hcju/isabelleps huggingface dataset:
inner_not_same_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"   and v :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis"   and "u \<noteq> v" shows "inner u v = 0"
that: fixes \<A> :: "'a set set" assumes "countable \<A>"   and "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A"   and "\<And>A. A \<in> \<A> \<Longrightarrow> open A"   and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S"   and "\<And>A B. \<lbrakk>A \<in> \<A>; B \<in> \<A>\<rbrakk> \<Longrightarrow> A \<inter> B \<in> \<A>" shows "thesis"
phl_cons2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and w :: "'c"   and x :: "'c"   and z :: "'c"   and y :: "'c" assumes "class.pre_dioid plus times less_eq less"   and "less_eq w x"   and "less_eq (times z y) (times y w)" shows "less_eq (times z y) (times y x)"
no_inter: shows "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb \<inter> (outstanding_refs is_volatile_Write\<^sub>s\<^sub>b (x # xs) \<union> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (x # xs) \<union> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (x # xs)) \<union> (outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (x # xs)) - RO) \<union> (\<O> \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (x # xs)))) = {}"
trans_iless_iadd1: fixes i :: "enat"   and j :: "enat"   and m :: "enat" assumes "i < j" shows "i < j + m"
nth_rest_power_bin: fixes k :: "nat"   and w :: "int"   and n :: "nat" shows "bit (((\<lambda>w. w div 2) ^^ k) w) n = bit w (n + k)"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a OR b) OR c = a OR b OR c"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "category C"   and "span_in_category_axioms C S" shows "span_in_category C S"
Lim_transform_eq: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and a :: "'c" assumes "((\<lambda>x. f x - g x) \<longlongrightarrow> (0::'c)) F" shows "(f \<longlongrightarrow> a) F = (g \<longlongrightarrow> a) F"
finite_lemma: fixes x :: "'b llist"   and A :: "'b set"   and B :: "'b set" assumes "x \<in> A\<^sup>\<star>"   and "x \<in> B\<^sup>\<infinity>" shows "x \<in> B\<^sup>\<star>"
ifthenelse_def: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rel_kat.ifthenelse p x y = rel_dioid_tests.t p ; x \<union> Id \<inter> - p ; y"
z12: shows "z \<in> b1 \<or> z \<in> b2"
mk_doubleton_image_vsubset_veqpoll: fixes X :: "V"   and Y :: "V" assumes "X \<subseteq>\<^sub>\<circ> Y" shows "mk_doubleton_image X X \<approx>\<^sub>\<circ> mk_doubleton_image X Y"
right_eq: shows "{p. \<exists>y. y \<in> cbox (g2 b) (g1 b) \<and> p = (b, y)} = (\<lambda>y. twoC (1, y)) ` {0..1}"
create_get: fixes x :: "'c \<Longrightarrow> 'd"   and v :: "'c" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (create\<^bsub>x\<^esub> v) = v"
foundation5: fixes \<tau> :: "'b state \<times> 'b state"   and P :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> P and Q" shows "\<tau> \<Turnstile> P \<and> \<tau> \<Turnstile> Q"
im_is_submodule: shows "LinearCombinations.submodule K imT W"
simps_1: fixes d :: "nat" shows "intv_const (Const d) = d"
FAssRedsVal: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>2 :: "char list exp"   and v :: "val"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "\<lfloor>(C, fs)\<rfloor> = h\<^sub>2 a" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>unit,(h\<^sub>2(a \<mapsto> (C, fs((F, D) \<mapsto> v))), l\<^sub>2)\<rangle>"
not_nonzero_memE: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "domain R"   and "a \<notin> nonzero R"   and "a \<in> carrier R" shows "a = \<zero>\<^bsub>R\<^esub>"
