hcju/isabelleps huggingface dataset:
induct: fixes P :: "nat rexp \<Rightarrow> nat rexp \<Rightarrow> bool"   and a0 :: "nat rexp"   and a1 :: "nat rexp" assumes "\<And>uu_. P Zero uu_"   and "P One Zero"   and "\<And>v. P (Atom v) Zero"   and "\<And>v va. P (Plus v va) Zero"   and "\<And>v va. P (Times v va) Zero"   and "\<And>v. P (Star v) Zero"   and "\<And>v. P (rexp.Not v) Zero"   and "\<And>v va. P (rexp.Inter v va) Zero"   and "P One One"   and "\<And>v. P One (Atom v)"   and "\<And>v va. P One (Plus v va)"   and "\<And>v va. P One (Times v va)"   and "\<And>v. P One (Star v)"   and "\<And>v. P One (rexp.Not v)"   and "\<And>v va. P One (rexp.Inter v va)"   and "\<And>v. P (Atom v) One"   and "\<And>v va. P (Plus v va) One"   and "\<And>v va. P (Times v va) One"   and "\<And>v. P (Star v) One"   and "\<And>v. P (rexp.Not v) One"   and "\<And>v va. P (rexp.Inter v va) One"   and "\<And>r s v. P s (Atom v) \<Longrightarrow> P (Times r s) (Atom v)"   and "\<And>r s v va. P s (Plus v va) \<Longrightarrow> P (Times r s) (Plus v va)"   and "\<And>r s v va. P s (Times v va) \<Longrightarrow> P (Times r s) (Times v va)"   and "\<And>r s v. P s (Star v) \<Longrightarrow> P (Times r s) (Star v)"   and "\<And>r s v. P s (rexp.Not v) \<Longrightarrow> P (Times r s) (rexp.Not v)"   and "\<And>r s v va. P s (rexp.Inter v va) \<Longrightarrow> P (Times r s) (rexp.Inter v va)"   and "\<And>v va. P (Atom v) (Atom va)"   and "\<And>v va vb. P (Atom v) (Plus va vb)"   and "\<And>v va vb. P (Atom v) (Times va vb)"   and "\<And>v va. P (Atom v) (Star va)"   and "\<And>v va. P (Atom v) (rexp.Not va)"   and "\<And>v va vb. P (Atom v) (rexp.Inter va vb)"   and "\<And>v va vb. P (Plus v va) (Atom vb)"   and "\<And>v va vb vc. P (Plus v va) (Plus vb vc)"   and "\<And>v va vb vc. P (Plus v va) (Times vb vc)"   and "\<And>v va vb. P (Plus v va) (Star vb)"   and "\<And>v va vb. P (Plus v va) (rexp.Not vb)"   and "\<And>v va vb vc. P (Plus v va) (rexp.Inter vb vc)"   and "\<And>v va. P (Star v) (Atom va)"   and "\<And>v va vb. P (Star v) (Plus va vb)"   and "\<And>v va vb. P (Star v) (Times va vb)"   and "\<And>v va. P (Star v) (Star va)"   and "\<And>v va. P (Star v) (rexp.Not va)"   and "\<And>v va vb. P (Star v) (rexp.Inter va vb)"   and "\<And>v va. P (rexp.Not v) (Atom va)"   and "\<And>v va vb. P (rexp.Not v) (Plus va vb)"   and "\<And>v va vb. P (rexp.Not v) (Times va vb)"   and "\<And>v va. P (rexp.Not v) (Star va)"   and "\<And>v va. P (rexp.Not v) (rexp.Not va)"   and "\<And>v va vb. P (rexp.Not v) (rexp.Inter va vb)"   and "\<And>v va vb. P (rexp.Inter v va) (Atom vb)"   and "\<And>v va vb vc. P (rexp.Inter v va) (Plus vb vc)"   and "\<And>v va vb vc. P (rexp.Inter v va) (Times vb vc)"   and "\<And>v va vb. P (rexp.Inter v va) (Star vb)"   and "\<And>v va vb. P (rexp.Inter v va) (rexp.Not vb)"   and "\<And>v va vb vc. P (rexp.Inter v va) (rexp.Inter vb vc)" shows "P a0 a1"
map_rewrite: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and k :: "nat"   and y :: "'b" shows "map f (rewrite xs k y) = rewrite (map f xs) k (f y)"
a1: shows "low_equal s1 s2 \<and> asi \<in> {8, 10}"
no_head_gen_set: fixes C :: "'b list set"   and b :: "'b list" shows "no_head_gen C b = {z \<cdot> b \<^sup>@ k |z k. z \<in> C \<and> z \<noteq> b}"
div_distrib: fixes base :: "nat"   and i :: "nat"   and n :: "nat"   and a :: "nat"   and b :: "nat" assumes "digits base"   and "i < n" shows "(a * base ^ n + b) div base ^ i mod base = b div base ^ i mod base"
return_needs_call: fixes a :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<exists>!a'. valid_edge a' \<and> (\<exists>Q r fs. kind a' = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs) \<and> a \<in> get_return_edges a'"
error: fixes j :: "nat" assumes "t (Suc j) \<le> t1" shows "norm (stepsize j *\<^sub>R s (stepsize j) (t j) (grid_function (discrete_evolution incrs) (x (t 0) + s0) t j)) \<le> B * stepsize j ^ (p + 1)"
