hcju/isabelleps huggingface dataset:
min_lem: fixes r :: "real"   and b :: "(real \<times> real \<times> real) list"   and c :: "(real \<times> real \<times> real) list"   and d :: "(real \<times> real \<times> real) list" assumes "\<exists>y'>r. \<forall>(d, e, f)\<in>set b. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f < 0"   and "\<exists>y'>r. \<forall>(d, e, f)\<in>set c. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"   and "\<exists>y'>r. \<forall>(d, e, f)\<in>set d. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0" shows "\<exists>x. (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)"
ih_t: assumes "wary t" shows "wt (subst \<rho> t) = wt t + extra_wt \<rho> t"
gauss_int_dvd_same_norm_imp_associated: fixes z1 :: "gauss_int"   and z2 :: "gauss_int" assumes "z1 dvd z2"   and "gauss_int_norm z1 = gauss_int_norm z2" shows "normalize z1 = normalize z2"
Ord_not_less: fixes x :: "V"   and y :: "V" assumes "Ord x"   and "Ord y" shows "(\<not> x \<subset>\<^sub>\<circ> y) = (y \<subseteq>\<^sub>\<circ> x)"
EncB_def: fixes SK :: "nat"   and K :: "nat"   and EK :: "nat"   and M :: "msg"   and b :: "msg" shows "EncB SK K EK M b = \<lbrace>Enc SK K EK \<lbrace>M, Hash b\<rbrace>, b\<rbrace>"
seqr_middle: fixes x :: "'b \<Longrightarrow> 'c"   and P :: "('d, 'c) urel"   and Q :: "('c, 'e) urel" assumes "vwb_lens x" shows "P ;; Q = (\<^bold>\<exists> v \<bullet> P\<lbrakk>\<guillemotleft>v\<guillemotright>/$x\<acute>\<rbrakk> ;; Q\<lbrakk>\<guillemotleft>v\<guillemotright>/$x\<rbrakk>)"
r_result1_diverg': fixes i :: "nat"   and x :: "nat"   and t :: "nat" assumes "eval r_phi [i, x] \<up>" shows "eval r_result1 [t, i, x] \<down>= 0"
exhaust_disc: fixes pac_step :: "('a, 'b, 'c) pac_step" obtains "is_Add pac_step"   | "is_Mult pac_step"   | "is_Extension pac_step"   | "is_Del pac_step"
polymul_norm: fixes p :: "'b poly"   and q :: "'b poly" assumes "isnpoly p"   and "isnpoly q" shows "isnpoly (p *\<^sub>p q)"
insertion_powsum_mpoly: fixes f :: "nat \<Rightarrow> 'a"   and X :: "nat set"   and k :: "nat" shows "insertion f (powsum_mpoly X k) = (\<Sum>i\<in>X. f i ^ k)"
init_z0: shows "local.adverse z i j \<triangleright> xmin = local.adverse 0 i j \<triangleright> xmin"
update_simps_1: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "ip \<notin> kD rt" shows "update rt ip r = rt(ip \<mapsto> r)"
eventually_natfloor: fixes P :: "nat \<Rightarrow> bool" assumes "eventually P sequentially" shows "\<forall>\<^sub>F x in at_top. P (nat \<lfloor>x\<rfloor>)"
F_i_analytically_valid: shows "analytically_valid Dy_pair (\<lambda>p. F p \<bullet> i) j"
chain_rel_prodD2: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Y :: "('c \<times> 'd) set" assumes "Complete_Partial_Order.chain (rel_prod orda ordb) Y" shows "Complete_Partial_Order.chain ordb (snd ` Y)"
refinesI: fixes e1 :: "db"   and e2 :: "db" assumes "\<And>C V1. C<e1> \<Down> V1 \<Longrightarrow> \<exists>V2. C<e2> \<Down> V2 \<and> V1 \<sim> V2" shows "e1 \<unlhd> e2"
Collect_on_transfer: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'a set"   and Y :: "'b set" assumes "rel_set R X Y" shows "rel_fun (rel_fun R (=)) (rel_set R) (\<lambda>P. {x \<in> X. P x}) (\<lambda>P. {y \<in> Y. P y})"
assms_3: shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child \<in> set ancestors"
inner_is_valid: fixes n :: "'b LDCFG_node" assumes "CFGExit_wf.inner_node n" shows "CFGExit_wf.valid_node n"
