hcju/isabelleps huggingface dataset:
can_mkarr_Can: fixes t :: "'c local.term" assumes "Can t" shows "can (mkarr t)"
c3: shows "graph_union (S j) (chain_sup S) = chain_sup S"
edge_begin_tup: fixes x :: "('a \<times> 'b) \<times> 'a \<times> 'c" shows "Pre_Port_Graph.edge_begin x = fst (fst x)"
arsinh_0: shows "arsinh (0::'b) = (0::'b)"
lens_plus_sub_assoc_2: fixes X :: "'d \<Longrightarrow> 'g"   and Y :: "'e \<Longrightarrow> 'g"   and Z :: "'f \<Longrightarrow> 'g" shows "(X +\<^sub>L Y) +\<^sub>L Z \<subseteq>\<^sub>L X +\<^sub>L Y +\<^sub>L Z"
simps_1: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and u :: "nat"   and v :: "nat" shows "len m u v [] = m u v"
set_lebesgue_integral_zero: fixes M :: "'e measure"   and A :: "'e set" shows "\<integral>x\<in>A. (0::'d)\<partial>M = (0::'d)"
td_b: shows "type_definition Rep_b Abs_b (Collect (Domainp B))"
disj1: shows "(set gs \<union> set bs) \<inter> set hs = {}"
UNION_disjoint: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "prod g (\<Union> (A ` I)) = (\<Prod>x\<in>I. prod g (A x))"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_on_left_moving_in_middle_B (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 rn. lm = lm1 @ lm2 \<and> (if lm1 = [] then l = Bk # ires else l = <rev lm1> @ Bk # Bk # ires) \<and> r = Bk # <lm2> @ Bk \<up> rn)"
adverse_at_01_2: fixes z :: "nat"   and i :: "nat"   and j :: "nat" shows "local.adverse z i j 1 \<down>= j"
recv_marker_means_snapshotted_1: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and ev :: "('a, 'b, 'c) event"   and i :: "nat"   and p :: "'a"   and q :: "'a"   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 c' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "ev = RecvMarker i p q"   and "distributed_system.next channel trans send recv c ev c'" shows "ps c' p \<noteq> None"
simps: fixes a1 :: "string_unop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'b OCL_Types.type" shows "string_unop_type a1 a2 a3 = (a1 = SizeOp \<and> a2 = String[1] \<and> a3 = Integer[1] \<or> a1 = CharactersOp \<and> a2 = String[1] \<and> a3 = Sequence String[1] \<or> a1 = ToUpperCaseOp \<and> a2 = String[1] \<and> a3 = String[1] \<or> a1 = ToLowerCaseOp \<and> a2 = String[1] \<and> a3 = String[1] \<or> a1 = ToBooleanOp \<and> a2 = String[1] \<and> a3 = Boolean[1] \<or> a1 = string_unop.ToIntegerOp \<and> a2 = String[1] \<and> a3 = Integer[1] \<or> a1 = ToRealOp \<and> a2 = String[1] \<and> a3 = Real[1])"
eint_plus_times: fixes d :: "eint"   and c :: "eint"   and k :: "int"   and b :: "eint"   and l :: "int" assumes "0 \<le> d"   and "c + eint k * d \<le> b"   and "l \<le> k" shows "c + eint l * d \<le> b"
subs_def: shows "subs \<equiv> \<lambda>x c d. pi_rec \<zero> (\<lambda>a b P. Output (a[c::=d]) (b[c::=d])) (\<lambda>P. Tau) (\<lambda>a x P. Input (a[c::=d]) x) (\<lambda>a b P. Match (a[c::=d]) (b[c::=d])) (\<lambda>a b P. Mismatch (a[c::=d]) (b[c::=d])) (\<lambda>P Q. (\<oplus>)) (\<lambda>P Q. (\<parallel>)) (\<lambda>x P. Res x) (\<lambda>P. Bang) x"
cod_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "cod (trg \<mu>) = trg \<mu>"
hyps: fixes a :: "nat"   and m' :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow> nat" assumes "a \<in> outstanding_refs is_Write\<^sub>s\<^sub>b sb" shows "flush sb m' a = flush sb m a"
