hcju/isabelleps huggingface dataset:
in_pairs: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {2 * m..Suc (2 * n)} = (\<Sum>i = m..n. g (2 * i) + g (Suc (2 * i)))"
red_chain_def: fixes G :: "('e, 'f) Group_scheme"   and m :: "nat"   and f :: "nat \<Rightarrow> 'e set" shows "red_chain G m f = (SOME g. g \<in> {h. red_ch_cd G f m h})"
mult_zero_right: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.mult_zero times zero" shows "times a zero = zero"
sel_2: fixes x1 :: "'b"   and x2 :: "'a"   and x3 :: "nat"   and x4 :: "('b, 'a) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.prio (SkewBinomialHeapStruc.Node x1 x2 x3 x4) = x2"
ord_iff_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat :: "'b \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "less x y = (rep_nat x < rep_nat y)"
arc_snd_in_verts: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "snd e \<in> pverts G"
B_def: shows "B \<equiv> {0} \<times> T \<union> {0..1} \<times> S"
prefs_undefined': fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'c" assumes "pref_profile_wf agents alts R"   and "i \<notin> agents" shows "R i = (\<lambda>_ _. False)"
image_g: shows "(-) (degree g) ` {0..<Suc (degree g)} = {0..<Suc (degree g)}"
continuous_map_axioms: shows "continuous_map S is_open S is_open (identity S)"
lubmap_belowI: fixes m :: "'a \<Rightarrow> 'b option"   and e :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and u :: "'c" assumes "\<And>k v. m k = Some v \<Longrightarrow> e k v \<sqsubseteq> u" shows "lub (mapCollect e m) \<sqsubseteq> u"
subst_notIn: fixes \<phi> :: "fmla"   and t :: "trm"   and x :: "name" assumes "\<phi> \<in> UNIV"   and "t \<in> UNIV"   and "x \<in> UNIV"   and "x \<notin> Fvars \<phi>" shows "\<phi>(x::=t) = \<phi>"
ide_a: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a" assumes "yoneda_functor_fixed_object C S \<phi> a" shows "partial_magma.ide C a"
IH1: assumes "wf_net_tree p1" shows "{(\<sigma>, snd (netgmap sr s)) |\<sigma> s. s \<in> init (pnet np p1) \<and> (\<forall>i. if i \<in> net_tree_ips p1 then \<sigma> i = the (fst (netgmap sr s) i) else \<sigma> i \<in> (fst \<circ> sr) ` init (np i))} \<subseteq> init (opnet onp p1)"
mpoly_map_vars_monom: fixes f :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and c :: "'b" assumes "bij f" shows "mpoly_map_vars f (MPoly_Type.monom m c) = MPoly_Type.monom (permutep (inv f) m) c"
term_to_nterm_induct: fixes P :: "name list \<Rightarrow> Term_Class.term \<Rightarrow> bool"   and a0 :: "name list"   and a1 :: "Term_Class.term" assumes "\<And>uu_ name. P uu_ (Const name)"   and "\<And>uv_ name. P uv_ (Free name)"   and "\<And>\<Gamma> n. P \<Gamma> (Bound n)"   and "\<And>\<Gamma> t. (\<And>x. P (x # \<Gamma>) t) \<Longrightarrow> P \<Gamma> (\<Lambda> t)"   and "\<And>\<Gamma> t\<^sub>1 t\<^sub>2. \<lbrakk>P \<Gamma> t\<^sub>1; \<And>x. P \<Gamma> t\<^sub>2\<rbrakk> \<Longrightarrow> P \<Gamma> (t\<^sub>1 $ t\<^sub>2)" shows "P a0 a1"
int_of_integer_of_int: fixes k :: "int" shows "int_of_integer (of_int k) = k"
FSC_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p" shows "A B C D FSC A' B' C' D' \<equiv> Col A B C \<and> A B C Cong3 A' B' C' \<and> Cong A D A' D' \<and> Cong B D B' D'"
add_aform'_def: fixes p :: "nat"   and x :: "(real \<times> real pdevs) \<times> real"   and y :: "(real \<times> real pdevs) \<times> real" shows "add_aform' p x y = (let z0 = trunc_bound_eucl p (fst (fst x) + fst (fst y)); z = trunc_bound_pdevs p (add_pdevs (snd (fst x)) (snd (fst y))) in ((fst z0, fst z), sum_list' p [snd z0, snd z, \<bar>snd x\<bar>, \<bar>snd y\<bar>]))"