par_skip: fixes par :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and skip :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice" assumes "parallel (par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (skip::'b::refinement_lattice)" shows "(par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (a::'b::refinement_lattice) (skip::'b::refinement_lattice) = a"
monad_state_stateT: assumes "monad return bind" shows "monad_state local.return_state local.bind_state local.get_state local.put_state"
arctan_upper_12_def: shows "arctan_lower_13 \<equiv> \<lambda>x. 3 * x / (x\<^sup>2 + 3)"
preliminary2: fixes H :: "'a set"   and H1 :: "'a set"   and K :: "'a set"   and K1 :: "'a set" assumes "subgroup H G"   and "H1 \<lhd> G\<lparr>carrier := H\<rparr>"   and "subgroup K G"   and "K1 \<lhd> G\<lparr>carrier := K\<rparr>" shows "H1 <#> H \<inter> K1 \<lhd> G\<lparr>carrier := H1 <#> H \<inter> K\<rparr>"
usubst_upd_var_id: fixes x :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x" shows "[x \<mapsto>\<^sub>s x] = id"
invert_side_of_triangle_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "partial_magma.arr C h"   and "C f g = h"   and "category.iso C f" shows "partial_magma.arr C (C (category.inv C f) h) \<and> g = C (category.inv C f) h"
term_cases_def: fixes if_const :: "name \<Rightarrow> 'c"   and if_free :: "name \<Rightarrow> 'c"   and if_app :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and otherwise :: "'c"   and t :: "'d" shows "term_cases if_const if_free if_app otherwise t = (case unconst t of None \<Rightarrow> case unfree t of None \<Rightarrow> case unapp t of None \<Rightarrow> otherwise | Some (t, u) \<Rightarrow> if_app t u | Some name \<Rightarrow> if_free name | Some name \<Rightarrow> if_const name)"
topo2: shows "top_on\<^sub>o (annos C2 ! i) (- vars (strip C2))"
val_of_spec: fixes e :: "'c exp"   and v :: "val" assumes "val_of e = \<lfloor>v\<rfloor>" shows "e = Val v"
vdw_imp_pos: fixes k :: "nat"   and l :: "nat"   and n :: "nat" assumes "vdw k l n"   and "0 < l" shows "0 < n"
GuardSwap: 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 R :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and g :: "'e set"   and f :: "'f" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A"   and "P \<subseteq> g \<inter> R" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Guard f g c Q,A"
distr_distr: fixes g :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and L :: "'d measure"   and f :: "'e \<Rightarrow> 'c"   and M :: "'e measure" assumes "g \<in> N \<rightarrow>\<^sub>M L"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "distr (distr M N f) L g = distr M L (g \<circ> f)"
finfun_snd_update: fixes f :: "('a, 'c \<times> 'b) finfun"   and a :: "'a"   and bc :: "'c \<times> 'b" shows "finfun_snd (finfun_update f a bc) = finfun_update (finfun_snd f) a (snd bc)"
nodes_of_phis_finite: fixes g :: "'g"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping"   and v :: "'val"   and ns :: "('node \<times> 'val) set" assumes "g \<turnstile> nodes_of_phis \<approx>\<^sub>\<phi> uninst_code.ssa.phiNodes_of phis g"   and "Mapping.lookup nodes_of_phis v = Some ns"   and "v \<in> snd ` dom (Mapping.lookup (phis g))" shows "finite ns"
wfp_def: fixes p :: "prog" shows "wfp p \<equiv> distinct (vars p) \<and> wfs (ctxt p) (body p)"
simps_1: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('d \<times> 'e) spmf"   and p :: "'e spmf"   and x :: "'f" shows "trace_callee callee p [] x = p \<bind> (\<lambda>s. map_spmf fst (callee s x))"
