hcju/isabelleps huggingface dataset:
remove: fixes A :: "'d set"   and x :: "'d"   and g :: "'d \<Rightarrow> 'e" assumes "finite A"   and "x \<in> A" shows "sum g A = g x + sum g (A - {x})"
a1: shows "low_equal s1 s2 \<and> fst (memory_read 8 addra s1) = fst (memory_read 8 addra s2)"
nodeToDigit_inv_1: fixes n :: "nat"   and nd :: "('a, 'b) Node" assumes "FingerTreeStruc.is_leveln_node (Suc n) nd" shows "FingerTreeStruc.is_leveln_digit n (FingerTreeStruc.nodeToDigit nd)"
degree_le_max_perm_degree: fixes A :: "('b poly, 'c) sq_matrix" shows "degree (Square_Matrix.det A) \<le> max_perm_degree A"
nth_default_Nil: fixes dflt :: "'a"   and n :: "nat" shows "nth_default dflt [] n = dflt"
max_point_rep: fixes B :: "'b set multiset"   and x :: "'b" shows "B rep x \<le> int (size B)"
lookup_map_entry': fixes x :: "'c"   and f :: "'b \<Rightarrow> 'b"   and m :: "('c, 'b) mapping"   and y :: "'c" shows "lookup (map_entry x f m) y = (if x = y then map_option f (lookup m y) else lookup m y)"
\<I>\<^sub>\<tau>_3: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau>"
inverse_minus_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse (uminus a) = uminus (inverse a)"
mset_sort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "mset (linorder.sort_key less_eq k xs) = mset xs"
fpxs_nth_of_fls: fixes f :: "'b fls"   and r :: "rat" shows "fpxs_nth (fpxs_of_fls f) r = (if r \<in> \<int> then fls_nth f \<lfloor>r\<rfloor> else (0::'b))"
U\<^sub>f_mult_without_empty_summands_sum_even: fixes i :: "nat"   and j :: "nat"   and A :: "complex Matrix.mat" assumes "i < dim_row U\<^sub>f"   and "j < dim_col A"   and "even i"   and "dim_col U\<^sub>f = dim_row A" shows "(\<Sum>k = 0..<dim_row A. U\<^sub>f $$ (i, k) * A $$ (k, j)) = (\<Sum>k\<in>{i, i + 1}. U\<^sub>f $$ (i, k) * A $$ (k, j))"
eval_wp_SetPC: fixes f :: "'c \<Rightarrow> bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and p :: "'b \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" shows "wp (SetPC f p) P = (\<lambda>s. \<Sum>a\<in>supp (p s). p s a * wp (f a) P s)"
max_eqL: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" shows "max a b = a"
inject_1: fixes x11 :: "'a"   and x12 :: "'b"   and x13 :: "'c"   and y11 :: "'a"   and y12 :: "'b"   and y13 :: "'c" shows "(NewThread x11 x12 x13 = NewThread y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
xortho: shows "x \<in> orthogonal_complement (f -` {0})"
new_element_is_l_new_element: shows "l_new_element DocumentClass.type_wf"
prems: shows "\<forall>b\<in>S. length (a # xs) \<le> b"
sqrtu_numeral: fixes x :: "real"   and n :: "num" shows "sqrtu x (numeral n) = (sqrtu x (pred_numeral n) + x / sqrtu x (pred_numeral n)) / 2"
proj2_Col_iff_set_Col: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" shows "proj2_Col p q r = proj2_set_Col {p, q, r}"
intro: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "total_preorder_on carrier le"   and "finite_total_preorder_on_axioms carrier" shows "finite_total_preorder_on carrier le"
map_simp_ide: fixes a :: "'c" assumes "Cop.ide a" shows "\<Phi> a = \<Phi>o a"
g_def: shows "g = Cn 4 r_ifeq [Cn 4 r_psi [recf.Id 4 2, recf.Id 4 0], Cn 4 r_nth [recf.Id 4 3, recf.Id 4 0], recf.Id 4 1, r_constn 3 1]"
sphere_def: fixes x :: "'a"   and e :: "real" shows "sphere x e = {y. dist x y = e}"
sublens_pres_indep': fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and Z :: "'g \<Longrightarrow> 'e" assumes "X \<subseteq>\<^sub>L Y"   and "Z \<bowtie> Y" shows "Z \<bowtie> X"
