hcju/isabelleps huggingface dataset:
that: fixes p :: "('a \<times> 'a) list" assumes "sG.awalk u p v" shows "thesis"
z_2: shows "\<Gamma> (Var z) = \<Gamma> s"
inverse_mat_base_case: shows "inverts_mat (mat_of_rows_list 2 [[1 / 2, 1 / 2], [1 / 2, - (1 / 2)]]) (mat_of_rows_list 2 [[1, 1], [1, - 1]])"
inf_right_unit: fixes x :: "'a" shows "- x \<sqinter> bot = bot"
irreducible\<^sub>d_factor: fixes p :: "'b poly" assumes "0 < degree p" shows "\<exists>q r. irreducible\<^sub>d q \<and> p = q * r \<and> degree r < degree p"
cases: fixes x :: "'c + 'a \<times> 'b" obtains r :: "'a" and h :: "'b" where "x = Inr (r, h)"   | uu_ :: "'c" where "x = Inl uu_"
bres_anti: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "y \<rightarrow> z \<le> x \<rightarrow> z"
CUT_SNa_aux: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and x :: "name"   and N :: "trm" assumes "<a>:M \<in> \<parallel><B>\<parallel>"   and "SNa M"   and "x:N \<in> \<parallel>(B)\<parallel>"   and "SNa N" shows "SNa (Cut <a>.M x.N)"
C: shows "C \<subseteq> {xy. ennreal (1 / real (Suc n)) \<le> f xy}"
GrpI: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a"   and A :: "'b set" assumes "f x = y"   and "x \<in> A" shows "Grp A f x y"
bind_assoc_pmf: fixes A :: "'f pmf"   and B :: "'f \<Rightarrow> 'e pmf"   and C :: "'e \<Rightarrow> 'd pmf" shows "A \<bind> B \<bind> C = A \<bind> (\<lambda>x. B x \<bind> C)"
ag_inc_zero: fixes A :: "('e, 'f) aGroup_scheme" assumes "aGroup A" shows "\<zero>\<^bsub>A\<^esub> \<in> carrier A"
inject_17: fixes x17 :: "'a exp"   and y17 :: "'a exp" shows "(throw x17 = throw y17) = (x17 = y17)"
snd_sym_preproc_pmdl: fixes gs :: "('a \<Rightarrow>\<^sub>0 'd) list"   and fs :: "('a \<Rightarrow>\<^sub>0 'd) list" shows "local.punit.pmdl (set gs \<union> set (snd (punit.sym_preproc gs fs))) = local.punit.pmdl (set gs \<union> set fs)"
mult_in_cancel_right: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "subgroup H G"   and "x \<in> carrier G"   and "y \<in> H"   and "Group.group G" shows "(x \<otimes>\<^bsub>G\<^esub> y \<in> H) = (x \<in> H)"
EX'_def: fixes f :: "'a set" shows "EX' f \<equiv> {s. \<exists>f0\<in>f. s \<rightarrow>\<^sub>i f0}"
SUP_le_iff: fixes f :: "V \<Rightarrow> V"   and A :: "V set"   and u :: "V" assumes "small (f ` A)" shows "(\<Squnion> (f ` A) \<le> u) = (\<forall>x\<in>A. f x \<le> u)"
Ln'_def: shows "Ln' \<equiv> \<lambda>z. Ln (- \<i> * z) + \<i> * complex_of_real pi / 2"
disc_10: fixes x2 :: "nat" shows "\<not> is_REL (WAIT x2)"
in_arcs_imp_in_arcs_ends: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs (subdivide G e w)" shows "(fst e, snd e) \<in> parcs (subdivide G e w)"
before_in_def: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "x < y in xs = (index xs x < index xs y \<and> y \<in> set xs)"
a2: shows "\<Gamma> \<turnstile> e\<^sub>2 : T'"
a1: shows "x ; 1 \<cdot> x\<^sup>T ; 1 \<cdot> y\<^sup>T ; 1 = 0"
dom_cod: fixes f :: "'a" assumes "L'.C.arr f" shows "L'.C.dom (L'.C.cod f) = L'.C.cod f"
hen_2: assumes "\<And>F. F \<in> set fs \<Longrightarrow> F \<in> set fs" shows "mset (map Mp gs) = mset fs"
assms: shows "\<bar>y\<bar> < \<bar>x\<bar>"
sgn_mult_self_eq: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sgn :: "'k \<Rightarrow> 'k"   and abs :: "'k \<Rightarrow> 'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and a :: "'k" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "times (sgn a) (sgn a) = zero_neq_one.of_bool one zero (a \<noteq> zero)"
Merge_PostConj_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and G :: "'f set"   and P' :: "'e set"   and R :: "'e set"   and X :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/G\<^esub> P' c R,X"   and "F \<subseteq> G"   and "P \<subseteq> P'" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c (Q \<inter> R),(A \<inter> X)"
multiplicity_characterization_int: fixes S :: "int set"   and f :: "int \<Rightarrow> nat"   and p :: "int"   and n :: "int" assumes "S = {p. 0 < f p}"   and "finite S"   and "\<forall>p\<in>S. prime p"   and "prime p"   and "n = (\<Prod>p\<in>S. p ^ f p)" shows "multiplicity p n = f p"