remove1_idem: fixes x :: "'g"   and xs :: "'g list" assumes "x \<notin> set xs" shows "remove1 x xs = xs"
obj_at_field_on_heapE: fixes P :: "'e \<Rightarrow> bool"   and r :: "'e"   and f :: "'f"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and s' :: "'i process_name \<Rightarrow> ('f, 'i, 'h, 'e) Model.local_state"   and P' :: "'e \<Rightarrow> bool" assumes "obj_at_field_on_heap P r f s"   and "sys_heap s' r = sys_heap s r"   and "\<And>r'. P r' \<Longrightarrow> P' r'" shows "obj_at_field_on_heap P' r f s'"
step: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
has_derivative_fst: fixes g :: "'c \<Rightarrow> 'd \<times> 'e"   and g' :: "'c \<Rightarrow> 'd \<times> 'e"   and F :: "'c filter" assumes "(g has_derivative g') F" shows "((\<lambda>x. fst (g x)) has_derivative (\<lambda>x. fst (g' x))) F"
fst_apfst: fixes f :: "'d \<Rightarrow> 'b"   and x :: "'d \<times> 'c" shows "fst (apfst f x) = f (fst x)"
hyps_5: fixes x :: "'a" assumes "x \<in> space borel" shows "(\<lambda>i. U i x) \<longlonglongrightarrow> f x"
inj_mod_subgroup: fixes f :: "'c \<Rightarrow> 'd"   and H :: "'c set"   and G :: "('c, 'e) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "weak_group_morphism f H G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "(f a = f b) = (a \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> b \<in> H)"
lih'_finite_reachable: shows "finite lih'.sa.reachable"
coinduct: fixes X :: "('sequent \<times> 'rule) tree \<Rightarrow> bool"   and x :: "('sequent \<times> 'rule) tree" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>t. x = t \<and> snd (tree.root t) \<in> R \<and> effStep (tree.root t) ((fst \<circ> tree.root) |`| cont t) \<and> (\<forall>x. x |\<in>| cont t \<longrightarrow> X x \<or> local.wf x)" shows "local.wf x"
roots1: fixes p :: "'a poly" assumes "degree p = 1" shows "{x. poly p x = (0::'a)} = {roots1 p}"
ind_sheaf_axioms: shows "Comm_Ring.ind_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str U"
star_inductl_var: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<cdot> z \<le> y"
integrable_circle: fixes n :: "nat" shows "f s contour_integrable_on part_circlepath 0 (R n) 0 pi"
updEnv_overwrite: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" shows "rho [x \<leftarrow> X]_xs [x \<leftarrow> X']_xs = rho [x \<leftarrow> X']_xs"
fps_radical0: fixes r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a fps" shows "fps_radical r 0 a = 1"
le_Limsup: fixes F :: "'a filter"   and l :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "F \<noteq> bot"   and "\<forall>\<^sub>F x in F. l \<le> f x" shows "l \<le> Limsup F f"
ord_of_nat_plus: fixes a :: "nat"   and b :: "nat" shows "a\<^sub>\<nat> + b\<^sub>\<nat> = (a + b)\<^sub>\<nat>"
subsetCE: fixes A :: "'h set"   and B :: "'h set"   and c :: "'h" assumes "A \<subseteq> B" obtains "c \<notin> A"   | "c \<in> B"
is_BoolE: fixes v :: "'a val" assumes "is_Bool v" obtains a :: "bool" where "v = Bool a"
vconn_implies_hpar_2: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "S.in_hom \<mu> f g" shows "S.trg \<mu> = S.trg f"
approx_mult_subst_star_of: fixes u :: "'a star"   and x :: "'a star"   and v :: "'a"   and y :: "'a star" assumes "u \<approx> x * star_of v"   and "x \<approx> y" shows "u \<approx> y * star_of v"
keys_monom_mult_subset: fixes c :: "'d"   and t :: "'c"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys (punit.monom_mult c t p) \<subseteq> punit.splus t ` keys p"
C: shows "(xs @ ys @ zs, X) \<in> failures P"