sum_list_inc_gt0': fixes u :: "nat list"   and y :: "nat list"   and i :: "nat"   and j :: "nat" assumes "length u = length y"   and "i < length y"   and "0 < y ! i"   and "j \<le> i" shows "0 < sum_list (inc y j u)"
cut_greater_subset: fixes I :: "'a set"   and t :: "'a" shows "I \<down>> t \<subseteq> I"
invert_side_of_triangle_1: fixes h :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "S.arr h"   and "f \<cdot>\<^sub>S g = h"   and "S.iso f" shows "S.seq (S.inv f) h \<and> g = S.inv f \<cdot>\<^sub>S h"
QReachImplReach: fixes cfg1 :: "('p, 'v, 's) configuration"   and Q :: "'p set"   and cfg2 :: "('p, 'v, 's) configuration" assumes "qReachable cfg1 Q cfg2" shows "reachable cfg1 cfg2"
on_circline_x_axis_I: fixes z :: "complex" assumes "is_real z" shows "on_circline x_axis (of_complex z)"
f: shows "linear_on S T (*\<^sub>R) (*\<^sub>R) f"
transition_cases_1: fixes T :: "('b ann_com option \<times> 'b set) list"   and s :: "'b"   and t :: "'b com \<times> 'b" assumes "(Parallel T, s) -P1\<rightarrow> t" obtains i :: "nat" and c :: "'b ann_com" and q :: "'b set" and r :: "'b ann_com option" and ta :: "'b" where "t = (Parallel (T[i := (r, q)]), ta)" and "i < length T" and "T ! i = (Some c, q)" and "(Some c, s) -1\<rightarrow> (r, ta)"
short_vector_def: fixes \<alpha> :: "rat"   and fs :: "int vec list" shows "LLL_Impl.short_vector \<alpha> fs = hd (LLL_Impl.reduce_basis \<alpha> fs)"
omega_complete_axioms_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" shows "omega_complete_axioms le lub \<equiv> (\<forall>f n. le (f n) (lub f)) \<and> (\<forall>f x. (\<forall>n. le (f n) x) \<longrightarrow> le (lub f) x)"
perm_wfactorsD: fixes as :: "'a list"   and bs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "as <~~> bs"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G" shows "a \<sim> b"
prefix_of_antimono: fixes \<pi> :: "'c prefix"   and \<pi>' :: "'c prefix"   and s :: "'c trace" assumes "\<pi> \<le> \<pi>'"   and "prefix_of \<pi>' s" shows "prefix_of \<pi> s"
TC_sup_distrib: fixes x :: "V"   and y :: "V" shows "TC (x \<squnion> y) = TC x \<squnion> TC y"
bex_triv_one_point1: fixes A :: "'f set"   and a :: "'f" shows "(\<exists>x\<in>A. x = a) = (a \<in> A)"
run_bind_option: fixes x :: "('a, 'm) optionT"   and f :: "'a \<Rightarrow> ('a, 'm) optionT" shows "run_option (local.bind_option x f) = bind (run_option x) (\<lambda>x. case x of None \<Rightarrow> return None | Some y \<Rightarrow> run_option (f y))"
SBHReadUnbuffered: fixes sb :: "'c memref list"   and a :: "nat"   and volatile :: "bool"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "buffered_val sb a = None" shows "(Read volatile a t # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>(t \<mapsto> m a), sb @ [Read\<^sub>s\<^sub>b volatile a t (m a)], m, \<D>, \<O>, \<R>, \<S>)"
cterms_not_choice: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "p \<in> cterms \<Gamma>" shows "not_choice p"
induced_map_conservative_mpt: fixes A :: "'a set" assumes "A \<in> sets M" shows "conservative_mpt (restrict_space M A) (induced_map A)"
KBasic2_5: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<rightarrow> (\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi>) in v]"
ide_in_hom: fixes a :: "'a2" shows "C2.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>2 a\<guillemotright>"
succ_inj: fixes i :: "'a"   and j :: "'a" assumes "local.aboveS i \<noteq> {}"   and "local.aboveS j \<noteq> {}" shows "(succ i = succ j) = (i = j)"