bind_nat: fixes p :: "nat spmf"   and S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a spmf"   and g :: "nat \<Rightarrow> 'b spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> S (f x) (g x)" shows "trace_eq_simcl S (p \<bind> f) (p \<bind> g)"
is_acc_def: fixes r :: "nat \<Rightarrow> 'Q" shows "is_acc r \<equiv> \<forall>n<num_acc. \<exists>\<^sub>\<infinity>i. n \<in> acc (r i)"
list_all2_NilR: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list" shows "list_all2 R xs [] = (xs = [])"
init_reds_sync_unit_throw': fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e' :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"   and "init_exp_of e\<^sub>0 = \<lfloor>unit\<rfloor>" shows "P \<turnstile> \<langle>init_switch e\<^sub>0 e',s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"
run_one_step_basic_const_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x23 :: "v"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $C x23) = (s', vs', res)" shows "\<exists>e. res = RSCrash e"
cross_ratio_unique: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2"   and t :: "proj2" assumes "cross_ratio_correct p q r s"   and "cross_ratio_correct p q r t"   and "cross_ratio p q r s = cross_ratio p q r t" shows "s = t"
that_1: shows "(f \<longlongrightarrow> l) (at x within Y)"
Prf_elims_4: fixes v :: "'b val" assumes "\<turnstile> v : One" obtains "v = Void"
alist_product_code: fixes xs :: "('a \<times> 'b) list"   and ys :: "('c \<times> 'd) list" shows "local.alist_product xs ys = rev (fold (\<lambda>(a, b). fold (\<lambda>(c, d). (#) ((a, c), f a b c d)) ys) xs [])"
\<alpha>_limit_set_in_compact_existence: fixes x :: "'a"   and K :: "'a set"   and y :: "'a" assumes "x \<in> X"   and "rev.trapped_backward x K"   and "compact K"   and "K \<subseteq> X"   and "y \<in> rev.\<alpha>_limit_set x" shows "rev.existence_ivl0 y = UNIV"
a: shows "abs.neval \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A n \<langle>c\<^sub>1\<^sub>A', mds\<^sub>A_of mds\<^sub>C', mem\<^sub>A_of mem\<^sub>1\<^sub>C'\<rangle>\<^sub>A \<and> (\<langle>c\<^sub>1\<^sub>A', mds\<^sub>A_of mds\<^sub>C', mem\<^sub>A_of mem\<^sub>1\<^sub>C'\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C) \<in> \<R> \<and> (\<forall>c\<^sub>2\<^sub>A' mem\<^sub>2\<^sub>A'. (\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> P \<and> abs.neval \<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A n \<langle>c\<^sub>2\<^sub>A', mds\<^sub>A_of mds\<^sub>C', mem\<^sub>2\<^sub>A'\<rangle>\<^sub>A \<longrightarrow> (\<exists>c\<^sub>2\<^sub>C' mem\<^sub>2\<^sub>C'. \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>2\<^sub>C', mds\<^sub>C', mem\<^sub>2\<^sub>C'\<rangle>\<^sub>C \<and> (\<langle>c\<^sub>2\<^sub>A', mds\<^sub>A_of mds\<^sub>C', mem\<^sub>2\<^sub>A'\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>C', mds\<^sub>C', mem\<^sub>2\<^sub>C'\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C', mds\<^sub>C', mem\<^sub>2\<^sub>C'\<rangle>\<^sub>C) \<in> P))"
heap_minimum: fixes l :: "int"   and r :: "int"   and heap :: "int \<Rightarrow> 'a" assumes "l < r"   and "is_heap (\<le>) heap l r" shows "heap l = Min_mset (arr_mset heap l r)"
Sup_inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and B :: "'b set"   and a :: "'b" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf (Sup B) a = Sup ((\<lambda>b. inf b a) ` B)"
JN_cong: fixes I :: "'c set"   and J :: "'c set"   and F :: "'c \<Rightarrow> 'd program"   and G :: "'c \<Rightarrow> 'd program" assumes "I = J"   and "\<And>i. i \<in> J \<Longrightarrow> F i = G i" shows "JOIN I F = JOIN J G"
wf_dickson_less_v_aux1: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and seq :: "nat \<Rightarrow> 't" assumes "dickson_grading d"   and "\<And>i. dickson_less_v d m (seq (Suc i)) (seq i)" obtains i :: "nat" where "\<And>j. i < j \<Longrightarrow> component_of_term (seq j) < component_of_term (seq i)"