cases: fixes a1 :: "nat list list"   and a2 :: "nat list list"   and a3 :: "nat list list"   and a4 :: "nat list list" assumes "merge a1 a2 a3 a4" obtains "a1 = []" and "a2 = []" and "a3 = []" and "a4 = []"   | as :: "nat list list" where "a1 = as__" and "a2 = []" and "a3 = [concat as__]" and "a4 = []" and "as__ \<noteq> []"   | as1 :: "nat list list" and bs1 :: "nat list list" and as2 :: "nat list list" and bs2 :: "nat list list" and as :: "nat list list" and bs :: "nat list list" where "a1 = as1 @ as2" and "a2 = bs1 @ bs2" and "a3 = concat as1 # as" and "a4 = concat bs1 # bs" and "as1 \<noteq> []" and "bs1 \<noteq> []" and "concat as1 < concat bs1" and "concat bs1 < concat as2" and "merge as2 bs2 as bs"
assms_1: shows "coincides_on Mkt Mkt2 (stocks Mkt)"
mult_nonneg_nonpos2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times b a) zero"
diamond_conjungation_aux: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "d p = p"   and "d q = q" shows "(\<langle>x| p \<le> ad q) = (q \<cdot> \<langle>x| p = (zero_class.zero::'a))"
alw_mp: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream"   and \<psi> :: "'b stream \<Rightarrow> bool" assumes "alw \<phi> xs"   and "alw (\<lambda>xs. \<phi> xs \<longrightarrow> \<psi> xs) xs" shows "alw \<psi> xs"
fps_nth_bernoulli_fps: fixes n :: "nat" shows "bernoulli_fps $ n = of_real (bernoulli n) / fact n"
sprod_vec_0: fixes v :: "'a vec"   and f :: "nat \<Rightarrow> 'a" shows "v \<bullet> vec 0 f = (0::'a)"
map_\<I>_swap_sum: fixes \<I>1 :: "('j, 'l) \<I>"   and \<I>2 :: "('i, 'k) \<I>" shows "map_\<I> swap_sum swap_sum (\<I>1 \<oplus>\<^sub>\<I> \<I>2) = \<I>2 \<oplus>\<^sub>\<I> \<I>1"
assert_s_def: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and a :: "'a \<times> 'b atom"   and s :: "('a, 'b) state" assumes "Incremental_State_Ops_Simplex assert_bound init check" shows "Incremental_State_Ops_Simplex.assert_s assert_bound a s = (let s' = assert_bound a s in if \<U> s' then Inl (the (\<U>\<^sub>c s')) else Inr s')"
add_mono_thms_linordered_field_3: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i < j \<and> k \<le> l" shows "i + k < j + l"
DAs_axioms_def: fixes alphabet1 :: "'b list"   and alphabet2 :: "'c list"   and letter_eq :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "DAs_axioms alphabet1 alphabet2 letter_eq \<equiv> \<forall>a b. letter_eq a b \<longrightarrow> (a \<in> set alphabet1) = (b \<in> set alphabet2)"
sas_plus_equivalent_to_strips_i_a_IX: fixes s :: "'c \<Rightarrow> 'd option"   and V :: "'c set"   and ops :: "('c, 'd) sas_plus_operator list" assumes "dom s \<subseteq> V"   and "\<forall>op\<in>set ops. \<forall>(v, a)\<in>set (effect_of op). v \<in> V" shows "dom (execute_parallel_operator_sas_plus s ops) \<subseteq> V"
dim_ty_sem_NoDimension_def: fixes uu :: "\<one> itself" shows "dim_ty_sem uu = 1"
Hoare_consequence: fixes P' :: "cell list \<times> cell list \<Rightarrow> bool"   and P :: "cell list \<times> cell list \<Rightarrow> bool"   and p :: "(action \<times> nat) list"   and Q :: "cell list \<times> cell list \<Rightarrow> bool"   and Q' :: "cell list \<times> cell list \<Rightarrow> bool" assumes "P' \<mapsto> P"   and "{P} p {Q}"   and "Q \<mapsto> Q'" shows "{P'} p {Q'}"
real_of_nat_eq_numeral_power_cancel_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and y :: "nat"   and x :: "num"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero y = power.power one times (numeral.numeral one plus x) n) = (y = numeral x ^ n)"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_in_subst: fixes p :: "'a strand_label \<times> ('b, 'c) Term.term \<times> ('b, 'c) Term.term"   and a :: "'a strand_label \<times> ('b, 'c) stateful_strand_step"   and \<delta> :: "'c \<Rightarrow> ('b, 'c) Term.term" assumes "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)" shows "\<exists>q\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p a. fst p = fst q \<and> snd p = snd q \<cdot>\<^sub>p rm_vars (set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (snd a))) \<delta>"