MClkReplyVal_def: fixes v :: "Vals" shows "MClkReplyVal v = (if v = RPCFailure then MemFailure else v)"
solution_iff_nth_solution: fixes D :: "nat"   and z :: "nat \<times> nat" assumes "pell D" shows "pell.solution D z = (z \<in> range (pell.nth_solution D))"
sub_branch_def: fixes f :: "'f \<Rightarrow> 'e"   and blocks :: "(('d, 'f) fm list \<times> 'f) list" shows "sub_branch f blocks \<equiv> map (sub_block f) blocks"
dd_commute_modulo_conv: fixes r :: "('b \<times> 'b) set" assumes "wf r"   and "trans r"   and "\<And>a b s t u. \<lbrakk>(s, t) \<in> L a; (s, u) \<in> R b\<rbrakk> \<Longrightarrow> \<exists>xs. local.lpeak r a b xs \<and> (t, u) \<in> local.lconv xs"   and "\<And>a b s t u. \<lbrakk>(s, t) \<in> L a; (s, u) \<in> E b\<rbrakk> \<Longrightarrow> \<exists>xs. local.lcliff r a b xs \<and> (t, u) \<in> local.lconv xs"   and "\<And>a b s t u. \<lbrakk>(s, t) \<in> (E a)\<inverse>; (s, u) \<in> R b\<rbrakk> \<Longrightarrow> \<exists>xs. local.rcliff r a b xs \<and> (t, u) \<in> local.lconv xs" shows "(\<Union> (range R) \<union> \<Union> (range E) \<union> (\<Union> (range L))\<inverse>)\<^sup>* \<subseteq> (\<Union> (range R))\<^sup>* O (\<Union> (range E))\<^sup>* O ((\<Union> (range L))\<inverse>)\<^sup>*"
inter_sorted_correct: fixes l1 :: "'b list"   and l2 :: "'b list" assumes "Sorted_Less.sorted (rev l1)"   and "Sorted_Less.sorted (rev l2)" shows "Sorted_Less.sorted (rev (inter_sorted_rev l1 l2)) \<and> set (inter_sorted_rev l1 l2) = set l1 \<inter> set l2"
image_Nset_Suc: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat" shows "f ` {j. j \<le> Suc n} = insert (f (Suc n)) (f ` {j. j \<le> n})"
that: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "r \<in> dgrad_sig_set' (length fs) d"   and "r \<in> dgrad_sig_set' j d"   and "p = rep_list r" shows "thesis"
sum_mult_pOp_closed: fixes A :: "'a set"   and B :: "'a set"   and a :: "'a"   and b :: "'a" assumes "A \<subseteq> carrier R"   and "B \<subseteq> carrier R"   and "a \<in> sum_mult R A B"   and "b \<in> sum_mult R A B" shows "a \<plusminus> b \<in> sum_mult R A B"
le_antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x = y"
index_size_conv: fixes xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "length xs = n" shows "(index xs x = n) = (x \<notin> set xs)"
cases: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "('test, 'atom) com \<times> 'state" assumes "a1 \<rightarrow>c a2" obtains c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1a__ ;; c2a__, sa__)" and "a2 = (c1'__ ;; c2a__, s'a__)" and "(c1a__, sa__) \<rightarrow>c (c1'__, s'a__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1b__ ;; c2b__, sb__)" and "a2 = (c2b__, s'b__)" and "(c1b__, sb__) \<rightarrow>t s'b__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tsta__ then c1c__ else c2c__, sc__)" and "a2 = (c1c__, sc__)" and "tval tsta__ sc__"   | tst :: "'test" and s :: "'state" and c1 :: "('test, 'atom) com" and c2 :: "('test, 'atom) com" where "a1 = (if tstb__ then c1d__ else c2d__, sd__)" and "a2 = (c2d__, sd__)" and "\<not> tval tstb__ sd__"   | tst :: "'test" and s :: "'state" and c :: "('test, 'atom) com" where "a1 = (while tst do c, se__)" and "a2 = (c ;; while tst do c, se__)" and "tval tst se__"   | c1 :: "('test, 'atom) com" and s :: "'state" and c1' :: "('test, 'atom) com" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1e__ | c2e__, sf__)" and "a2 = (c1' | c2e__, s'c__)" and "(c1e__, sf__) \<rightarrow>c (c1', s'c__)"   | c2 :: "('test, 'atom) com" and