is_INS_def: fixes operation :: "'a operation" shows "is_INS operation = (\<exists>x11 x12 x13. operation = INS x11 x12 x13)"
keys_conv_dom_lookup: fixes t :: "('b, 'c) trie" assumes "ID cbl \<noteq> None" shows "keys t = dom (lookup t)"
subspace_image: fixes S :: "'b set" assumes "m1.subspace S" shows "m2.subspace (f ` S)"
wf_trm_subst_rangeD: fixes \<theta> :: "'a \<Rightarrow> ('fun, 'a) Term.term"   and x :: "'a" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)"
elims_2: fixes x :: "nat"   and xa :: "(nat \<times> nat) list"   and xb :: "nat" assumes "cf.isPath x xa xb" obtains u :: "nat" and v :: "nat" where "x = u__" and "xa = []" and "xb = v__" and "u__ = v__"   | u :: "nat" and xc :: "nat" and y :: "nat" and p :: "(nat \<times> nat) list" and v :: "nat" where "x = u" and "xa = (xc, y) # p" and "xb = v" and "u = xc \<and> (xc, y) \<in> cf.E \<and> cf.isPath y p v"
max_fun_diff_le_eq: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "(t.max_fun_diff f g, x) \<in> t"   and "f \<noteq> g"   and "f \<in> rt.FINFUNC"   and "g \<in> rt.FINFUNC"   and "x \<noteq> t.max_fun_diff f g" shows "f x = g x"
norm_stack_def: fixes Finca_empty :: "'a"   and Finca_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef option"   and Finca_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef \<Rightarrow> 'a"   and Finca_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f) Inca.instr fundef) list"   and Fubx_empty :: "'g"   and Fubx_get :: "'g \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef option"   and Fubx_add :: "'g \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef \<Rightarrow> 'g"   and Fubx_to_list :: "'g \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f, 'h, 'i, 'j) Ubx.instr fundef) list"   and heap_empty :: "'k"   and heap_get :: "'k \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'k \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'k"   and heap_to_list :: "'k \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and box_ubx1 :: "'i \<Rightarrow> 'c"   and unbox_ubx1 :: "'c \<Rightarrow> 'i option"   and box_ubx2 :: "'j \<Rightarrow> 'c"   and unbox_ubx2 :: "'c \<Rightarrow> 'j option"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'f \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'f option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'e"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'h \<Rightarrow> ('c, 'i, 'j) unboxed list \<Rightarrow> ('c, 'i, 'j) unboxed option"   and \<UU>\<bb>\<xx> :: "'f \<Rightarrow> Unboxed.type option list \<Rightarrow> 'h option"   and \<BB>\<oo>\<xx> :: "'h \<Rightarrow> 'f"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'h \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and \<Sigma> :: "('c, 'i, 'j) unboxed list" assumes "inca_to_ubx_simulation Finca_empty Finca_get Finca_add Finca_to_list Fubx_empty Fubx_get Fubx_add Fubx_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>" shows "inca_to_ubx_simulation.norm_stack box_ubx1 box_ubx2 \<Sigma> \<equiv> map (unboxedval.norm_unboxed box_ubx1 box_ubx2) \<Sigma>"
unifiers_insert_VarD_2: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and x :: "'a"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" assumes "\<sigma> \<in> unifiers (insert (t, Var x) E)" shows "subst x t \<circ>\<^sub>s \<sigma> = \<sigma>"
latestActless: fixes n\<^sub>s :: "nat"   and n :: "nat"   and nid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>n'\<ge>n\<^sub>s. latestAct_cond nid t n n'" shows "n\<^sub>s \<le> \<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>"