simps_1: fixes h :: "nat"   and n_ops :: "nat"   and t :: "nat"   and k :: "nat" shows "cnf_to_dimacs.valid_state_var h n_ops (Operator t k) = (t < h \<and> k < n_ops)"
assms_12: fixes v :: "int poly" assumes "v \<in> set vs" shows "Mp v = v"
null_sets_completion_iff: fixes N :: "'a set"   and M :: "'a measure" assumes "N \<in> sets M" shows "(N \<in> null_sets (completion M)) = (N \<in> null_sets M)"
polychain_of_segments_of_aform1: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "length (segments_of_aform X) = 1" shows "False"
IH: shows "\<forall>var L. freeIn var \<phi> \<longrightarrow> (\<exists>x. eval \<phi> (L[var := x])) = (\<forall>x. eval \<phi> (L[var := x]))"
count_image_mset: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d multiset"   and x :: "'c" shows "count (image_mset f A) x = sum (count A) (f -` {x} \<inter> set_mset A)"
sl9: shows "b\<^sup>2 * e\<^sup>2 \<in> radical_sqrt"
WTrt_elim_cases_7: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and F :: "char list"   and D :: "char list"   and T :: "ty" assumes "P,E,h \<turnstile> e\<bullet>F{D} : T" obtains C :: "char list" where "P,E,h \<turnstile> e : Class C" and "P \<turnstile> C has F:T in D"   | "P,E,h \<turnstile> e : NT"
in_these_eq: fixes x :: "'c"   and A :: "'c option set" shows "(x \<in> Option.these A) = (Some x \<in> A)"
sees_method_sub_Obj: fixes P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> C \<preceq>\<^sup>* Object"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "\<langle>'b\<rangle>\<^sub>\<bottom>" shows "(case option of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (if option = \<bottom> then f1 else f2 (the option))"
perm_distinct_iff: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs <~~> ys" shows "distinct xs = distinct ys"
euclidean_eqI: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and x :: "'c"   and y :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "\<And>b. b \<in> Basis \<Longrightarrow> inner x b = inner y b" shows "x = y"
assms_1: shows "\<Psi> \<rhd> P \<doteq> Q"
finfun_Diag_apply: fixes f :: "('a, 'b) finfun"   and g :: "('a, 'c) finfun" shows "finfun_apply (finfun_Diag f g) = (\<lambda>x. (finfun_apply f x, finfun_apply g x))"
seqr_left_unit: fixes P :: "('b, 'c) urel" shows "II ;; P = P"
contractible_space_alt: fixes X :: "'a topology" shows "contractible_space X = (\<forall>a\<in>topspace X. homotopic_with (\<lambda>x. True) X X id (\<lambda>x. a))"
fmrelI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and n :: "('c, 'e) fmap" assumes "\<And>x. rel_option R (fmlookup m x) (fmlookup n x)" shows "fmrel R m n"
eheight_exp_def: fixes n :: "nat" shows "eheight_exp n = measure_pmf.expectation (eheight_rbst n) real"
l_cancel: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c + a = c + b" shows "a = b"
assms: shows "<\<nu>x>\<tau>.P \<longmapsto> a[b] \<prec> P'"
mult1_def_lemma: fixes M :: "'c multiset"   and M0 :: "'c multiset"   and a :: "'c"   and N :: "'c multiset"   and K :: "'c multiset"   and r :: "('c \<times> 'c) set" assumes "M = M0 + {#a#} \<and> N = M0 + K \<and> (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)" shows "(N, M) \<in> mult1 r"
arr_map: fixes gf :: "'c \<times> 'c" assumes "CopxC.arr gf" shows "S.arr (local.map gf)"
subgroup_in_rcosets: assumes "Group.group G" shows "H \<in> rcosets H"