ffb_sup_dual: shows "sup_dual fb\<^sub>\<F>"
distinct_3: fixes x1 :: "'b"   and x3 :: "'b formula" shows "Atom x1 \<noteq> \<^bold>\<not> x3"
rel_var_res_def: fixes P :: "'b hrel"   and x :: "'c \<Longrightarrow> 'b" shows "P \<restriction>\<^sub>\<alpha> x = (\<exists> $x \<bullet> \<exists> $x\<acute> \<bullet> P)"
correct_fin_inf_append1: fixes s1 :: "nat \<Rightarrow> 'a"   and x :: "'a"   and s :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "s1 = fin_inf_append [x] s" shows "s1 (Suc i) = s i"
mult_smult_left: fixes s :: "'b"   and p :: "'b mpoly"   and q :: "'b mpoly" shows "MPoly_Type.smult s p * q = MPoly_Type.smult s (p * q)"
mono_imp_mono_on: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "mono f" shows "mono_on f A"
h1: shows "\<forall>x y. f x \<rightarrow> f y = x \<rightarrow> f y"
compat: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "y \<succ> z" shows "x \<succ> z"
hseqI': fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "hseq \<nu> \<mu>"
prems_5: shows "\<forall>mx\<in>set ms. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
elims: fixes x :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and xa :: "'e option"   and xb :: "'f option"   and y :: "'d option" assumes "lift2_option x xa xb = y" obtains uu_ :: "'e \<Rightarrow> 'f \<Rightarrow> 'd" and uv_ :: "'f option" where "xa__ = uu_" and "xa = None" and "xb = uv_" and "ya__ = None"   | uw_ :: "'e \<Rightarrow> 'f \<Rightarrow> 'd" and v :: "'e" where "xa__ = uw_" and "xa = Some v" and "xb = None" and "ya__ = None"   | f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd" and x :: "'e" and y :: "'f" where "xa__ = f" and "xa = Some x" and "xb = Some y" and "ya__ = Some (f x y)"
simps_2: fixes l :: "'a tree"   and a :: "'a"   and r :: "'a tree" shows "height \<langle>l, a, r\<rangle> = max (height l) (height r) + 1"
select_convs_3: fixes hosts :: "'b set"   and flows_fix :: "('b \<times> 'b) set"   and flows_state :: "('b \<times> 'b) set"   and more :: "'c" shows "flows_state \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = flows_state"
polyn_expr0: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "polyn_expr R X 0 c = snd c 0"
cnorm_eq_square: fixes x :: "'a"   and a :: "real" shows "(norm x = a) = (0 \<le> a \<and> cinner x x = complex_of_real (a\<^sup>2))"
going_to_within_def: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter"   and A :: "'a set" shows "f going_to F within A = inf (filtercomap f F) (principal A)"
to_f: fixes p :: "nat \<Rightarrow> nat" assumes "p permutes {0..<n}" shows "pderiv (signof p * (\<Prod>i = 0..<n. Sum $$ (i, p i))) = sum (f p) {0..<n}"
simps_2: fixes P :: "'a \<Rightarrow> nat \<Rightarrow> 'b + unit"   and i :: "nat"   and x :: "'a"   and xs :: "'a list" shows "??.Error_Monad.forallM_index_aux P i (x # xs) = (try P x i catch (\<lambda>xa. Inl ((x, i), xa))) \<bind> (\<lambda>_. ??.Error_Monad.forallM_index_aux P (Suc i) xs)"
between_congs: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a"   and vs' :: "'a list" assumes "pre_between vs ram1 ram2"   and "vs \<cong> vs'" shows "between vs ram1 ram2 = between vs' ram1 ram2"
ab_semigroup_add_axioms_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.ab_semigroup_add_axioms plus \<equiv> \<forall>a b. plus a b = plus b a"