simps_2: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "QE_dnf opt step (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = or (QE_dnf opt step \<phi>\<^sub>1) (QE_dnf opt step \<phi>\<^sub>2)"
m_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
intro: fixes pair_of_term :: "'c \<Rightarrow> 'b \<times> 'd"   and term_of_pair :: "'b \<times> 'd \<Rightarrow> 'c" assumes "\<And>t u. term_powerprod.splus pair_of_term term_of_pair t u = splus (term_of_pair (t, the_min)) u" shows "gd_nat_term_axioms pair_of_term term_of_pair"
isometry_on_lipschitz: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "isometry_on X f" shows "1-lipschitz_on X f"
gram_schmidt_def: fixes n :: "nat"   and ws :: "'c vec list" shows "gram_schmidt n ws = rev (gram_schmidt_sub n [] ws)"
d: fixes C :: "'a set set" assumes "subset.chain \<B> C" shows "subset.chain {B. \<not> B \<turnstile> \<phi>} C"
list_slice_div_eq_Suc: fixes xs :: "'b list"   and k :: "nat"   and n :: "nat" assumes "length xs div k = Suc n" shows "list_slice xs k = list_slice (xs \<down> (n * k)) k @ [xs \<up> (n * k) \<down> k]"
simps: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and F :: "atom fm"   and z :: "nat" shows "linear_substitution_fm_helper var b c F z = liftmap (\<lambda>x A. fm.Atom (linear_substitution (var + x) (liftPoly 0 x b) (liftPoly 0 x c) A)) F z"
regions_part_ex_1: fixes u :: "'c \<Rightarrow> real" assumes "u \<in> V" shows "u \<in> [u]\<^sub>\<R>"
simps_1: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option" shows "lex_ord_pair f [] [] = Some Eq"
max2_def: fixes a :: "'b \<Rightarrow> 'a"   and b :: "'b \<Rightarrow> 'a" shows "o.max2 a b \<equiv> if (a, b) \<in> oexp then b else a"
similar'_up: fixes n :: "nat"   and d :: "Value"   and e :: "CValue" assumes "d \<triangleleft>\<triangleright>\<^bsub>n\<^esub> e" shows "\<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>d \<triangleleft>\<triangleright>\<^bsub>Suc n\<^esub> \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>e"
ASSOC: fixes X :: "freeExp"   and Y :: "freeExp"   and Z :: "freeExp" shows "PLUS X (PLUS Y Z) \<sim> PLUS (PLUS X Y) Z"
FeqG: shows "insertAssertion F \<Psi> \<simeq>\<^sub>F insertAssertion G \<Psi>"
sink_rev_step_2: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and q' :: "'d"   and i :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "\<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q"   and "q = \<delta> q' (w i)" shows "\<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q'"
unique_n_valuation: fixes P :: "('a \<Rightarrow> ant) set"   and P' :: "('a \<Rightarrow> ant) set" assumes "P \<in> Pds"   and "P' \<in> Pds" shows "(P = P') = (\<nu>\<^bsub>K P\<^esub> = \<nu>\<^bsub>K P'\<^esub>)"
zip_eq_Nil_iff: fixes xs :: "'d list"   and ys :: "'e list" shows "(zip xs ys = []) = (xs = [] \<or> ys = [])"
ReplId: fixes A :: "ZF" shows "Repl A id = A"
src_some_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_src (some_src \<mu>) = some_src \<mu>"
assms: shows "sat_precond_as (fmrestrict_set vs s) as"