iter_dom_invariant: fixes ss0 :: "nat list list"   and w0 :: "nat list"   and ss' :: "nat list list"   and w' :: "nat list" assumes "wf_dom ss0 w0" shows "iter f step ss0 w0 = (ss', w') \<longrightarrow> wf_dom ss' w'"
simps_1: fixes x :: "'b" shows "preprm_apply [] x = x"
le_Star: fixes s :: "'a language" shows "s \<le> Star s"
return_def: fixes x :: "'c" shows "return x = Heap_Monad.heap (Pair x)"
matches_simps_2: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c"   and m' :: "'b match_expr" assumes "matches \<gamma> m a p" shows "matches \<gamma> (MatchAnd m' m) a p = matches \<gamma> m' a p"
assms_2: fixes p :: "'s1 spmf"   and q :: "'s2 spmf"   and a :: "'a" assumes "S p q"   and "a \<in> A" shows "p \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q \<bind> (\<lambda>s. map_spmf fst (callee2 s a))"
residual_network_def: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "countable_network \<Delta>" shows "countable_network.residual_network \<Delta> f = \<lparr>edge = \<lambda>x y. edge \<Delta> x y \<or> edge \<Delta> y x \<and> y \<noteq> source \<Delta>, capacity = \<lambda>(x, y). if edge \<Delta> x y then capacity \<Delta> (x, y) - f (x, y) else if y = source \<Delta> then 0 else f (y, x), source = source \<Delta>, sink = sink \<Delta>, \<dots> = network.more \<Delta>\<rparr>"
fold_set_fold_remdups: fixes y :: "'b fset set"   and xs :: "'b fset set list" shows "Finite_Set.fold (\<otimes>) y (set xs) = fold (\<otimes>) (remdups xs) y"
div_0: fixes a :: "'a" shows "(0::'a) / a = (0::'a)"
vector_derivative_within: fixes x :: "real"   and S :: "real set"   and f :: "real \<Rightarrow> 'a"   and y :: "'a" assumes "at x within S \<noteq> bot"   and "(f has_vector_derivative y) (at x within S)" shows "vector_derivative f (at x within S) = y"
gen_closed: shows "a \<in> carrier (G\<lparr>carrier := generate G {a}\<rparr>)"
of_nat_le_0_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) zero = (m = 0)"
that: fixes ivl1 :: "float interval"   and ivl2 :: "float interval" assumes "a = Some ivl1"   and "b = Some ivl2"   and "f ivl1 ivl2 = Some ivl" shows "thesis"
int_and_1: fixes x :: "int" shows "x AND 1 = x mod 2"
is_OK_returns_heap_I: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and h' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'" shows "h \<turnstile> ok f"
return_time_isCont_outside: fixes S :: "'a set"   and s :: "'a \<Rightarrow> real"   and x :: "'a"   and Ds :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" assumes "returns_to {x \<in> S. s x = 0} x"   and "closed S"   and "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"   and "\<And>x. isCont Ds x"   and "blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)) \<noteq> 0"   and "\<forall>\<^sub>F x in at (poincare_map {x \<in> S. s x = 0} x). s x = 0 \<longrightarrow> x \<in> S"   and "x \<notin> S \<or> s x \<noteq> 0" shows "isCont (return_time {x \<in> S. s x = 0}) x"
superPicker_eval: fixes amount :: "nat"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "amount \<le> var + 1" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (superPicker amount var L F) (xs @ \<Gamma>))"
sliden_def: fixes i :: "nat"   and j :: "nat" shows "sliden i j \<equiv> j - i"
comp_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "VoV.comp g f = (if VoV.arr f \<and> VoV.arr g \<and> VxV.seq g f then VxV.comp g f else VxV.null)"
\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c_approximating_bigstep_iff_iptables_bigstep: fixes rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<alpha> :: "action \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and s :: "state"   and t :: "state"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "\<forall>r\<in>set rs. \<forall>c. get_action r \<noteq> Call c" shows "(\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c \<gamma>, \<alpha>),p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