skew_product_conservative: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. real_cond_exp M Invariants f x = 0" shows "conservative_mpt (M \<Otimes>\<^sub>M lborel) (\<lambda>(x, y). (T x, y + f x))"
monofun_cfun_arg: fixes x :: "'d"   and y :: "'d"   and f :: "'d \<rightarrow> 'e" assumes "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> f\<cdot>y"
asm: shows "\<sigma> \<in> {\<sigma>. is_unifiers \<sigma> AAA}"
eq1: fixes i :: "nat" shows "LBINT x=ereal (l i)..ereal (u i). g' x *\<^sub>R f (g x) = interval_lebesgue_integral lborel (ereal (g (l i))) (ereal (g (u i))) f"
map_to_set_empty: shows "map_to_set Map.empty = {}"
Vitali_covering_lemma_cballs_balls: fixes K :: "'b set"   and r :: "'b \<Rightarrow> real"   and B :: "real"   and a :: "'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> K \<Longrightarrow> 0 < r i \<and> r i \<le> B" obtains C :: "'b set" where "countable C" and "C \<subseteq> K" and "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C" and "\<And>i. i \<in> K \<Longrightarrow> \<exists>j. j \<in> C \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and> cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
preserves_cod: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr f" shows "Span.cod (Span.src f) = Span.src (Span.cod f)"
to_nat_from_nat_id: fixes i :: "nat" assumes "i < CARD('a)" shows "to_nat (from_nat i) = i"
simps: fixes ap :: "abc_inst list" shows "tm_of ap = concat (tms_of ap)"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> es @ [e] @ es' : ts _> ts'"
eq_iff_3: fixes nat :: "nat"   and nata :: "nat" shows "(DBInd nat = DBInd nata) = (nat = nata)"
psubstEnv_idEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" shows "idEnv &[rho] = rho"
optimal_for_women_def: fixes \<mu> :: "(M \<times> W) set" shows "Non_Strict.optimal_for_women \<mu> = (Non_Strict.stable \<mu> \<and> (\<forall>\<mu>''. Non_Strict.stable \<mu>'' \<longrightarrow> Non_Strict.weakly_preferred_by_women \<mu> \<mu>''))"
finite_M_bounded_by_nat: fixes P :: "nat \<Rightarrow> bool"   and i :: "nat" shows "finite {k. P k \<and> k < i}"
show_nat_def: fixes i :: "nat" shows "show_nat i = show i"
ccpo_Sup_upper: fixes A :: "'b set"   and x :: "'b" assumes "Complete_Partial_Order.chain ord' A"   and "x \<in> A" shows "ord' x (lub' A)"
sc_def: fixes C :: "nat set"   and X :: "'a set" shows "sc C X = (C \<subseteq> {1..m} \<and> \<Union> (S ` C) = X)"
subspaceI: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set" assumes "module scale"   and "(0::'e) \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> scale c x \<in> S" shows "module.subspace scale S"
cofinite_closure_infinite: fixes X :: "'a cofinite set" assumes "infinite X" shows "closure X = UNIV"
ft: fixes P :: "'a \<Rightarrow> real" assumes "unitary P" shows "unitary (t P)"
fps_of_poly_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "fps_of_poly (p * q) = fps_of_poly p * fps_of_poly q"
eval_fps_def: fixes f :: "'a fps"   and z :: "'a" shows "eval_fps f z = (\<Sum>n. fps_nth f n * z ^ n)"
Ly: fixes y :: "'a" shows "(\<lambda>n. dist y ((f ^^ n) y) / real n) \<longlonglongrightarrow> Inf {dist y ((f ^^ n) y) / real n |n. 0 < n}"
sdrop_siterate: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "sdrop n (siterate f x) = siterate f ((f ^^ n) x)"
assms: shows "s' = snd (fst (div_comp instr rs1 rd operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0"
mult2_alt_nsI: fixes X :: "'a multiset"   and X1 :: "'a multiset"   and X2 :: "'a multiset"   and Y :: "'a multiset"   and Y1 :: "'a multiset"   and Y2 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "X = X1 + X2"   and "Y = Y1 + Y2"   and "(X1, Y1) \<in> multpw ns"   and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)" shows "(X, Y) \<in> mult2_alt_ns ns s"
e3: shows "pdevs_val e (fst (trunc_bound_pdevs p (scaleR_pdevs (fst (fst X)) (snd (fst Y))))) = pdevs_val e (scaleR_pdevs (fst (fst X)) (snd (fst Y))) + e3"