BSD_implies_D: assumes "BSD \<V> Tr\<^bsub>(induceES SES)\<^esub>" shows "D \<V> Tr\<^bsub>(induceES SES)\<^esub>"
cbox_def: fixes a :: "'a"   and b :: "'a" shows "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"
Suc: shows "y \<le> y \<cdot> x\<^bsup>Suc n\<^esup> + z \<cdot> x\<^sup>\<star>"
ltake_lmap: fixes n :: "enat"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "ltake n (lmap f xs) = lmap f (ltake n xs)"
tendsto_divide_zero: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" assumes "(f \<longlongrightarrow> (0::'b)) F" shows "((\<lambda>x. f x / c) \<longlongrightarrow> (0::'b)) F"
simps: fixes queue_\<alpha> :: "'a \<Rightarrow> 'b list"   and queue_invar :: "'a \<Rightarrow> bool"   and queue :: "'a"   and n :: "nat"   and T :: "'b set" shows "round_robin_base.round_robin_invar queue_\<alpha> queue_invar (queue, n) T = (queue_invar queue \<and> Round_Robin.round_robin_invar (queue_\<alpha> queue, n) T)"
pmf_eqI: fixes M :: "'b pmf"   and N :: "'b pmf" assumes "\<And>i. pmf M i = pmf N i" shows "M = N"
Wbis_transC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "c \<approx>w d"   and "s \<approx> t"   and "(c, s) \<rightarrow>c (c', s')" obtains d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t'__)" and "s' \<approx> t'__" and "c' \<approx>w d'"   | t' :: "'state" where "(d, t) \<rightarrow>*t t'" and "s' \<approx> t'" and "discr c'"
simps_2: fixes f :: "('b \<Rightarrow> 'a) stream"   and x :: "'b stream" shows "stl (f \<diamondop> x) = stl f \<diamondop> stl x"
dfs_distinct_member: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and xs :: "('d, 'e) bintree option list"   and x :: "('d, 'e) bintree" assumes "distinct (dfs f xs)"   and "Some x \<in> set xs" shows "distinct (bt_dfs f x)"
next_subset_dfs: fixes S :: "'b"   and xs :: "'a list"   and x :: "'a" assumes "invariant S"   and "list_all isNode xs"   and "x \<in> set xs" shows "memb x (dfs S xs)"
continuous_on_closed_Un: fixes s :: "'b set"   and t :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "closed s"   and "closed t"   and "continuous_on s f"   and "continuous_on t f" shows "continuous_on (s \<union> t) f"
valid_def: fixes g :: "vname gexp" shows "valid g \<equiv> \<forall>s. gval g s = true"
simps_2: fixes \<mu> :: "'c" shows "Arr \<^bold>\<langle>\<mu>\<^bold>\<rangle> = C.arr \<mu>"
sum_rw: fixes g :: "(nat \<Rightarrow> nat) \<Rightarrow> 'b" shows "sum g local.F_inj = (\<Sum>y\<in>local.F_strict. sum g {x \<in> local.F_inj. strict_from_inj n x = y})"
distinct_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (set xs))"
real_of_ereal_le_0: fixes x :: "ereal" shows "(real_of_ereal x \<le> 0) = (x \<le> 0 \<or> x = \<infinity>)"
generat: shows "generat \<in> set_spmf (the_gpv gpv)"
0: shows "(a \<ominus> b) k = a k \<ominus>\<^bsub>residue_ring (p ^ k)\<^esub> b k"
nn_integral_eq_simple_integral: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "simple_function M f" shows "integral\<^sup>N M f = integral\<^sup>S M f"
step_ge_x0: fixes t :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "t \<in> set ts"   and "x\<^sub>1 \<le> x" shows "x\<^sub>0 \<le> t x"
x_over_ln_mono: fixes e :: "real"   and x :: "real"   and y :: "real" assumes "0 < e"   and "exp e < x"   and "x \<le> y" shows "x / ln x powr e \<le> y / ln y powr e"
k_def: shows "k = local.Min ((fst ` set xs \<union> fst ` set ys) \<inter> {k. f k (lookup_pair xs k) (lookup_pair ys k) \<noteq> Some Eq})"