less_example_peers_def: shows "(<) \<equiv> rec_example_peers (case_example_peers False True True) (case_example_peers False False True) (case_example_peers False False False)"
pcp_lim_in: fixes C :: "'b formula set set"   and S :: "'b formula set" assumes "pcp (C::'b formula set set)"   and "(S::'b formula set) \<in> (C::'b formula set set)"   and "subset_closed (C::'b formula set set)"   and "finite_character (C::'b formula set set)" shows "pcp_lim (C::'b formula set set) (S::'b formula set) \<in> C"
simps_2: fixes r :: "real" shows "type (Rv r) = Rty"
remainder_def: shows "remainder = (\<lambda>s. \<Sum>n. fds_nth h (n + Suc N) / nat_power (n + Suc N) s)"
H1_H2_commute: shows "H1 \<circ> H2 = H2 \<circ> H1"
sdg_new_parcs: shows "{(u, w), (v, w), (w, u), (w, v)} \<subseteq> parcs (subdivide G e w)"
obj_simps_2: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.src a = a"
downset_faithful: fixes x :: "'a"   and y :: "'a" assumes "\<down> x \<subseteq> \<down> y" shows "x \<le> y"
tuples_def: shows "tuples = (\<lambda>X. {f \<in> perms \<rightarrow>\<^sub>E Roots. image_mset f (mset_set perms) = X})"
enn2ereal_range: shows "e2ennreal ` {0..} = UNIV"
cut_cut_ge: fixes i :: "'a set"   and a :: "'a"   and b :: "'a" shows "i \<down>\<ge> a \<down>\<ge> b = i \<down>\<ge> max a b"
add_numeral_left: fixes v :: "num"   and w :: "num"   and z :: "'a" shows "numeral v + (numeral w + z) = numeral (v + w) + z"
ext_inject: fixes precondition_of :: "('c \<times> 'd) list"   and effect_of :: "('c \<times> 'd) list"   and more :: "'e"   and precondition_of' :: "('c \<times> 'd) list"   and effect_of' :: "('c \<times> 'd) list"   and more' :: "'e" shows "(\<lparr>sas_plus_operator.precondition_of = precondition_of, effect_of = effect_of, \<dots> = more\<rparr> = \<lparr>sas_plus_operator.precondition_of = precondition_of', effect_of = effect_of', \<dots> = more'\<rparr>) = (precondition_of = precondition_of' \<and> effect_of = effect_of' \<and> more = more')"
has_locksE: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and n :: "nat" assumes "has_locks l t = n" obtains "l = None" and "n = 0"   | n' :: "nat" where "l = \<lfloor>(t, n'__)\<rfloor>" and "Suc n'__ = n"   | t' :: "'b" and n' :: "nat" where "l = \<lfloor>(t', n')\<rfloor>" and "t' \<noteq> t" and "n = 0"
even_succ_div_two: 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 division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "divide (plus a one) (numeral.numeral one plus (num.Bit0 num.One)) = divide a (numeral.numeral one plus (num.Bit0 num.One))"
distinct_157: fixes x81 :: "expr"   and x82 :: "char list"   and x83 :: "char list list"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "x81\<bullet>x82{x83} \<noteq> Call x101 x102 x103 x104"
simps_2: fixes \<alpha> :: "game option" shows "Choiceo \<alpha> undefg = undefg"
finite_Diff_insert: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" shows "finite (A - insert a B) = finite (A - B)"
assms_1: shows "a \<subseteq>\<^sub>\<circ> A"
of_nat_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m ^ n) = power.power one times (semiring_1.of_nat one plus zero m) n"
surjective: fixes r :: "('d, 'e, 'f) Category_scheme" shows "r = \<lparr>Obj = obj\<^bsub>r\<^esub>, Mor = mor\<^bsub>r\<^esub>, Dom = Dom r, Cod = Cod r, Id = Category.Id r, Comp = (;;\<^bsub>r\<^esub>), \<dots> = Category.more r\<rparr>"
measure_pmf_eq_density: fixes p :: "'e pmf" shows "measure_pmf p = density (count_space UNIV) (\<lambda>x. ennreal (pmf p x))"
inv\<^sub>2E_2: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "inv\<^sub>2 T A j" shows "j \<le> n"
expr_list_inj_term: fixes x :: "expr list"   and y :: "expr list" shows "\<langle>x\<rangle> = \<langle>y\<rangle> \<equiv> x = y"
insert_remove: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "mset_set (insert x A) = add_mset x (mset_set (A - {x}))"
set_child_nodes_locs_def: shows "Shadow_DOM.set_child_nodes_locs \<equiv> l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes_locs"
