hcju/isabelleps huggingface dataset:
basis_expand: fixes B :: "(real, 'a) vec set"   and v :: "(real, 'a) vec" assumes "is_basis B" shows "\<exists>c. v = (\<Sum>w\<in>B. c w *\<^sub>R w)"
lens_plus_comm: fixes X :: "'c \<Longrightarrow> 'd"   and Y :: "'e \<Longrightarrow> 'd" assumes "X \<bowtie> Y" shows "X +\<^sub>L Y \<approx>\<^sub>L Y +\<^sub>L X"
sum_list_map_aezfun_apply: fixes f :: "'c \<Rightarrow> ('a, 'b) aezfun"   and as :: "'c list"   and x :: "'b" shows "aezfun (sum_list (map f as)) x = (\<Sum>a\<leftarrow>as. aezfun (f a) x)"
sep_conjE: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "(P ** Q) s" obtains p :: "'a" and q :: "'a" where "P p" and "Q q" and "p \<oplus> q = Some s"
new_shadow_root_known_ptr: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "ShadowRootClass.known_ptr (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^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_shadow_root_ptr)"
l_def: shows "l = Lcm ((snd \<circ> quotient_of \<circ> \<beta>) ` X)"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b \<squnion> (a \<squnion> c) = a \<squnion> (b \<squnion> c)"
assms_3: fixes n :: "nat"   and p :: "'a poly"   and x :: "'a" assumes "0 < n"   and "poly p x \<noteq> (0::'a)"   and "P p" shows "P ([:- x, 1::'a:] ^ n * p)"
closure_convex_hull: fixes S :: "'c set" assumes "compact S" shows "closure (convex hull S) = convex hull S"
signed_rconjaction_by_order2_eq_lconjaction: fixes s :: "'a" assumes "s + s = (0::'a)" shows "signed_rconjaction s = signed_lconjaction s"
simps_2: fixes delta :: "'c"   and inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs option" shows "qGood (qOp delta inp binp) = (liftAll qGood inp \<and> liftAll qGoodAbs binp \<and> |{i. inp i \<noteq> None}| <o |UNIV| \<and> |{i. binp i \<noteq> None}| <o |UNIV|)"
rcs_eq: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G" shows "(b \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> a \<in> H) = (H \<bullet>\<^bsub>G\<^esub> a = H \<bullet>\<^bsub>G\<^esub> b)"
RDirProd_def: fixes R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" shows "RDirProd R S = monoid.extend (R \<times>\<times> S) \<lparr>zero = \<one>\<^bsub>add_monoid R \<times>\<times> add_monoid S\<^esub>, add = (\<otimes>\<^bsub>add_monoid R \<times>\<times> add_monoid S\<^esub>)\<rparr>"
restrict_codomain_to_image: shows "ChamberComplexMorphism X (g \<turnstile> X) g"
assms: shows "\<forall>i\<ge>n. i < f i \<and> P (S i) (S (f i))"
dom_simp: fixes f :: "'a" assumes "L.arr f" shows "L.dom f = local.dom f"
ineq_model_def: fixes \<I> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" shows "ineq_model \<I> X F \<equiv> \<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> list_ex (\<lambda>f. fst f \<cdot> \<delta> \<circ>\<^sub>s \<I> \<noteq> snd f \<cdot> \<delta> \<circ>\<^sub>s \<I>) F"
arr_cod: fixes f :: "'a" assumes "C.arr f" shows "C.arr (C.cod f)"
mod_ring_mod32: shows "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (\<lambda>x y. if y = 0 then x else 0) (mod)"
alllsts_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "A\<^sup>\<infinity> \<subseteq> B\<^sup>\<infinity>"
pCons_0_0: shows "[:0::'a:] = 0"
cod_simp: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.cod f = VxV.cod f"
distinct_7: fixes x2 :: "'b match_expr"   and x31 :: "'b match_expr"   and x32 :: "'b match_expr" shows "MatchNot x2 \<noteq> MatchAnd x31 x32"