elims: fixes x :: "'a ifex"   and y :: "('a \<times> bool) list option" assumes "ifex_sat_list x = y" obtains "x = Trueif" and "y = Some []"   | "x = Falseif" and "y = None"   | v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e" and "y = (case ifex_sat_list e of None \<Rightarrow> case ifex_sat_list t of None \<Rightarrow> None | Some a \<Rightarrow> Some ((v, True) # a) | Some a \<Rightarrow> Some ((v, False) # a))"
fps_compose_inj_right: fixes a :: "'a fps"   and b :: "'a fps"   and c :: "'a fps" assumes "fps_nth a 0 = (0::'a)"   and "fps_nth a 1 \<noteq> (0::'a)" shows "(b oo a = c oo a) = (b = c)"
a2: shows " s1 \<longrightarrow>\<^sub>\<beta>\<^sup>* s2"
select_convs_4: fixes igWls :: "'g \<Rightarrow> 'c \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'g \<Rightarrow> 'd \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and igAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd"   and igOp :: "'h \<Rightarrow> ('e \<Rightarrow> 'c option) \<Rightarrow> ('f \<Rightarrow> 'd option) \<Rightarrow> 'c"   and igFresh :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSwapAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> 'd"   and igSubst :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubstAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'i" shows "igAbs \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igAbs"
inv_inv_eq: fixes f :: "'b \<Rightarrow> 'c" assumes "bij f" shows "inv (inv f) = f"
cod_comp: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'c"   and f :: "'c" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.cod C (C g f) = partial_magma.cod C g"
abs_eq: fixes x :: "'c set" shows "ccspan x = Abs_clinear_space (closure (cspan x))"
inverse_arrows_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C" shows "category.inverse_arrows C f g \<equiv> partial_magma.ide C (C g f) \<and> partial_magma.ide C (C f g)"
Bernstein_coeffs_01_add: fixes P :: "real poly"   and p :: "nat"   and Q :: "real poly"   and i :: "nat" assumes "degree P \<le> p"   and "degree Q \<le> p" shows "nth_default 0 (Bernstein_coeffs_01 p (P + Q)) i = nth_default 0 (Bernstein_coeffs_01 p P) i + nth_default 0 (Bernstein_coeffs_01 p Q) i"
coplanar_perm_16: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet C D A B"
inducts: fixes P :: "unit \<Rightarrow> bool"   and unit :: "unit" assumes "P ()" shows "P unit"
VSGeneralBlocks: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (VSGeneralBlocks \<phi>) xs = eval \<phi> xs"
length_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'a list" shows "length (permute_list f xs) = length xs"
assms: shows "\<Gamma> \<turnstile> e : \<tau>"
induct: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'b list"   and x2 :: "'b list"   and P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "lexext gt x1 x2"   and "\<And>y ys. P (y # ys) []"   and "\<And>y x ys xs. gt y x \<Longrightarrow> P (y # ys) (x # xs)"   and "\<And>ys xs x. \<lbrakk>lexext gt ys xs; P ys xs\<rbrakk> \<Longrightarrow> P (x # ys) (x # xs)" shows "P x1 x2"
b: shows "b = cmod \<langle>\<psi>, A \<phi>\<rangle> / (norm \<psi> * norm \<phi>)"
mat_kernelD_2: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> mat_kernel A" shows "A *\<^sub>v v = 0\<^sub>v nr"
M_perp_def: fixes l :: "proj2_line"   and m :: "proj2_line" shows "M_perp l m \<equiv> proj2_incident (pole l) m"
piecewise_continuous_on_finite_superset: fixes a :: "'c"   and b :: "'c"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and J :: "'c set" assumes "piecewise_continuous_on a b I f"   and "I \<subseteq> J"   and "finite J" shows "piecewise_continuous_on a b J f"
m_st_anti_mono: fixes S1 :: "ivl Abs_State.st"   and S2 :: "ivl Abs_State.st" assumes "S1 \<sqsubseteq> S2" shows "m_st m_ivl S2 \<le> m_st m_ivl S1"
simpfm: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm" shows "Ifm vs bs (simpfm p) = Ifm vs bs p"
ivl_disj_un_two_8: fixes l :: "'b"   and m :: "'b"   and u :: "'b" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m<..u} = {l..u}"