indexed_eval_aux_monomial_degree_le: fixes Ps :: "('d multiset \<Rightarrow> 'a) list"   and i :: "'d"   and m :: "'d multiset" assumes "list_all carrier_coeff Ps"   and "list_all (\<lambda>P. index_free P i) Ps"   and "indexed_eval_aux Ps i m \<noteq> \<zero>" shows "count m i \<le> degree Ps"
integral_swap_2dim: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "continuous_on (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y)" shows "integral (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y) = integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)"
lbvc_axioms: shows "lbvc A r f \<top> \<bottom> step \<tau>s"
lprod_subset: fixes S :: "('b \<times> 'b) set"   and R :: "('b \<times> 'b) set" assumes "S \<subseteq> R" shows "lprod S \<subseteq> lprod R"
homeomorphic_translation: fixes S :: "'b set"   and a :: "'b" shows "S homeomorphic (+) a ` S"
intros_2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "r x y"   and "star r y z" shows "star r x z"
power_mult_distrib: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d"   and n :: "nat" assumes "class.comm_monoid_mult times one" shows "power.power one times (times a b) n = times (power.power one times a n) (power.power one times b n)"
Gpq: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "GAUSS p (int q)"
lm017: fixes G :: "'c set"   and N :: "'d set" shows "injectionsUniverse \<inter> {a. Domain a \<in> all_partitions G \<and> Range a \<subseteq> N} \<subseteq> possible_allocations_rel G N"
InvTempUpright_def: fixes c :: "('c, 'd, 'e) configuration_scheme" shows "InvTempUpright c = (\<forall>p. upright (c_temp c p))"
list_all2_flip: fixes P :: "'h \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list" shows "list_all2 (flip P) xs ys = list_all2 P ys xs"
dghm_singleton_ObjMap_vsv: fixes j :: "V"   and \<CC> :: "V" shows "vsv (dghm_singleton j \<CC>\<lparr>ObjMap\<rparr>)"
fps_divide_times: fixes h :: "'a fps"   and g :: "'a fps"   and f :: "'a fps" assumes "subdegree h \<le> subdegree g" shows "f * g / h = f * (g / h)"
hcomp_trg_ide: fixes f :: "('a E.term, 'a) arr" assumes "ide f" shows "trg f \<star> f = f"
cnjE: fixes x :: "complex" assumes "x \<noteq> 0" shows "cnj x = cor ((cmod x)\<^sup>2) / x"
pochhammer_prod_rev: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a n = comm_monoid_mult.prod times one (\<lambda>i. plus a (semiring_1.of_nat one plus zero (n - i))) {1..n}"
ord_isom_Pre1: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and a :: "'c"   and f :: "'c \<Rightarrow> 'e" assumes "Worder D"   and "Worder E"   and "a \<in> carrier D"   and "ExPre D a"   and "ord_isom D E f" shows "ExPre E (f a)"
orelse_thms_4: fixes y :: "tr" shows "(y orelse FF) = y"
bound_intersect_2d_eq_None: fixes prec :: "nat"   and Xs :: "((real \<times> real) \<times> real \<times> real) list"   and x :: "real"   and X :: "(real \<times> real) \<times> real \<times> real" assumes "bound_intersect_2d prec Xs x = None"   and "X \<in> set Xs" shows "intersect_segment_xline prec X x = None"
reducible_def: fixes g :: "'g" shows "old.reducible g \<equiv> \<forall>n ns. g \<turnstile> n-ns\<rightarrow>n \<longrightarrow> (\<exists>m\<in>set ns. \<forall>n\<in>set ns. old.dominates g m n)"
a_type_wf_def: fixes h :: "(_) heap" shows "DocumentClass.type_wf h = (CharacterDataClass.type_wf h \<and> (\<forall>document_ptr\<in>fset (document_ptr_kinds h). get document_ptr h \<noteq> None))"
simps: fixes x :: "'b"   and l :: "'b tree"   and uu :: "'b"   and r :: "'b tree" shows "replace_min x \<langle>l, uu, r\<rangle> = (if le_root x l \<and> le_root x r then \<langle>l, x, r\<rangle> else let a = value l in if le_root a r then \<langle>replace_min x l, a, r\<rangle> else \<langle>l, value r, replace_min x r\<rangle>)"