lookup_bs_env: fixes \<rho> :: "(nat \<times> bval) list"   and \<rho>' :: "(nat \<times> exp) list"   and x :: "nat"   and w :: "bval" assumes "bs_env \<rho> \<rho>'"   and "lookup \<rho> x = Some w" shows "\<exists>v. lookup \<rho>' x = Some v \<and> bs_val w v"
distinct_keys: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "distinct (RBT_Impl.keys t)"
hd_rotate_conv_nth: fixes xs :: "'b list"   and n :: "nat" assumes "xs \<noteq> []" shows "hd (rotate n xs) = xs ! (n mod length xs)"
2: shows "deriv f \<alpha> \<otimes> h \<oplus> c \<otimes> h [^] 2 = \<zero>"
sumlist_snoc: fixes xs :: "rat vec list"   and x :: "rat vec" assumes "set xs \<subseteq> Rn"   and "x \<in> Rn" shows "gs.M.sumlist (xs @ [x]) = gs.M.sumlist xs + x"
integrable_continuous_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'e" assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
prime_power_inj'_1: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "power.power one times p m = power.power one times q n"   and "0 < m"   and "0 < n" shows "p = q"
s_delayed_has_vector_derivative': fixes t :: "real" assumes "\<delta> \<le> t" shows "(ego2.s \<circ> \<tau> has_vector_derivative (ego2.s' \<circ> \<tau>) t) (at t within {\<delta>..})"
ordinal_0_le: fixes x :: "ordinal" shows "0 \<le> x"
write_cpu_def: fixes w :: "word32"   and cr :: "CPU_register" shows "write_cpu w cr \<equiv> do modify (cpu_reg_mod w cr); return () od"
simb': fixes as :: "'a"   and cs :: "'c"   and l :: "'l" assumes "A.reachable as"   and "C.reachable cs"   and "R as cs" shows "C.can_step l cs = A.can_step l as"
filterlim_iff: fixes f :: "'b \<Rightarrow> 'd"   and F2 :: "'d filter"   and F1 :: "'b filter" shows "filterlim f F2 F1 = (\<forall>P. eventually P F2 \<longrightarrow> (\<forall>\<^sub>F x in F1. P (f x)))"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case Neg x2 of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = f2 x2"
exhaust: fixes y :: "'a le_constraint" obtains x1 :: "le_rel" and x2 :: "linear_poly" and x3 :: "'a" where "y = Le_Constraint x1 x2 x3"
is_ground_lit_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and L :: "'a literal" shows "substitution_ops.is_ground_lit subst_atm L = substitution_ops.is_ground_atm subst_atm (atm_of L)"
word_power_less_0_is_0: fixes x :: "'b word"   and a :: "'b word" assumes "x < a ^ 0" shows "x = 0"
integrable_0: fixes S :: "'a set" shows "(\<lambda>x. 0::'b) integrable_on S"
joinableD: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'c) set" assumes "(x, y) \<in> joinable R" shows "\<exists>z. (x, z) \<in> R \<and> (y, z) \<in> R"
cases: fixes a1 :: "'action list"   and a2 :: "'state" assumes "path a1 a2" obtains p :: "'state" where "a1 = []" and "a2 = pa__"   | a :: "'action" and p :: "'state" and r :: "'action list" where "a1 = a # r" and "a2 = p" and "en a p" and "path r (ex a p)"
findaugmenting_edge: fixes E1 :: "'a uprod set"   and u :: "'a"   and p :: "'a uprod list"   and v :: "'a"   and E2 :: "'a uprod set" assumes "epath E1 u p v"   and "\<nexists>p. epath E2 u p v" shows "\<exists>a b. (a, b) \<notin> uconnected E2 \<and> Upair a b \<notin> E2 \<and> Upair a b \<in> E1"
rel_intros: fixes R :: "'n \<Rightarrow> 't \<Rightarrow> bool"   and x :: "(('n, 'o, 'p, 'q, 'r, 's) T, 'n, 'o, 'p, 'q, 'r, 's) G"   and y :: "(('t, 'o, 'p, 'q, 'r, 's) T, 't, 'o, 'p, 'q, 'r, 's) G" assumes "rell_G (rell_T R) R x y" shows "rell_T R (C_T x) (C_T y)"
sgn1_zero_eq_arg_zero: fixes x :: "nat" shows "(sgn1 x = 0) = (x = 0)"
part_eq: shows "P x y = (PP x y \<or> x = y)"
