbIV" shows "\<exists>A. Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"
simps: fixes x :: "'b extN" shows "x\<^sup>T = x"
goal3_3: shows "Rmi_g (Suc (Suc n)) (IF v t e) bdd"
simps_2: fixes S :: "nat \<Rightarrow> typ"   and n :: "nat" shows "$ S (BVar n) = BVar n"
assms_1: shows "\<forall>cx\<in>CX. interpret_slp (slp_of_fas fas) (env cx) \<in> R"
interrupt_actions_ok'_all_nthI: fixes ias :: "'b interrupt_action list"   and "is" :: "'b set" assumes "\<And>n. n < length ias \<Longrightarrow> interrupt_action_ok' (redT_updIs is (take n ias)) (ias ! n)" shows "interrupt_actions_ok' is ias"
chain: shows "Complete_Partial_Order.chain (ord_spmf (=)) Y"
succsr_def: fixes succs :: "'a \<Rightarrow> 'b list" shows "succsr succs = {(x, y). y \<in> set (succs x)}"
vcg_post_imp: fixes Q :: "('c, 'd, 'e, 'f, 'g) system_state \<Rightarrow> bool"   and R :: "('c, 'd, 'e, 'f, 'g) system_state \<Rightarrow> bool"   and coms :: "'e \<Rightarrow> ('c, 'd, 'f, 'g) com"   and p :: "'e"   and aft :: "'g \<Rightarrow> 'd set"   and P :: "('c, 'd, 'e, 'f, 'g) system_state \<Rightarrow> bool"   and c :: "('c, 'd, 'f, 'g) com" assumes "\<And>s. Q s \<Longrightarrow> R s"   and "CIMP_vcg_rules.valid_syn coms p aft P c Q" shows "CIMP_vcg_rules.valid_syn coms p aft P c R"
replicate_iff: fixes l :: "color list" shows "(\<forall>i<length l. l ! i = R) = (\<exists>n. l = replicate n R)"
LT17: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (F \<or> G \<leadsto> H) = ((F \<leadsto> H) \<and> (G \<leadsto> H))"
list_wset_rel_br_eq: fixes a :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "\<langle>br a I\<rangle>list_wset_rel = br (\<lambda>xs. a ` set xs) (\<lambda>xs. \<forall>x\<in>set xs. I x)"
assms_3: assumes "p \<in> sterms \<Gamma> p1" shows "P"
all_partial_equivs_on_def: fixes A :: "'b set" shows "all_partial_equivs_on A = {0..card A} \<bind> (\<lambda>k. {A'. A' \<subseteq> A \<and> card A' = k} \<bind> (\<lambda>A'. {R. equiv A' R}))"