Lcm_mono: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "\<And>x. x \<in> A \<Longrightarrow> dvd.dvd times (f x) (g x)" shows "dvd.dvd times (Lcm (f ` A)) (Lcm (g ` A))"
partition2_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and as :: "'c list"   and bs :: "'d list" shows "partition2 P as bs \<equiv> (filter2 P as bs, filter2 (\<lambda>a b. \<not> P a b) as bs)"
pw_bind_inres: fixes M :: "'d nres"   and f :: "'d \<Rightarrow> 'c nres" shows "inres (M \<bind> f) = (\<lambda>x. nofail M \<longrightarrow> (\<exists>y. inres M y \<and> inres (f y) x))"
1: shows "\<forall>a\<in>allAllocations ({seller} \<union> N) (set \<Omega>). finite a \<and> (\<forall>X. X \<in> Range a \<longrightarrow> b (seller, X) = (0::'b))"
upd_rows_upd_row_swap: fixes i :: "'c"   and I :: "'c set"   and M :: "('d, 'c) sq_matrix"   and x :: "('d, 'c) vec"   and f :: "'c \<Rightarrow> ('d, 'c) vec" assumes "i \<notin> I" shows "upd_rows (upd_row M i x) I f = upd_row (upd_rows M I f) i x"
correctness_rsa: fixes n :: "nat"   and m1 :: "bool \<times> bool"   and m2 :: "bool" shows "etp.OT_12.correctness n m1 m2"
isInitialState_def: fixes state :: "State"   and F0 :: "Literal list list" shows "isInitialState state F0 \<equiv> getM state = []"
t_split: fixes ly :: "nat list"   and aprog :: "abc_inst list"   and as :: "nat"   and ins :: "abc_inst" assumes "ly = layout_of aprog"   and "abc_fetch as aprog = Some ins" shows "\<exists>tp1 tp2. concat (tms_of aprog) = tp1 @ ci ly (start_of ly as) ins @ tp2 \<and> tp1 = concat (take as (tms_of aprog)) \<and> tp2 = concat (drop (Suc as) (tms_of aprog))"
prm_def: shows "(\<cdot>) \<equiv> map_fun id (map_fun rep_trm abs_trm) (\<bullet>)"
bounded_closed_interval: fixes a :: "'b"   and b :: "'b" shows "bounded {a..b}"
vfsequence_at_not_last: fixes xs :: "V"   and i :: "V"   and x :: "V" assumes "vfsequence xs"   and "i \<in>\<^sub>\<circ> vcard xs" shows "(xs #\<^sub>\<circ> x)\<lparr>i\<rparr> = xs\<lparr>i\<rparr>"
Memop: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "unit"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "unit"   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 "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow> (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')" shows "direct_computation.concurrent_step (ts, m, \<S>) (ts[i := (p, is', \<theta>', sb', \<D>', \<O>', \<R>')], m', \<S>')"
sup_aci_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
assms_1: fixes x :: "nat \<Rightarrow> 'a" shows "v x \<in> carrier_vec n"
Polys_closed_uminus: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set" assumes "p \<in> P[X]" shows "- p \<in> P[X]"
get_WIM_bit_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "get_WIM_bit (nat ((uint (fst (fst (get_curr_win () s1))) - 1) mod NWINDOWS)) (cpu_reg_val WIM (snd (fst (get_curr_win () s1)))) = get_WIM_bit (nat ((uint (fst (fst (get_curr_win () s2))) - 1) mod NWINDOWS)) (cpu_reg_val WIM (snd (fst (get_curr_win () s2))))"
legs_form_cospan_1: shows "C.cospan \<mu>.leg0 \<nu>.leg1"
gauss_sum_int_conv_gauss_sum: fixes k :: "int" shows "gauss_sum_int k = gauss_sum (nat (k mod int n))"
part_reflexivity: fixes x :: "i" assumes "MM3" shows "P x x"
star_of_approx_iff: fixes x :: "'d"   and y :: "'d" shows "(star_of x \<approx> star_of y) = (x = y)"
mkarr_memb_ARR: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f"   and "t \<in> f" shows "free_monoidal_category.mkarr C t = f"
differentiable_eqI: fixes g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "g differentiable at x"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x"   and "x \<in> X"   and "open X" shows "f differentiable at x"