max_mono: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x" shows "max z y \<le> max z x"
drop_n_lem: fixes n :: "nat"   and l :: "'c list"   and x :: "'c" assumes "Suc n \<le> length l" shows "drop (Suc n) (l[n := x]) = drop (Suc n) l"
in_gamma_sup_UpI: fixes s :: "char list \<Rightarrow> int"   and S1 :: "'av st option"   and S2 :: "'av st option" assumes "s \<in> \<gamma>\<^sub>o S1 \<or> s \<in> \<gamma>\<^sub>o S2" shows "s \<in> \<gamma>\<^sub>o (S1 \<squnion> S2)"
fmi'_def: fixes s :: "bdd" shows "fmi' s \<equiv> oassert (bdd_sane s) \<bind> (\<lambda>_. Some (fmi s))"
EqOnUOut_eqOnUOut: fixes env :: "pvar \<Rightarrow> nat"   and p :: "pvar"   and i :: "nat"   and p' :: "pvar"   and i' :: "nat"   and u :: "'U"   and ou :: "'Out" assumes "env p = i"   and "env p' = i'" shows "sem (EqOnUOut p p' u ou) env = eqOnUOut i i' u ou"
gmnl_gmft: fixes nl :: "('b, 'c) Node list" assumes "\<forall>x\<in>set nl. is_measured_node x" shows "gmft (nlistToTree nl) = gmnl nl"
action_loc_read_action_singleton: fixes r :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and adal :: "'b \<times> addr_loc"   and P :: "'c prog"   and adal' :: "'b \<times> addr_loc" assumes "r \<in> read_actions E"   and "adal \<in> action_loc P E r"   and "adal' \<in> action_loc P E r" shows "adal = adal'"
blinding_of_rt_F\<^sub>m_mono: fixes boa :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and boa' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bob :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and bob' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ha :: "'c \<Rightarrow> 'e"   and hb :: "'d \<Rightarrow> 'f" assumes "boa \<le> boa'"   and "bob \<le> bob'" shows "blinding_of_blindable (map_prod ha (map hb)) (rel_prod boa (list_all2 bob)) \<le> blinding_of_blindable (map_prod ha (map hb)) (rel_prod boa' (list_all2 bob'))"
vec_first_smult: fixes m :: "nat"   and n :: "nat"   and x :: "'b vec"   and c :: "'b" assumes "m \<le> n"   and "x \<in> carrier_vec n" shows "vec_first (c \<cdot>\<^sub>v x) m = c \<cdot>\<^sub>v vec_first x m"
preprm_ext_uncompose: fixes \<pi> :: "('b \<times> 'b) list"   and \<sigma> :: "('b \<times> 'b) list"   and P :: "('b \<times> 'b) list"   and S :: "('b \<times> 'b) list" assumes "\<pi> =p \<sigma>"   and "preprm_compose \<pi> P =p preprm_compose \<sigma> S" shows "P =p S"
subseqs_Cons: fixes x :: "'b"   and xs :: "'b list" shows "subseqs (x # xs) = map ((#) x) (subseqs xs) @ subseqs xs"
R4_3: shows "pmf (sds R4) c = 1 - pmf (sds R4) a"
rep_eq: fixes x :: "'b fpxs"   and xa :: "'b fpxs" shows "fpxs_nth (x + xa) = (\<lambda>xb. fpxs_nth x xb + fpxs_nth xa xb)"
ide_dom: fixes f :: "'a" assumes "S.arr f" shows "S.ide (S.dom f)"
discovered_not_stack_imp_finished: fixes x :: "'a" assumes "x \<in> dom (discovered s)"   and "x \<notin> set (stack s)" shows "x \<in> dom (finished s)"
vseq_implies_hpar_1: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.seq \<nu> \<mu>" shows "Span.src \<nu> = Span.src \<mu>"
flow_in_domain: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> existence_ivl0 x0" shows "flow0 x0 (t - t0) \<in> X"
nprt_le_zero: fixes 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 inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq (lattice_ab_group_add.nprt zero inf a) zero"
equality: fixes r :: "('a, 'b, 'c) configuration_scheme"   and r' :: "('a, 'b, 'c) configuration_scheme" assumes "c_work r = c_work r'"   and "c_pts r = c_pts r'"   and "c_imp r = c_imp r'"   and "more r = more r'" shows "r = r'"