slength_bottom_iff: fixes xs :: "[:'a:]" shows "(slength\<cdot>xs = \<bottom>) = (xs = \<bottom>)"
lcp_rulers: fixes r :: "'b list"   and s :: "'b list"   and r' :: "'b list"   and s' :: "'b list" assumes "r \<le>p s"   and "r' \<le>p s'" shows "r \<bowtie> r' \<or> r \<and>\<^sub>p r' = s \<and>\<^sub>p s'"
lepoll_refl: fixes A :: "'a set" shows "A \<lesssim> A"
simps_3: fixes v :: "'a"   and va :: "'a" shows "first_srcNet (DenyAllFromTo v va) = srcNet (DenyAllFromTo v va)"
cooper: fixes p :: "fm"   and bbs :: "bool list"   and bs :: "int list" assumes "qfree p" shows "(\<exists>x. Ifm bbs (x # bs) p) = Ifm bbs bs (cooper p) \<and> qfree (cooper p)"
min_def: fixes a :: "'c"   and b :: "'c" shows "ord_class.min a b = (if a \<le> b then a else b)"
new_tv_Cons: fixes n :: "nat"   and x :: "'b"   and l :: "'b list" shows "new_tv n (x # l) = (new_tv n x \<and> new_tv n l)"
alg_def: fixes ts :: "((nat \<times> nat) \<times> int) list"   and x :: "nat \<Rightarrow> int" shows "alg ts x = fold_mset (\<lambda>((r, c), v) y. y(c := y c + x r * v)) (\<lambda>_. 0) (mset ts)"
negligible_cmult: fixes c :: "real"   and f :: "nat \<Rightarrow> real" shows "negligible (\<lambda>\<eta>. c * f \<eta>) = (negligible f \<or> c = 0)"
trace_hide_union: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a event list" shows "trace_hide t (ev ` (A \<union> B)) = trace_hide (trace_hide t (ev ` A)) (ev ` B)"
singleton_intersection: fixes A :: "'c set"   and B :: "'c set" assumes "card A = 1"   and "card B = 1"   and "A \<noteq> B" shows "A \<inter> B = {}"
power_inject_base: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat"   and b :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "power.power one times a (Suc n) = power.power one times b (Suc n)"   and "less_eq zero a"   and "less_eq zero b" shows "a = b"
prems_1: shows "coll 0 a (sum_list (x # xs))"
norm_sq_mtx_def: fixes A :: "'a sq_mtx" shows "\<parallel>A\<parallel> = \<parallel>to_vec A\<parallel>\<^sub>o\<^sub>p"
set_rel_br: fixes a :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" shows "\<langle>br a I\<rangle>set_rel = br ((`) a) (\<lambda>X. Ball X I)"
initial_abs: shows "\<alpha> ` initialc \<subseteq> wa_initial WAA"
hyps: fixes y :: "'a" assumes "(y, r) \<in> M"   and "I y" shows "\<exists>sr'. (Some y, sr') \<in> option_while' C B"
intros_7: fixes cl :: "cl"   and t1s :: "t list"   and t2s :: "t list"   and f :: "host"   and ves :: "e list"   and vcs :: "v list"   and n :: "nat"   and m :: "nat"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "cl = Func_host (t1s _> t2s) f"   and "ves = $$* vcs"   and "length vcs = n"   and "length t1s = n"   and "length t2s = m" shows "\<lparr>s;vs;ves @ [Callcl cl]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Trap]\<rparr>"
subdegree_div': fixes p :: "'a fps"   and q :: "'a fps" assumes "\<exists>k. p = k * q" shows "subdegree (p / q) = subdegree p - subdegree q"
rq_rev_lq: fixes v :: "'b list"   and u :: "'b list" shows "rev v\<^sup><\<inverse>rev u  = rev (u\<inverse>\<^sup>>v)"
compact_space: fixes X :: "'a topology" shows "compact_space X = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> \<Union> \<U> = topspace X \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> \<Union> \<F> = topspace X))"
that: fixes k :: "nat \<Rightarrow> nat" assumes "strict_mono k"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>l. (\<lambda>n. f (k n) x) \<longlonglongrightarrow> l" shows "thesis"
injI: fixes f :: "'ag \<Rightarrow> 'ah" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