s :: "'state" and c2' :: "('test, 'atom) com" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1f__ | c2f__, sg__)" and "a2 = (c1f__ | c2', s'd__)" and "(c2f__, sg__) \<rightarrow>c (c2', s'd__)"   | c1 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c2 :: "('test, 'atom) com" where "a1 = (c1g__ | c2g__, sh__)" and "a2 = (c2g__, s'e__)" and "(c1g__, sh__) \<rightarrow>t s'e__"   | c2 :: "('test, 'atom) com" and s :: "'state" and s' :: "'state" and c1 :: "('test, 'atom) com" where "a1 = (c1 | c2, s)" and "a2 = (c1, s')" and "(c2, s) \<rightarrow>t s'"
invariant3p_IntI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'a set"   and J :: "'a set" assumes "invariant3p r I"   and "invariant3p r J" shows "invariant3p r (I \<inter> J)"
map_comp_def: fixes f :: "'e \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'e option" shows "f \<circ>\<^sub>m g = (\<lambda>k. case g k of None \<Rightarrow> None | Some v \<Rightarrow> f v)"
map_poly_Re_power: fixes p :: "complex poly"   and n :: "nat" assumes "set (coeffs p) \<subseteq> \<real>" shows "map_poly Re (p ^ n) = map_poly Re p ^ n"
rel_cong: fixes x :: "'c dual"   and ya :: "'c dual"   and y :: "'d dual"   and xa :: "'d dual"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set_dual ya; yb \<in> set_dual xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_dual R x y = rel_dual Ra ya xa"
exhaust: fixes y :: "val" obtains x1 :: "nat" where "y = VNat x1"   | x2 :: "(val \<times> val) fset" where "y = VFun x2"
bind_def: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'e set" shows "Set.bind A f = {x. \<exists>B\<in>f ` A. x \<in> B}"
sheaf_of_rings_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 "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> sheaf_of_rings_axioms S is_open \<FF> \<rho> zero_str"
that: fixes r :: "'a list"   and n :: "nat" assumes "r <p t"   and "t \<^sup>@ n \<cdot> r = u" shows "thesis"
less_eq_mask_iff_take_bit_eq_self: fixes w :: "'b word"   and n :: "nat" shows "(w \<le> mask n) = (take_bit n w = w)"
unfold_congs_1: fixes r :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and r' :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and v' :: "'f \<Rightarrow> 'f \<times> 'g"   and f :: "('f \<Rightarrow> 'f \<times> 'g) \<Rightarrow> 'f \<Rightarrow> 'f \<times> 'g"   and f' :: "('f \<Rightarrow> 'f \<times> 'g) \<Rightarrow> 'f \<Rightarrow> 'f \<times> 'g" assumes "r = r'"   and "pending r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "pending_update f r = pending_update f' r'"
emeasure_embed_measure': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and A :: "'d set" assumes "inj_on f (space M)"   and "A \<in> sets (embed_measure M f)" shows "emeasure (embed_measure M f) A = emeasure M (f -` A \<inter> space M)"
case_prodE': fixes c :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool"   and p :: "'e \<times> 'f"   and z :: "'g" assumes "(case p of (a, b) \<Rightarrow> c a b) z" obtains x :: "'e" and y :: "'f" where "p = (x, y)" and "c x y z"
simps: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'd Monomorphic_Monad.id" shows "bind_id (return_id x) f = f x"
distinct_101: fixes x3 :: "val"   and x17 :: "'a exp" shows "Val x3 \<noteq> throw x17"
H: shows "\<Gamma> \<turnstile> (\<lambda>X<:S. s) : (\<forall>X<:U. U')"
image_def: fixes f :: "'r \<Rightarrow> 'q"   and A :: "'r set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
ivl_disj_un_singleton_1: fixes l :: "'a" shows "{l} \<union> {l<..} = {l..}"