inject_2: fixes x21 :: "typ"   and x22 :: "typ"   and y21 :: "typ"   and y22 :: "typ" shows "(x21 -> x22 = y21 -> y22) = (x21 = y21 \<and> x22 = y22)"
map_def: fixes f :: "'c local.term set" shows "E.map f \<equiv> if \<F>C.arr f then \<lbrace>\<F>C.rep f\<rbrace> else D.null"
eventually_sequentially: fixes P :: "nat \<Rightarrow> bool" shows "evseq P = (\<exists>N. \<forall>n\<ge>N. P n)"
hyps_4: shows "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) u (Fun (Fu f) T)"
distrib: fixes g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "finite {a. g a \<noteq> (0::'c)}"   and "finite {a. h a \<noteq> (0::'c)}" shows "(\<Sum>a. g a + h a) = Sum_any g + Sum_any h"
poly_lift_pCons: fixes a :: "'c"   and p :: "'c poly" shows "poly_lift (pCons a p) = pCons [:a:] (poly_lift p)"
ipresFreshAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'i, 'd, 'j) model_scheme" shows "ipresFreshAbs hA MOD \<equiv> \<forall>ys y us s A. wlsAbs (us, s) A \<longrightarrow> freshAbs ys y A \<longrightarrow> igFreshAbs MOD ys y (hA A)"
grounding: shows "\<G>_I_q q \<iota>' \<noteq> None \<and> \<iota> \<in> the (\<G>_I_q q \<iota>')"
dgrad_max_def: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and d :: "'a \<Rightarrow> nat" shows "qpm_inf_term.dgrad_max fs d = Max (d ` insert (0::'a) (Keys (set fs)))"
int_pow_subgroup_generated: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and S :: "'b set"   and n :: "int" assumes "Group.group G"   and "x \<in> carrier (subgroup_generated G S)" shows "x [^]\<^bsub>subgroup_generated G S\<^esub> n = x [^]\<^bsub>G\<^esub> n"
REC_rule_old: fixes body :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and \<Phi> :: "'b \<Rightarrow> bool"   and x :: "'b"   and M :: "'b \<Rightarrow> 'c" assumes "trimono body"   and "\<Phi> x"   and "\<And>f x. \<lbrakk>\<And>x. \<Phi> x \<Longrightarrow> f x \<le> M x; \<Phi> x; f \<le> REC body\<rbrakk> \<Longrightarrow> body f x \<le> M x" shows "REC body x \<le> M x"
runit_agreement: fixes a :: "'a" assumes "ide a" shows "runit a = \<r>[a]"
chain_boundary_simplicial_cone_of': fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and p :: "nat"   and l :: "nat \<Rightarrow> nat \<Rightarrow> real"   and v :: "nat \<Rightarrow> real" assumes "f = oriented_simplex p l" shows "chain_boundary (Suc p) (simplicial_cone p v (frag_of f)) = frag_of f - (if p = 0 then frag_of (\<lambda>u\<in>standard_simplex p. v) else simplicial_cone (p - 1) v (chain_boundary p (frag_of f)))"
simul_def: fixes Retr :: "'b \<Rightarrow> 'b"   and theta :: "'b" shows "simul Retr theta \<equiv> theta \<le> Retr theta"
lcp_self: fixes w :: "'b list" shows "w \<and>\<^sub>p w = w"
caratheodory: fixes p :: "'a set" shows "convex hull p = {x. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('a) + 1 \<and> x \<in> convex hull S}"
l_cancel_one: fixes x :: "nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "a \<in> carrier Zp" shows "(x \<oplus> a = x) = (a = \<zero>)"
smallI: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<And>c. 0 < c \<Longrightarrow> \<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)" shows "f \<in> \<omega>[F](g)"
eval_wlp_Apply: fixes f :: "'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> real" shows "wlp (Apply f) P = P \<circ> f"
simps_6: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) form"   and q :: "('b, 'c) form" shows "eval e f g (Impl p q) = (eval e f g p \<longrightarrow> eval e f g q)"