lunit_in_vhom: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and b :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "trg f = b" shows "partial_magma.in_hom V (bicategory.lunit V H \<a> \<i> src trg f) (H b f) f"
inext_in_imp_in: fixes n :: "nat"   and I :: "nat set" assumes "inext n I \<in> I" shows "n \<in> I"
atLeastLessThan_eq_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a < b"   and "c < d" shows "({a..<b} = {c..<d}) = (a = c \<and> b = d)"
top_assn_def: shows "true \<equiv> Abs_assn in_range"
module_plus_2: shows "proj_poly_syz (length bs) p = sum_list (map2 (\<odot>) (cofactor_list_syz (length bs) p) bs)"
not_final_thread_final_thread_conv: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "thr s t \<noteq> None" shows "(\<not> final_thread s t) = not_final_thread s t"
mapping_of_inverse: fixes x :: "'q mpoly" shows "MPoly (mapping_of x) = x"
imply_append: fixes ps :: "'b fm list"   and ps' :: "'b fm list"   and q :: "'b fm" shows "imply (ps @ ps') q = imply ps (imply ps' q)"
add_mset_subseteq_single_iff: fixes a :: "'f"   and M :: "'f multiset"   and b :: "'f" shows "(add_mset a M \<subseteq># {#b#}) = (M = {#} \<and> a = b)"
add_mset_add_single: fixes a :: "'k"   and A :: "'k multiset" shows "add_mset a A = A + {#a#}"
boundedI: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> a \<le> x" shows "Max A \<le> x"
1: shows "f * g = fls_shift (- (fls_subdegree f + fls_subdegree g)) (fps_to_fls (fps_shift (nat (fls_subdegree f) + nat (fls_subdegree g)) (fls_regpart f * fls_regpart g)))"
strict_mono_sets_imp_disjoint: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "strict_mono_sets A f" shows "pairwise (\<lambda>x y. disjnt (f x) (f y)) A"
nonzero: shows "\<forall>x. insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 2) \<noteq> 0 \<or> insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 1) \<noteq> 0 \<or> insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 0) \<noteq> 0"
mem_read_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and asi :: "word8"   and address :: "word32" assumes "low_equal s1 s2 \<and> asi \<in> {8, 10}" shows "fst (memory_read asi address s1) = fst (memory_read asi address s2)"
left_idem: fixes a :: "'c"   and b :: "'c" shows "a + (a + b) = a + b"
comp_in_homE: fixes g :: "'d"   and f :: "'d"   and a :: "'d"   and c :: "'d" assumes "\<guillemotleft>g \<cdot>\<^sub>D f : a \<Rightarrow>\<^sub>D c\<guillemotright>" obtains b :: "'d" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" and "\<guillemotleft>g : b \<Rightarrow>\<^sub>D c\<guillemotright>"
\<D>_4: fixes K :: "'a set" assumes "K \<in> \<D>" shows "\<exists>a b. K = cbox a b"
monom_mult_assoc: fixes c :: "'d"   and s :: "'a"   and d :: "'d"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'd" shows "monom_mult c s (monom_mult d t p) = monom_mult (c * d) (s + t) p"
finite_maxsimp: fixes x :: "'a set" assumes "maxsimp x" shows "finite x"
f'_def: shows "f' \<equiv> smult (((1::'a) / lead_coeff g) ^ (Suc (degree f) - degree g)) f"
intros_2: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "(\<Gamma>::'b formula set) \<turnstile>\<^sub>H (F::'b formula)"   and "(\<Gamma>::'b formula set) \<turnstile>\<^sub>H (F::'b formula) \<^bold>\<rightarrow> (G::'b formula)" shows "(\<Gamma>::'b formula set) \<turnstile>\<^sub>H (G::'b formula)"
whole_ideal: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "ideal R (carrier R)"
basis_finite: fixes B :: "(real, 'a) vec set" assumes "is_basis B" shows "finite B"
outgoing_def: fixes v :: "nat" shows "lo.cf.outgoing v \<equiv> {(v, u) |u. (v, u) \<in> lo.cf.E}"
ide_char: fixes a :: "'a1" shows "C1.ide a = (C1.arr a \<and> C1.dom a = a \<and> C1.cod a = a)"
the_\<theta>_simps_2: shows "src \<chi>.the_\<theta> = src s\<^sub>0"
cblinfun_apply_1_left: fixes y :: "'a" shows "1 *\<^sub>V y = y"
abs_eq: fixes x :: "nat \<Rightarrow> rat" assumes "realrel x x" shows "inverse (Real x) = Real (if vanishes x then \<lambda>n. 0 else (\<lambda>n. inverse (x n)))"