commute: fixes a :: "'a"   and b :: "'a" shows "a \<^bold>\<squnion> b = b \<^bold>\<squnion> a"
rel_fundefs_Some2: fixes f :: "'a \<Rightarrow> ('dyn, 'b, 'c, 'd, 'opinl) Inca.instr fundef option"   and g :: "'a \<Rightarrow> ('dyn, 'b, 'c, 'd, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef option"   and x :: "'a"   and y :: "('dyn, 'b, 'c, 'd, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "rel_fundefs f g"   and "g x = Some y" shows "\<exists>z. f x = Some z \<and> rel_fundef (\<lambda>x y. x = norm_instr y) z y"
rtrancl_induct: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P a"   and "\<And>y z. (a, y) \<in> r\<^sup>* \<and> (y, z) \<in> r \<and> P y \<Longrightarrow> P z" shows "P b"
smult_nsg_sg: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "G \<guillemotright> N \<diamondop>\<^bsub>G\<^esub> H "
pdevs_apply_unop_pdevs: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c pdevs"   and i :: "nat" shows "pdevs_apply (unop_pdevs f x) i = (if pdevs_apply x i = (0::'c) then 0::'b else f (pdevs_apply x i))"
arev_permutes_arcs: shows "edge_rev M permutes arcs G"
between_antisym: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "between (b, c) a"   and "between (a, c) b" shows "a = b"
fmlookup_restrict_fset: fixes A :: "'d fset"   and m :: "'d \<rightharpoonup>\<^sub>f 'c"   and x :: "'d" shows "fmlookup (fmrestrict_fset A m) x = (if x |\<in>| A then fmlookup m x else None)"
runit_tensor: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<r>\<^sub>1[a \<otimes>\<^sub>1 b] = (a \<otimes>\<^sub>1 \<r>\<^sub>1[b]) \<cdot> \<a>\<^sub>1[a, b, \<I>\<^sub>1]"
prodag_sameTr4: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and B :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)"   and "\<forall>k\<in>I. A k = B k" shows "prod_zero I A = prod_zero I B"
exists_long_division: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])"   and "q \<noteq> []" obtains b :: "'a list" and r :: "'a list" where "b \<in> carrier (K [X])" and "r \<in> carrier (K [X])" and "long_divides p q (b, r)"
unique_midpoint_x_axis: fixes x :: "complex"   and y :: "complex" assumes "is_real x"   and "- 1 < Re x"   and "Re x < 1"   and "is_real y"   and "- 1 < Re y"   and "Re y < 1"   and "x \<noteq> y" shows "\<exists>!z. - 1 < Re z \<and> Re z < 1 \<and> is_real z \<and> poincare_distance (of_complex z) (of_complex x) = poincare_distance (of_complex z) (of_complex y)"
GSubspace_Ker: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FGModuleHom G smult V smult' T" shows "RModule (aezfun_setspan G) smult (ker T \<inter> V) \<and> ker T \<inter> V \<subseteq> V"
prems: shows "op_tdghm \<NN> = op_tdghm \<NN>'"
abs_fresh_2: fixes x :: "'b"   and b :: "name"   and a :: "name" assumes "finite (supp x)" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
R_eq_finite: fixes A :: "'b set"   and k :: "nat" shows "R A k = finite A"
inject_2: fixes x21 :: "'d"   and x22 :: "('d, 'e) Term.term list"   and y21 :: "'d"   and y22 :: "('d, 'e) Term.term list" shows "(Fun x21 x22 = Fun y21 y22) = (x21 = y21 \<and> x22 = y22)"
pwd_spy_sep_map_def: fixes n :: "nat" shows "pwd_spy_sep_map n \<equiv> case pwd_owner_iv n of (s, S, A, B, C, D) \<Rightarrow> (insert (Spy, PubKey A) s, S, A, B, C, D)"
assms_1: shows "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* u"
make_cert_def: fixes step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and \<tau>s :: "'a list"   and B :: "'a" shows "make_cert step \<tau>s B = map (\<lambda>pc. if is_target step \<tau>s pc then \<tau>s ! pc else B) [0..<length \<tau>s] @ [B]"