assms_2: shows "wf\<^sub>t\<^sub>r\<^sub>m s"
assms_4: shows "\<not> poincare_on_line a x1 x2"
dense_repr_replicate_zero: fixes n :: "nat"   and p :: "'a list" shows "dense_repr (replicate n \<zero> @ p) = dense_repr p"
simps_3: fixes S1 :: "com"   and S2 :: "com"   and P :: "complex mat" shows "wlp (S1;; S2) P = wlp S1 (wlp S2 P)"
simps: fixes P :: "((('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'a set \<times> ('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set \<times> (('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'a set \<times> ('a ltl \<Rightarrow> ('a ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set) set" shows "local.combine_pairs' P = (\<Union> (fst ` P), snd ` P)"
abs_endofun_def: shows "abs_endofun \<equiv> quot_type.abs (=) Abs_endofun"
assms_5: shows "pb \<in> proj_set (f b) G"
abs_mult_sgn: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "times (abs a) (sgn a) = a"
lQuot_compl: fixes a :: "'b"   and A :: "'b list set" shows "lQuot a (- A) = - lQuot a A"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_add_one m lm (l, r) = (\<exists>rn. l = [] \<and> (r = <m # lm> @ Bk \<up> rn \<or> r = Bk # <m # lm> @ Bk \<up> rn))"
fst_conv: fixes x1 :: "'a"   and x2 :: "'d" shows "fst (x1, x2) = x1"
word_diff_ls'_4: fixes y :: "'b word"   and w :: "'b word"   and x :: "'b word" assumes "y \<le> w + x"   and "x \<le> y" shows "y - x \<le> w"
image_mset_is_empty_iff: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset" shows "(image_mset f M = {#}) = (M = {#})"
u_max: fixes x :: "real" shows "u x \<le> u_max"
UN_extend_simps_7: fixes A :: "'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "A - \<Inter> (B ` C) = (\<Union>x\<in>C. A - B x)"
simps_5: fixes A :: "'b"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula" shows "(G \<^bold>\<rightarrow> H)[F/A] = G[F/A] \<^bold>\<rightarrow> H[F/A]"
map_add_empty: fixes m :: "'a \<Rightarrow> 'b option" shows "\<emptyset> \<Oplus> m = m"
comp_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(f \<cdot> g) x = f (g x)"
bprv_eqv_all_not_PPf_imp_\<phi>G: shows "bprv (imp (all yy (neg (PPf (Var yy) \<langle>\<phi>G\<rangle>))) \<phi>G)"
isomorphic_implies_ide_2: fixes f :: "'a"   and g :: "'a" assumes "f \<cong>\<^sub>C g" shows "C.ide g"
A3: shows "Esigma (3\<^sup>2 * 7 * 13) = Esigma (3\<^sup>2 * 7) * Esigma 13"
unif_imp_converse_unifless: fixes x :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and y :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "(x, y) \<in> unif" shows "(y, x) \<in> unifless"
dense_lattice_char_2: shows "(\<forall>x y. x \<sqinter> y = bot \<longrightarrow> x = bot \<or> y = bot) = (\<forall>x. regular x \<longrightarrow> x = bot \<or> x = top)"
Jacobi_eq_0_not_coprime: fixes n :: "int"   and a :: "int" assumes "n \<noteq> 0"   and "\<not> coprime a n" shows "Jacobi a n = 0"
comp_arr_ide: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and f :: "'a" assumes "partial_magma C"   and "partial_magma.ide C a"   and "partial_magma.arr C (C f a)" shows "C f a = f"
frequently_mono: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "\<forall>x. P x \<longrightarrow> Q x"   and "frequently P F" shows "frequently Q F"
lift_def: fixes i :: "nat" shows "lift i \<equiv> rename (lift_map i)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "inj_comm_ring_hom hom" shows "map_poly_inj_comm_ring_hom hom"