assms_2: shows "\<forall>i'\<ge>i. \<not> prerecording_event t i'"
equality_option_simps_1: fixes eq\<^sub>'\<^sub>a :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "equality_option eq\<^sub>'\<^sub>a None None = True"
purgeIdle_set_isState: fixes s :: "('a, 'b, 'c) state"   and sl :: "('a, 'b, 'c) state list" assumes "s \<in> set (GM_sec_model.purgeIdle sl)" shows "isState s"
aux6: fixes a1 :: "'b set set"   and a2 :: "'b set set"   and a :: "'b set set"   and b :: "'b set set" assumes "twoNetsDistinct a1 a2 a b" shows "dom (deny_all_from_to a1 a2) \<inter> dom (deny_all_from_to a b) = {}"
Pair_below_iff: fixes a :: "'b"   and b :: "'c"   and c :: "'b"   and d :: "'c" shows "((a, b) \<sqsubseteq> (c, d)) = (a \<sqsubseteq> c \<and> b \<sqsubseteq> d)"
simps_1: fixes p1 :: "net_tree"   and p2 :: "net_tree" shows "net_tree_ips (p1 \<parallel> p2) = net_tree_ips p1 \<union> net_tree_ips p2"
f0: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "a < b"   and "b < c"   and "c < d" shows "f0 {a, b, c, d} = f {(a, b), (c, d)}"
gamma_minus_ivl: fixes n1 :: "int"   and i1 :: "ivl"   and n2 :: "int"   and i2 :: "ivl" assumes "n1 \<in> \<gamma>_ivl i1"   and "n2 \<in> \<gamma>_ivl i2" shows "n1 - n2 \<in> \<gamma>_ivl (minus_ivl i1 i2)"
Sum_any_when_equal: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> 'a"   and b :: "'b" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.Sum_any plus zero (\<lambda>a. zero.when zero (f a) (a = b)) = f b"
sset_sdrop: fixes x :: "'a"   and i :: "nat"   and xs :: "'a stream" assumes "x \<in> sset (sdrop i xs)" shows "x \<in> sset xs"
ans_d_def: fixes x :: "'a" shows "d x = ad (ad x)"
1: fixes s :: "'a option \<Rightarrow>\<^sub>0 nat"   and t :: "'a option \<Rightarrow>\<^sub>0 nat" assumes "lookup s None = lookup t None"   and "restrict_indets_pp s = restrict_indets_pp t" shows "s = t"
fresh_at_base_2: fixes a :: "atom"   and b :: "'b" shows "a \<sharp> b = (a \<noteq> atom b)"
prefix_length_aux_correct: fixes P :: "'b \<Rightarrow> bool"   and acc :: "nat"   and xs :: "'b list" shows "prefix_length_aux P acc xs = prefix_length P xs + acc"
monom_mult_dist_left_minus: fixes c :: "'c"   and d :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult (c - d) t p = monom_mult c t p - monom_mult d t p"
wf_mstate_mstep: fixes \<phi> :: "Formula.formula"   and \<pi> :: "(String.literal \<times> event_data list) prefix"   and R :: "event_data list set"   and st :: "('a, 'b) mstate"   and tdb :: "(String.literal \<times> event_data list) set \<times> nat" assumes "wf_mstate \<phi> \<pi> R st"   and "last_ts \<pi> \<le> snd tdb" shows "wf_mstate \<phi> (psnoc \<pi> tdb) R (snd (mstep (map_prod mk_db id tdb) st))"
ev_block_def: fixes n :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.ev_block n A = (\<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> A $$ (i, i) = A $$ (j, j))"
givenBy_def: fixes f :: "'d \<Rightarrow> 'e" shows "givenBy f = range ((-`) f)"
finite_nth_default_neq_default: fixes dflt :: "'b"   and xs :: "'b list" shows "finite {k. nth_default dflt xs k \<noteq> dflt}"