mod_mult_mult1: 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 c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times c a) (times c b) = times c (modulo a b)"
Inr_Inl_False: fixes x :: "'n"   and y :: "'m" shows "(Inr x = Inl y) = False"
simptm_allpolys_npoly: fixes p :: "tm" shows "allpolys isnpoly (simptm p)"
cases: fixes x :: "trilean" obtains "x = true"   | "x = false"   | "x = invalid"
wgraph_specs_5: fixes g :: "'g"   and u :: "'v"   and v :: "'v"   and d :: "nat" assumes "invar g"   and "(u, v) \<notin> edges (\<alpha>g g)"   and "u \<noteq> v" shows "invar (add_edge (u, v) d g)"
ab_2: shows "0 \<le> b \<and> b \<le> 2 \<and> cos b = 0"
prv_neg_exi_eqv_all_neg: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "prv (eqv (neg (exi x \<phi>)) (all x (neg \<phi>)))"
U_ACTIVE: shows "u \<in> {v \<in> V. (0::'a) < excess f v}"
zip_inj: fixes a :: "'a list"   and b :: "'b list"   and a' :: "'a list"   and b' :: "'b list" assumes "length a = length b"   and "length a' = length b'"   and "zip a b = zip a' b'" shows "a = a' \<and> b = b'"
P: fixes n :: "'b" assumes "n \<in> I" shows "{x \<in> space M. P n x} \<in> events"
sum_upto_dirichlet_prod: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and x :: "real" shows "sum_upto (dirichlet_prod f g) x = sum_upto (\<lambda>d. f d * sum_upto g (x / real d)) x"
tuple_ABC_eq_ABC'_1: shows "tuple_BC = tuple_BC'"
split_sel_asm: fixes P :: "'e \<Rightarrow> bool"   and f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e"   and step :: "('g, 'f) step" shows "P (case step of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = (\<not> (step = Done \<and> \<not> P f1 \<or> step = Skip (un_Skip step) \<and> \<not> P (f2 (un_Skip step)) \<or> step = Yield (un_Yield1 step) (un_Yield2 step) \<and> \<not> P (f3 (un_Yield1 step) (un_Yield2 step))))"
elims_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list"   and y :: "bool" assumes "class.linorder less_eq less"   and "linorder.sorted_descending less_eq x = y" obtains "x = []" and "y"   | a :: "'a" and as :: "'a list" where "x = a # as" and "y = ((\<forall>x\<in>set as. less_eq x a) \<and> linorder.sorted_descending less_eq as)"
preserves_reflects_arr: fixes f :: "'b \<times> 'b" shows "D.arr (GoH.cmp f) = B.VV.arr f"
Stop_is_action: shows "R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> wait A') \<in> {p. is_CSP_process p}"
less_atm_stable: fixes A :: "'a"   and B :: "'a"   and \<sigma> :: "'s" assumes "less_atm A B" shows "less_atm (A \<cdot>a \<sigma>) (B \<cdot>a \<sigma>)"
VA1: shows "Vagree \<nu> \<nu>' (FVT \<theta>\<^sub>1)"
ltake_LCons: fixes i :: "nat"   and x :: "'d"   and t :: "'d llist" assumes "0 < i" shows "Co_Snapshot.ltake i (LCons x t) = x # Co_Snapshot.ltake (i - 1) t"
in_restricted_global_in_unrestricted_global_2: fixes l :: "'d"   and s :: "'e \<Rightarrow> (('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr) option"   and r :: "'e" assumes "l \<in> LID\<^sub>G (s(r := None))" shows "l \<in> LID\<^sub>G s"
bounded_minus_comp: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "bounded (f ` S)"   and "bounded (g ` S)" shows "bounded ((\<lambda>x. f x - g x) ` S)"
sep_conj_empty: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_empty zero) = P"
renaming_preserves_strictly_maximal_literal: fixes C :: "'at Literal set"   and L :: "'at Literal"   and A :: "'at set" assumes "strictly_maximal_literal C L" shows "strictly_maximal_literal (rename_clause A C) (rename_literal A L)"
assms: shows "inf_last_ti dt (Suc (Suc t)) \<noteq> []"