dom_fmlookup_finite: fixes m :: "('c, 'd) fmap" shows "finite (dom (fmlookup m))"
weak_ranking_altdef: shows "weak_ranking (\<lambda>_ _. False) = weak_ranking_aux {}"
split_list: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs"
sum_sum_distr: fixes X :: "'b set"   and Y :: "'b set" assumes "finite X"   and "finite Y" shows "\<Sum> ((\<cdot>) (\<Sum> X) ` Y) = \<Sum> {x \<cdot> y |x y. x \<in> X \<and> y \<in> Y}"
lt_set_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set" shows "lt_set F = lt ` (F - {0})"
bourbaki_witt_fixpoint_restrict_rel: fixes leq :: "('a \<times> 'a) set"   and A :: "'a set"   and lub :: "'a set \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "Partial_order leq"   and "\<And>M. \<lbrakk>M \<in> Chains (restrict_rel A leq); M \<noteq> {}\<rbrakk> \<Longrightarrow> lub M \<in> A"   and "\<And>M z. \<lbrakk>M \<in> Chains leq; M \<noteq> {}; \<And>x. x \<in> M \<Longrightarrow> (x, z) \<in> leq\<rbrakk> \<Longrightarrow> (lub M, z) \<in> leq"   and "\<And>M z. \<lbrakk>M \<in> Chains leq; z \<in> M\<rbrakk> \<Longrightarrow> (z, lub M) \<in> leq"   and "\<And>x. \<lbrakk>x \<in> A; x \<in> Field leq\<rbrakk> \<Longrightarrow> (x, f x) \<in> leq \<and> f x \<in> A" shows "bourbaki_witt_fixpoint lub (restrict_rel A leq) f"
array_length_rule': fixes xs :: "'a list"   and n :: "nat"   and p :: "'a dynamic_array" shows "<dyn_array_raw (xs, n) p> array_length p <\<lambda>r. dyn_array_raw (xs, n) p * \<up> (r = n)>"
map_def: fixes f :: "'a" shows "local.map f = (if A.arr f then \<tau> (A.cod f) \<cdot>\<^sub>B \<sigma> f else B.null)"
deformation_retraction_imp_isomorphic_homology_groups: fixes X :: "'a topology"   and Y :: "'b topology"   and r :: "'a \<Rightarrow> 'b"   and s :: "'b \<Rightarrow> 'a"   and p :: "int" assumes "retraction_maps X Y r s"   and "homotopic_with (\<lambda>h. True) X X (s \<circ> r) id" shows "homology_group p X \<cong> homology_group p Y"
quantale_closed_unital_Sup_maps_def: shows "quantale_closed_unital_Sup_maps = {f. (\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y)) \<and> Sup_pres f \<and> (1::'d) \<le> f (1::'c)}"
mult_less_cancel_left_disj: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times c a) (times c b) = (less zero c \<and> less a b \<or> less c zero \<and> less b a)"
deriv: fixes x\<^sub>0 :: "real" assumes "poly p x\<^sub>0 = 0" shows "\<forall>\<^sub>F x in at x\<^sub>0. sgn (poly (p * ps ! 1) x) = (if x\<^sub>0 < x then 1 else - 1)"
ordinal_rec0_0: fixes z :: "'a"   and s :: "ordinal \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" shows "ordinal_rec0 z s l 0 = z"
distrib_right_numeral: fixes a :: "'d"   and b :: "'d"   and v :: "num" shows "(a + b) * numeral v = a * numeral v + b * numeral v"
fisher_yates_alt_def: fixes xs :: "'a list" shows "fisher_yates_alt xs = fisher_yates_alt_aux (length xs - 1) xs"
disjoint_family_on_bisimulation: fixes f :: "'c \<Rightarrow> 'd set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'e set" assumes "disjoint_family_on f S"   and "\<And>n m. \<lbrakk>n \<in> S; m \<in> S; n \<noteq> m; f n \<inter> f m = {}\<rbrakk> \<Longrightarrow> g n \<inter> g m = {}" shows "disjoint_family_on g S"
rel_pmf_bindI: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'c pmf"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd pmf"   and g :: "'c \<Rightarrow> 'e pmf" assumes "rel_pmf R p q"   and "\<And>x y. R x y \<Longrightarrow> rel_pmf P (f x) (g y)" shows "rel_pmf P (p \<bind> f) (q \<bind> g)"