cases: fixes a1 :: "('d, 'e, 'f) com"   and a2 :: "('d, 'e, 'f) com" assumes "com_relp a1 a2" obtains c1 :: "('d, 'e, 'f) com" and c2 :: "('d, 'e, 'f) com" where "a1 = c1a__" and "a2 = Seq c1a__ c2a__"   | c2 :: "('d, 'e, 'f) com" and c1 :: "('d, 'e, 'f) com" where "a1 = c2b__" and "a2 = Seq c1b__ c2b__"   | c1 :: "('d, 'e, 'f) com" and b :: "'d set" and c2 :: "('d, 'e, 'f) com" where "a1 = c1c__" and "a2 = Cond b__ c1c__ c2c__"   | c2 :: "('d, 'e, 'f) com" and b :: "'d set" and c1 :: "('d, 'e, 'f) com" where "a1 = c2d__" and "a2 = Cond ba__ c1d__ c2d__"   | c :: "('d, 'e, 'f) com" and b :: "'d set" where "a1 = ca__" and "a2 = While b ca__"   | c :: "'d \<Rightarrow> ('d, 'e, 'f) com" and x :: "'d" where "a1 = cb__ x" and "a2 = DynCom cb__"   | c :: "('d, 'e, 'f) com" and f :: "'f" and g :: "'d set" where "a1 = c" and "a2 = Guard f g c"   | c1 :: "('d, 'e, 'f) com" and c2 :: "('d, 'e, 'f) com" where "a1 = c1e__" and "a2 = Catch c1e__ c2e__"   | c2 :: "('d, 'e, 'f) com" and c1 :: "('d, 'e, 'f) com" where "a1 = c2" and "a2 = Catch c1 c2"
leq_path1_wfD: fixes Cs :: "char list list"   and Cs' :: "char list list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P,C \<turnstile> Cs \<sqsubset>\<^sup>1 Cs'"   and "wf_prog wf_md P" shows "Cs \<noteq> Cs' \<and> (Cs', Cs) \<notin> (leq_path1 P C)\<^sup>+"
type: shows "type_definition Rep_bit0 Abs_bit0 {0..<int CARD('a bit0)}"
add_rem_commute: fixes i :: "'c"   and "is" :: "'c set"   and e1 :: "'d"   and e2 :: "'d" assumes "i \<notin> is" shows "\<langle>Add i e1\<rangle> \<rhd> \<langle>Rem is e2\<rangle> = \<langle>Rem is e2\<rangle> \<rhd> \<langle>Add i e1\<rangle>"
span_base: fixes a :: "'a \<Rightarrow>\<^sub>0 'c"   and S :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "a \<in> S" shows "a \<in> local.punit.pmdl S"
atc_io_reduction_on_sets_from_obs: fixes M1 :: "('d, 'e, 'f) FSM"   and T :: "'d list set"   and M2 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set" assumes "M1 \<preceq>\<lbrakk>T\<rbrakk> M2"   and "(\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>) \<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>)" shows "M1 \<preceq>\<lbrakk>T.\<Omega>\<rbrakk> M2"
assms_2: fixes h :: "'a"   and g :: "'a" assumes "h \<in> H"   and "g \<in> carrier G" shows "g \<otimes> h \<otimes> inv g \<in> H"
leI: fixes x :: "'b word"   and y :: "'b word" assumes "\<not> x <s y" shows "y \<le>s x"
rScope: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'" shows "Prop (\<lparr>\<nu>x\<rparr>P')"
leq_add_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq x (plus x y)"
ord_isom_surj: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e"   and b :: "'e" assumes "Order D"   and "Order E"   and "ord_isom D E f"   and "b \<in> carrier E" shows "\<exists>a\<in>carrier D. b = f a"
radj_prop: fixes f :: "'c \<Rightarrow> 'b" shows "radj f = Sup \<circ> (-`) f \<circ> \<down>"
same_cardinality_post_swap_2: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and j :: "nat"   and i :: "nat"   and n :: "nat" assumes "computation channel trans send recv init final"   and "computation.prerecording_event channel trans send recv init t j"   and "computation.postrecording_event channel trans send recv init t i"   and "i < j"   and "j < length t"   and "computation.count_violations channel trans send recv init t = Suc n"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "distributed_system.trace channel trans send recv init t final" shows "card {k \<in> {i..<j + 1}. computation.prerecording_event channel trans send recv init t k} = card {k \<in> {i..<j + 1}. computation.prerecording_event channel trans send recv init (swap_events i j t) k}"
Dube_def: shows "local.Dube = (if n \<le> 1 \<or> d = 0 then d else local.Dube_aux 1)"