simps_1: fixes uu :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and uv :: "'c" shows "OF_match_linear uu [] uv = NoAction"
zero_diff: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and minus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and a :: "'f" assumes "class.comm_monoid_diff plus minus zero" shows "minus zero a = zero"
takeWhile_eq_Nil_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(takeWhile P xs = []) = (xs = [] \<or> \<not> P (hd xs))"
DERIV_even_real_root: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "even n"   and "x < 0" shows "(root n has_real_derivative inverse (- real n * root n x ^ (n - Suc 0))) (at x)"
vrat_of_rat_in_vratE: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" obtains b :: "rat" where "b\<^sub>\<rat> = a"
gen_g_impl_rel_ext_def: fixes Rm :: "('ab \<times> 'ad) set"   and Rv :: "('y \<times> 'ac set) set"   and Re :: "('z \<times> ('ac \<times> 'ac) set) set"   and Rv0 :: "('aa \<times> 'ac set) set" shows "\<langle>Rm, Rv, Re, Rv0\<rangle>gen_g_impl_rel_ext \<equiv> {(\<lparr>gi_V = Vi, gi_E = Ei, gi_V0 = V0i, \<dots> = mi\<rparr>, \<lparr>g_V = V, g_E = E, g_V0 = V0, \<dots> = m\<rparr>) |Vi Ei V0i mi V E V0 m. (Vi, V) \<in> Rv \<and> (Ei, E) \<in> Re \<and> (V0i, V0) \<in> Rv0 \<and> (mi, m) \<in> Rm}"
dimension_independent: fixes K :: "'a set"   and Us :: "'a list" assumes "local.independent K Us" shows "dimension (length Us) K (Span K Us)"
build_ip_partition_distinct': fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "distinct (build_ip_partition c rs)"
l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order get_attribute get_attribute_locs get_element_by_id get_elements_by_class_name get_elements_by_tag_name DocumentClass.type_wf"
sum: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "additive f" shows "f (sum g A) = (\<Sum>x\<in>A. f (g x))"
Keys_hom_components: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "Keys (hom_components p) = keys p"
div_minus_minus: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "divide (uminus a) (uminus b) = divide a b"
rep_eq: fixes x :: "'g \<Rightarrow> 'e"   and xa :: "'g set['f]" shows "set_bset (map_bset x xa) = x ` set_bset xa"
Limsup_const: fixes F :: "'a filter"   and c :: "'b" assumes "F \<noteq> \<bottom>" shows "Limsup F (\<lambda>x. c) = c"
delete_edge_correct: fixes v :: "'b"   and v' :: "'b"   and G :: "'b list_graph" shows "FiniteGraph.delete_edge v v' (list_graph_to_graph G) = list_graph_to_graph (FiniteListGraph.delete_edge v v' G)"
vdomain_mono: fixes r :: "V"   and s :: "V" assumes "r \<subseteq>\<^sub>\<circ> s" shows "\<D>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> s"
called_c'_def: fixes rs :: "'a rule list" shows "called_c' rs = set (concat (map (\<lambda>r. map (\<lambda>c. c) (case get_action r of Call c \<Rightarrow> [c] | _ \<Rightarrow> [])) rs))"
prems: shows "is_path_undir (add_edge a w b G) v (e # p') v'"
Range_snd: fixes r :: "('b \<times> 'a) set" shows "Range r = snd ` r"
f'_def: shows "f' \<equiv> \<lambda>x. if x \<in> cbox w z then f x else x"
position_Left_only_subst: fixes p :: "dir list"   and \<rho> :: "'d \<Rightarrow> ('c, 'd) tm"   and w :: "('c, 'd) tm"   and d :: "dir" assumes "list_all (\<lambda>x. x = dir.Left) p"   and "position_of (subst \<rho> w) (p @ [d])"   and "num_args (subst \<rho> w) = num_args w" shows "position_of w (p @ [d])"