quantify_nfa_well_formed_aut: fixes N :: "bool list bdd list \<times> bool list"   and n :: "nat"   and v :: "nat" assumes "wf_nfa N (Suc n)"   and "v \<le> n" shows "wf_nfa (quantify_nfa v N) n"
simps_3: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_sat_list (IF v t e) = (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))"
tdghm_NTMap_is_arr: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<GG>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"
correctCompositionDiffLevelsA82: shows "correctCompositionDiffLevels sA82"
cong: fixes a :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "(a, a') \<in> Ring_Congruence"   and "(b, b') \<in> Ring_Congruence" shows "(a \<cdot> b, a' \<cdot> b') \<in> Ring_Congruence"
assms_7: shows "3 \<le> rk {A, B, C, D}"
fmeasurableD: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> fmeasurable M" shows "A \<in> sets M"
test_mult_comp: fixes x :: "'a" shows "at x \<cdot> at (at x) = (0::'a)"
neg_cn: fixes P :: "pol" assumes "isnorm P" shows "isnorm (neg P)"
finite_interleavings: fixes xs :: "'a list"   and ys :: "'a list" shows "finite {zs. interleavings xs ys zs}"
ge_ge_tpoly_trans: fixes r :: "('v pvar, hmultiset) tpoly"   and q :: "('v pvar, hmultiset) tpoly"   and p :: "('v pvar, hmultiset) tpoly" assumes "r \<ge>\<^sub>p q"   and "q \<ge>\<^sub>p p" shows "r \<ge>\<^sub>p p"
qAbs_alphaAbs_qSwap_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "qGood X"   and "qFresh xs x' X" shows "qAbs xs x X $= qAbs xs x' (X #[[x' \<and> x]]_xs)"
preallocatedD: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and C :: "char list" assumes "preallocated h"   and "C \<in> sys_xcpts" shows "\<exists>fs. h (addr_of_sys_xcpt C) = \<lfloor>(C, fs)\<rfloor>"
hom_restrict0_freeword_funlift: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "GroupHom (FreeGroup S) (res_freeword_funlift f S)"
elims_2: fixes x :: "('b \<times> bal) tree" assumes "avl x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> bal) tree" and a :: "'b" and b :: "bal" and r :: "('b \<times> bal) tree" where "x = \<langle>l, (a, b), r\<rangle>" and "(case b of Lh \<Rightarrow> height l = height r + 1 | Bal \<Rightarrow> height r = height l | Rh \<Rightarrow> height r = height l + 1) \<and> avl l \<and> avl r"
maxsimpchainD_maxsimp: fixes xs :: "'a set list"   and x :: "'a set" assumes "gallery xs"   and "x \<in> set xs" shows "chamber x"
eigen_vector_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and v :: "('d, 'e) Finite_Cartesian_Product.vec"   and ev :: "'d" shows "eigen_vector A v ev = (v \<noteq> 0 \<and> A *v v = ev *s v)"
match_ex_table_stack_xlift_eq_Some_conv: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and n :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and pc' :: "nat"   and d :: "nat" shows "(match_ex_table P C pc (stack_xlift n xt) = \<lfloor>(pc', d)\<rfloor>) = (n \<le> d \<and> match_ex_table P C pc xt = \<lfloor>(pc', d - n)\<rfloor>)"
interv_is_compl_latt: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "interval r a b \<noteq> {}" shows "\<lparr>pset = interval r a b, order = induced (interval r a b) r\<rparr> \<in> CompleteLattice"
degree_sum_le: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c poly"   and n :: "nat" assumes "\<And>x. x \<in> A \<Longrightarrow> degree (f x) \<le> n" shows "degree (sum f A) \<le> n"