combine_infmatrix_commute: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "commutative f" shows "commutative (combine_infmatrix f)"
cinner_sup_norm_cblinfun: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "norm A = (SUP (\<psi>, \<phi>). cmod (\<psi> \<bullet>\<^sub>C (A *\<^sub>V \<phi>)) / (norm \<psi> * norm \<phi>))"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "(if x \<in> T then g' x else (\<lambda>x. 0::'b)) (b1 + b2) = (if x \<in> T then g' x else (\<lambda>x. 0::'b)) b1 + (if x \<in> T then g' x else (\<lambda>x. 0::'b)) b2"
b6: fixes x :: "'b" shows "\<parallel>x\<parallel> * \<parallel>A ((A\<^sup>\<dagger>) x)\<parallel> \<le> \<parallel>x\<parallel> * M * \<parallel>(A\<^sup>\<dagger>) x\<parallel>"
new_character_data_no_child_nodes: fixes h :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr) \<rightarrow>\<^sub>r []"
gamma_s_top: shows "\<gamma>\<^sub>s \<top> = UNIV"
pure_fold: shows "(\<lambda>a c. \<up> (c = a)) = id_assn"
seteq: shows "{i. i < length xs \<and> i \<in> A} = {i \<in> A. i < length xs}"
plus_and_or: fixes x :: "int"   and y :: "int" shows "(x AND y) + (x OR y) = x + y"
bound_of_least: fixes b :: "real"   and P :: "'a \<Rightarrow> real" assumes "bounded_by b P" shows "bound_of P \<le> b"
constant_rules_4: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and c :: "'b"   and c' :: "'b" assumes "standard' C G"   and "c \<in> C"   and "c' \<in> C"   and "c \<noteq> c'" shows "maintained (const_disj c c') G"
dvd_prodD: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b"   and a :: "'a" assumes "finite A"   and "prod f A dvd x"   and "a \<in> A" shows "f a dvd x"
singular_boundary_eq_cycle_singleton: fixes X :: "'b topology"   and a :: "'b"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "topspace X = {a}"   and "1 \<le> p" shows "singular_relboundary p X {} c = singular_relcycle p X {} c"
test_inf_commutative: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "x \<sqinter> y = y \<sqinter> x"
simps_2: fixes p :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list"   and q :: "('a, 'b) fm" shows "(p # ps \<^bold>\<leadsto> q) = (p \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> q)"
squarefree_part_finite: fixes n :: "nat" shows "finite (squarefree_part n)"
elims_3: fixes p :: "'b \<Rightarrow> bool"   and xa :: "'b list \<times> 'b set" assumes "\<not> While_Combinator.rtrancl_while_test p xa" obtains ws :: "'b list" and uu_ :: "'b set" where "xa = (ws, uu_)" and "\<not> (ws \<noteq> [] \<and> p (hd ws))"
is_extensional: fixes f :: "'b \<times> 'b \<times> 'b" assumes "\<not> VVV.arr f" shows "\<alpha>' f = null"
caratheodory': fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "countably_additive M f" shows "\<exists>\<mu>. (\<forall>s\<in>M. \<mu> s = f s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>"
intro: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "Group G"   and "RModuleHom (aezfun_setspan G) smult V smult' T" shows "FGModuleHom G smult V smult' T"
HighIn3: shows "the (\<sigma>\<^sub>3 ''highIn'') = highIn"
abs_0_eq: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "(zero = abs a) = (a = zero)"
induct: fixes P :: "'a \<Rightarrow> 'var \<Rightarrow> 'node list \<Rightarrow> 'node \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'var"   and a2 :: "'node list"   and a3 :: "'node" assumes "\<And>g v unvisited n. (\<And>x. \<lbrakk>\<not> (n \<notin> set unvisited \<or> v \<in> defs2 g n); x \<in> set (i.predecessors g n)\<rbrakk> \<Longrightarrow> P g v (removeAll n unvisited) x) \<Longrightarrow> P g v unvisited n" shows "P a0 a1 a2 a3"
incseq_def: fixes X :: "nat \<Rightarrow> 'd" shows "incseq X = (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
subspace_inter: fixes A :: "'d set"   and B :: "'d set" assumes "csubspace A"   and "csubspace B" shows "csubspace (A \<inter> B)"