s0_init_def: shows "s0_init \<equiv> s0_secrecy"
cyclic_group: fixes \<eta> :: "nat" shows "cyclic_group (\<G> \<eta>)"
lr_def: fixes F'' :: "'a filter"   and g'' :: "'a \<Rightarrow> real" shows "lr F'' g'' = {f. \<forall>c>0. \<forall>\<^sub>F x in F''. R (norm (f x)) (c * norm (g'' x))}"
type_definition_Quotient_not_empty: fixes P :: "'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Quotient (eq_onp P) Abs Rep T" shows "\<exists>x. P x"
fmap_of_list_simps_2: fixes k :: "'c"   and v :: "'d"   and kvs :: "('c \<times> 'd) list" shows "fmap_of_list ((k, v) # kvs) = fmupd k v (fmap_of_list kvs)"
conv_many_strongly_connected: fixes x :: "'a" shows "many_strongly_connected x = many_strongly_connected (x\<^sup>T)"
reflects_iso: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and f :: "'b"   and a' :: "'b"   and a :: "'b" assumes "fully_faithful_functor A B F"   and "partial_magma.in_hom A f a' a"   and "category.iso B (F f)" shows "category.iso A f"
Mapping_inject: fixes x :: "'a \<Rightarrow> 'b option"   and y :: "'a \<Rightarrow> 'b option" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(mapping.Mapping x = mapping.Mapping y) = (x = y)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "P 0 0"   and "\<And>k. P 0 (Suc k)"   and "\<And>n. P (Suc n) 0"   and "\<And>n k. \<lbrakk>P n (Suc k); P n k\<rbrakk> \<Longrightarrow> P (Suc n) (Suc k)" shows "P a0 a1"
unitary_elim: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat"   and P :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "P \<in> carrier_mat n n"   and "unitary P"   and "P * A * adjoint P = P * B * adjoint P" shows "A = B"
safe_combined_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('d, 'c, 'a) Combined.configuration" assumes "dataflow_topology summary results_in" shows "dataflow_topology.safe_combined summary results_in c \<equiv> \<forall>loc1 loc2 t s p. 0 < zcount (cri.records (exchange_config c)) (loc1, t) \<and> s \<in>\<^sub>A graph.path_weight summary loc1 loc2 \<and> init c p \<longrightarrow> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp (prop_config c p) loc2) \<and> t' \<le> results_in t s)"
mat_inv_mult_sm: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" assumes "k \<noteq> 0" shows "mat_inv (k *\<^sub>s\<^sub>m A) = (1 / k) *\<^sub>s\<^sub>m mat_inv A"
parts_crypt: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" shows "parts (insert (Crypt K X) H) = insert (Crypt K X) (parts (insert X H))"
d_fro: shows "a - complex_of_real d \<in> frontier (inside (path_image p))"
linear_independent_injective_image: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b set" assumes "linear (*s) (*s) f"   and "independent s"   and "inj_on f (span s)" shows "independent (f ` s)"
residue_factor_unique: fixes k :: "nat"   and x :: "nat \<Rightarrow> int"   and u :: "int"   and m :: "nat" assumes "0 < k"   and "x \<in> carrier Zp"   and "u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k)" shows "u = (THE u. u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k))"
sum_list_Basis_list: fixes f :: "'c \<Rightarrow> 'b" shows "sum_list (map f Basis_list) = sum f Basis"
lossless_S1: fixes msgs :: "bool \<times> bool" shows "lossless_spmf (S1 msgs ())"
encode_exp_0: fixes e :: "nat"   and n :: "nat" shows "local.encode e n = local.encode 0 (base ^ e * n)"
alg_mult_def: fixes p :: "'a list list"   and x :: "'a list" shows "UP.alg_mult p x = (if p = [] then 0 else if x \<in> carrier (poly_ring R) then GREATEST n. [\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> x] [^]\<^bsub>poly_ring (poly_ring R)\<^esub> n pdivides\<^bsub>poly_ring R\<^esub> p else 0)"