subgroup_def: fixes G :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "subgroup G M composition unit \<equiv> submonoid G M composition unit \<and> Group_Theory.group G composition unit"
Skip': fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "P = (F \<and>* $ 1)" shows "\<turnstile>\<^sub>3\<^sub>' {P} SKIP { F}"
safe_distance_4r_def: shows "safe_distance_4r = (v\<^sub>o + a\<^sub>o * \<delta> - v\<^sub>e)\<^sup>2 / 2 / (a\<^sub>o - a\<^sub>e) - v\<^sub>o * \<delta> - 1 / 2 * a\<^sub>o * \<delta>\<^sup>2 + v\<^sub>e * \<delta>"
subst_atmss_id_subst: fixes AAA :: "'a set set" shows "AAA \<cdot>ass id_subst = AAA"
edom_fun_upd_subset: fixes h :: "'a \<Rightarrow> 'b"   and x :: "'a"   and v :: "'b" shows "edom (h(x := v)) \<subseteq> insert x (edom h)"
tendsto_e2ennrealI: fixes f :: "'a \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>i. e2ennreal (f i)) \<longlongrightarrow> e2ennreal l) F"
eq_bottom_iff: fixes x :: "'a"   and y :: "'a" shows "(eq\<cdot>x\<cdot>y = \<bottom>) = (x = \<bottom> \<or> y = \<bottom>)"
nn_integral_embed_measure': fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'c \<Rightarrow> ennreal" assumes "inj_on f (space M)"   and "g \<in> borel_measurable (embed_measure M f)" shows "integral\<^sup>N (embed_measure M f) g = \<integral>\<^sup>+ x. g (f x) \<partial>M"
dim_eq: shows "dim_vec (row (submatrix A I UNIV) i) = dim_vec (row A (pick I i))"
map_def: fixes f :: "'b" shows "local.map f = (if arr f then f else null)"
prob_space: fixes M :: "'b pmf" shows "measure_pmf.prob M (space (measure_pmf M)) = 1"
split_conv: fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and a :: "'b"   and b :: "'d" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
assms: shows "dep_var_set PROB vs (vs' \<union> vs'')"
weakenSimE: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<leadsto>\<^sub>w<Rel> Q"   and "\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P" shows "\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel"
U_prefix_complete_imp_total_strategy: fixes U :: "(nat \<Rightarrow> nat option) set"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and s :: "nat \<Rightarrow> nat option" assumes "\<And>vs. 0 < length vs \<Longrightarrow> \<exists>f\<in>U. prefix f (length vs - 1) = vs"   and "learn_cons \<psi> U s" shows "\<exists>t. total1 t \<and> learn_cons \<psi> U t"
z_def: shows "z \<equiv> Re (zeta (complex_of_real s))"
resource_def: fixes rest :: "('j, 'event, 'f, 'g, 'h, 'i, 'j) rest_scheme" shows "resource rest = RES (fuse rest) (core_init, rinit rest)"
simIFresh: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and x :: "name"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and C :: "'e" assumes "eqvt Rel"   and "x \<sharp> \<Psi>"   and "x \<sharp> P"   and "x \<sharp> Q"   and "\<And>\<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* subject \<alpha>; distinct (bn \<alpha>); bn \<alpha> \<sharp>* C; x \<sharp> \<alpha>; x \<sharp> Q'\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
aNDSubset: fixes a :: "('a set set, 'b) Combinators list"   and b :: "('a set set, 'b) Combinators list" assumes "singleCombinators a"   and "set a \<subseteq> set b"   and "allNetsDistinct b" shows "allNetsDistinct a"
star_conv_concat: fixes A :: "'a list set" shows "star A = {concat ws |ws. set ws \<subseteq> A}"
Mp_id: fixes f :: "int poly" shows "q.Mp f =m f"
P5: shows "Bet E F F0 \<and> Cong F F0 B C"