zero_neq_numeral: fixes n :: "num" shows "(0::'b) \<noteq> numeral n"
eq_on_subst_other_2: fixes x :: "'a"   and X :: "'a set"   and s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b"   and v :: "'b" assumes "x \<notin> X" shows "(s = t(x := v) on X) = (s = t on X)"
Re_winding_number_half_left: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "\<forall>p\<in>path_image \<gamma>. Re p \<le> Re z"   and "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "Re (winding_number \<gamma> z) = (Im (Ln (z - pathfinish \<gamma>)) - Im (Ln (z - pathstart \<gamma>))) / (2 * pi)"
cadd_assoc: fixes i :: "V"   and j :: "V"   and k :: "V" shows "i \<oplus> j \<oplus> k = i \<oplus> (j \<oplus> k)"
a_inv_closed: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "additive_subgroup H G"   and "x \<in> H" shows "\<ominus>\<^bsub>G\<^esub> x \<in> H"
elims_2: fixes x :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" assumes "BsSkewBinomialHeapStruc.elem_invar x" obtains e :: "'a" and a :: "'b" and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" where "x = BsSkewBinomialHeapStruc.Element e a q" and "(\<forall>x. x \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)) \<longrightarrow> a \<le> BsSkewBinomialHeapStruc.eprio x \<and> BsSkewBinomialHeapStruc.elem_invar x) \<and> SkewBinomialHeapStruc.invar (map BsSkewBinomialHeapStruc.bsmapt q)"
step: fixes w :: "'a" assumes "u \<rightarrow> w"   and "w \<rightarrow>\<^sup>* v" shows "thesis"
a: shows "(\<Union>n. (T ^^ n) --` A) \<in> sets M"
inputRemoveTau: fixes P :: "pi"   and u :: "name"   and P'''' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "\<tau>.P \<Longrightarrow>\<^sub>lu in P''''\<rightarrow>a<x> \<prec> P'" shows "P \<Longrightarrow>\<^sub>lu in P''''\<rightarrow>a<x> \<prec> P'"
superset_preserves_redundancy: fixes C :: "'a Literal set"   and S :: "'a Literal set set"   and S' :: "'a Literal set set" assumes "redundant C S"   and "S \<subseteq> S'" shows "redundant C S'"
assms_2: fixes B :: "'a tensor" assumes "B \<in> set BS" shows "dims B = ds"
numeral_Bit0_div_2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and n :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 num.One)) = numeral.numeral one plus n"
cong: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "simple_function M f"   and "simple_function M g"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"   and "P f" shows "P g"
simps_1: fixes p :: "perm"   and \<alpha> :: "'a" shows "p \<bullet> S_action.Act \<alpha> = S_action.Act (p \<bullet> \<alpha>)"
rep_trm_def: shows "rep_trm \<equiv> quot_type.rep Rep_trm"
S_edge_reachable: fixes u :: "'v"   and v :: "'v" assumes "u \<in> S A"   and "(u, v) \<in> edges g" shows "(u, v) \<in> edges rg"
lv_ivl_sings: fixes x :: "'c"   and y :: "'c" shows "lv_ivl [x] [y] = (\<lambda>x. [x]) ` {x..y}"
leD: fixes y :: "'a Word.word"   and x :: "'a Word.word" assumes "y \<le>s x" shows "\<not> x <s y"
clop_closure: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "clop f" shows "(x \<in> range f) = (f x = x)"
SUP_bot: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup ((\<lambda>x. bot) ` A) = bot"
one_aezfun_apply_neq: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "aezfun 1 x = (0::'b)"