simps_20: fixes f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and x41 :: "('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool"   and x42 :: "'e"   and x43 :: "('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option" shows "(case x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = f4 x41 x42 x43"
swap_col_to_front_det: fixes A :: "'b mat"   and n :: "nat"   and I :: "nat" assumes "A \<in> carrier_mat n n"   and "I < n" shows "det (swap_col_to_front A I) = (- (1::'b)) ^ I * det A"
AXIOMS_eqvt_name_2: fixes pi :: "(name \<times> name) list"   and B :: "ty" shows "pi \<bullet> AXIOMSc B = AXIOMSc B"
cospan_\<mu>\<nu>: shows "C.cospan f.leg0 src.leg1"
Aexp_fresh_bot: fixes v :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "atom v \<sharp> e" shows "\<A>\<^bsub>a\<^esub> e v = \<bottom>"
hyps_1: shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
eq_f: shows "functions_of A M = domain_permutation A B `` {f}"
Q'aP': shows "four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (n - 1)) (0\<^sub>m (n - 1) 1) Q' = adjoint (four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (n - 1)) (0\<^sub>m (n - 1) 1) P')"
keys_subset_deg_le_sectI: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and d :: "nat" assumes "p \<in> P[X]"   and "poly_deg p \<le> d" shows "keys p \<subseteq> deg_le_sect X d"
substRes2: fixes b :: "name"   and P :: "pi"   and a :: "name" assumes "b \<sharp> P" shows "<\<nu>a>P = <\<nu>b>P[a::=b]"
prems_1: shows "\<Gamma>,\<gamma>,p\<turnstile> a # rs\<^sub>3 \<Rightarrow>\<^sub>r Undecided"
"25_2": assumes "even n" shows "(\<Sum>k = 1..n - 1. cmod (f (int k))) \<le> 2 * (\<Sum>k = 1..(n - 2) div 2. cmod (f (int k))) + cmod (f (int (n div 2)))"
cong_midpoint__image: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and X :: "'a"   and Y :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A X A Y"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B X Y" shows "Tarski_neutral_dimensionless.Reflect Bet Cong Y X A B"
distinct_1: fixes x11 :: "String.literal"   and x12 :: "typ list"   and x21 :: "variable"   and x22 :: "String.literal set" shows "Ty x11 x12 \<noteq> Tv x21 x22"
one_fls_def: shows "1 \<equiv> Abs_fls (\<lambda>k. if k = 0 then 1::'a else (0::'a))"
greatest_Lower_cong_l: fixes x :: "'a"   and x' :: "'a"   and A :: "'a set" assumes "x .=\<^bsub>division_rel G\<^esub> x'"   and "x \<in> carrier (division_rel G)"   and "x' \<in> carrier (division_rel G)" shows "is_glb (division_rel G) x A = is_glb (division_rel G) x' A"
prefix_fininf_eq_inf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item stream"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>2 =\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
intro: fixes v0 :: "'c"   and G :: "('c, 'd) Graph_scheme"   and v1 :: "'c" assumes "v0 \<in> V\<^bsub>G\<^esub>"   and "v1 \<in> V\<^bsub>G\<^esub>"   and "(v0, v1) \<notin> E\<^bsub>G\<^esub>"   and "v0 \<noteq> v1" shows "v0_v1_Digraph_axioms G v0 v1"
x_def: shows "x = [a, b]\<^sub>\<circ>"
AssignIdx: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and i :: "aexp"   and a :: "aexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (x[i] ::= a, s) \<Rightarrow> s(x := (s x)(aval i s := aval a s))"
mult_nonneg_nonpos2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "Rings.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"
functor_axioms: shows "functor C.CC.comp (\<cdot>\<^sub>C) T\<^sub>C"
mem: shows "(s, fst x, snd x) \<in> ars"
ladder_shift_n_cons: fixes N :: "nat"   and x :: "nat \<times> nat \<times> nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "ladder_shift_n N (x # L) = (fst x - N, snd x) # ladder_shift_n N L"
all_conj_distrib: fixes P :: "'n \<Rightarrow> bool"   and Q :: "'n \<Rightarrow> bool" shows "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))"
cone_\<chi>: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and a :: "'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah" assumes "limit_cone J C D a \<chi>" shows "\<chi> \<in> {\<chi>. Limit.cone J C D a \<chi>}"