simps: fixes as :: "nat"   and lm :: "nat list"   and m :: "nat"   and n :: "nat"   and initlm :: "nat list" shows "mv_box_inv (as, lm) m n initlm = (let plusa = initlm ! m + initlm ! n in max m n < length initlm \<and> m \<noteq> n \<and> (if as = 0 then \<exists>k l. lm = initlm[m := k, n := l] \<and> k + l = plusa \<and> k \<le> initlm ! m else if as = 1 then \<exists>k l. lm = initlm[m := k, n := l] \<and> k + l + 1 = plusa \<and> k < initlm ! m else if as = 2 then \<exists>k l. lm = initlm[m := k, n := l] \<and> k + l = plusa \<and> k \<le> initlm ! m else if as = 3 then lm = initlm[m := 0, n := plusa] else False))"
typeII_twoCubeImg: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" assumes "typeII_twoCube twoC" shows "\<exists>a b g1 g2. a < b \<and> (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and> cubeImage twoC = {(y, x). x \<in> {a..b} \<and> y \<in> {g2 x..g1 x}} \<and> twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b)) \<and> g1 piecewise_C1_differentiable_on {a..b} \<and> g2 piecewise_C1_differentiable_on {a..b}"
AE_count_space: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b set" shows "almost_everywhere (count_space A) P = (\<forall>x\<in>A. P x)"
rel_induct: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b list"   and y :: "'c list"   and Q :: "'b list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "list_all2 R x y"   and "Q [] []"   and "\<And>a21 a22 b21 b22. \<lbrakk>R a21 b21; Q a22 b22\<rbrakk> \<Longrightarrow> Q (a21 # a22) (b21 # b22)" shows "Q x y"
moebius_mu_squarefree_eq': fixes n :: "nat" assumes "squarefree n" shows "moebius_mu n = (- (1::'b)) ^ size (prime_factorization n)"
congruence_scale_m: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and k :: "complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "congruence M (k *\<^sub>s\<^sub>m H) = k *\<^sub>s\<^sub>m congruence M H"
dualization_2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<not> \<diamond>F = \<box>\<not> F"
inf_cases: fixes x :: "'a"   and y :: "'a" shows "- x = (- x \<squnion> - y) \<sqinter> (- x \<squnion> - - y)"
bounds_2: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "g x \<le> g b"
TS_xr'_2: fixes x :: "nat"   and y :: "nat"   and qs :: "nat list"   and h :: "nat list" assumes "x \<noteq> y"   and "qs \<in> lang (question (Atom x))"   and "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)" shows "(\<exists>hs. rev qs @ h = [x, x] @ hs) \<or> rev qs @ h = [x] \<or> rev qs @ h = []"
sup_bot_left: fixes x :: "'c multiset" shows "{#} \<union># x = x"
deriv_in_subterms_4: fixes p1 :: "('e, 'f, 'g, 'h) seqp"   and p2 :: "('e, 'f, 'g, 'h) seqp"   and p :: "('e, 'f, 'g, 'h) seqp" assumes "p1 \<oplus> p2 \<in> subterms p" shows "p2 \<in> subterms p"
InducedFinGroupRepresentation_axioms: shows "InducedFinGroupRepresentation H G (\<cdot>) V (\<currency>)"
simps_1: fixes V :: "'c"   and v :: "val" shows "lass_val_of (V:=Val v) = \<lfloor>(V, v)\<rfloor>"
B: shows "(evsFR4, S, A, U) \<in> protocol"
unsigned_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero 0 = zero"
bounded_by_def: fixes xs :: "real list"   and vs :: "float interval option list" shows "bounded_by xs vs = (\<forall>i<length vs. case vs ! i of None \<Rightarrow> True | Some ivl \<Rightarrow> xs ! i \<in>\<^sub>r ivl)"
birkhoff_theorem_AE_nonergodic: fixes f :: "'a \<Rightarrow> real" assumes "integrable MA f" shows "AE x in MA. (\<lambda>n. A.birkhoff_sum f n x / real n) \<longlonglongrightarrow> real_cond_exp MA A.Invariants f x"
CAs0_is: shows "CAs0 \<cdot>\<cdot>cl \<eta>s = CAs"