hcju/isabelleps huggingface dataset:
span_zero: fixes S :: "'d set" shows "(0::'d) \<in> span S"
distinct_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and x1 :: "'c"   and x21 :: "'c"   and x22 :: "'c" assumes "antiparallel_edges \<Delta>" shows "Vertex x1 \<noteq> Edge x21 x22"
FG_advice_minus_size: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and Y :: "'b ltln set" assumes "size \<phi> \<le> size \<psi>"   and "\<phi> \<noteq> \<psi>" shows "\<phi>[Y - {\<psi>}]\<^sub>\<mu> = \<phi>[Y]\<^sub>\<mu>"
card_atLeastAtMost_word: fixes s :: "'b word"   and e :: "'b word" shows "card {s..e} = Suc (unat e) - unat s"
rep_eq: fixes x :: "'b uprod" shows "set_uprod x = (case rep_uprod x of (a, b) \<Rightarrow> {a, b})"
list_of_pdevs_nonzero: fixes x :: "'a"   and xs :: "'a pdevs" assumes "x \<in> set (map snd (list_of_pdevs xs))" shows "x \<noteq> (0::'a)"
derivg_nonneg: fixes x :: "real" assumes "x \<in> {a..b}" shows "0 \<le> g' x"
composable: shows "src (\<mu> \<star> \<nu>) = trg \<pi>"
step: fixes F :: "'b \<Rightarrow> ennreal"   and s :: "'a" assumes "F \<in> borel_measurable N" shows "integral\<^sup>N (M s) (f F) = g (\<lambda>s. integral\<^sup>N (M s) F) s"
assms_1: shows "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"
at_within_eq_bot_iff: fixes c :: "'a"   and A :: "'a set" shows "(at c within A = bot) = (c \<notin> closure (A - {c}))"
assms: shows "cs\<^bsup>\<pi>\<^esup> i = xs @ [x] @ ys @ [\<pi> i]"
induct: fixes x1 :: "fm set"   and x2 :: "fm"   and P :: "fm set \<Rightarrow> fm \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>A H. A \<in> H \<Longrightarrow> P H A"   and "\<And>H. P H extra_axiom"   and "\<And>A H. A \<in> boolean_axioms \<Longrightarrow> P H A"   and "\<And>A H. A \<in> equality_axioms \<Longrightarrow> P H A"   and "\<And>A H. A \<in> special_axioms \<Longrightarrow> P H A"   and "\<And>A H. A \<in> HF_axioms \<Longrightarrow> P H A"   and "\<And>A H. A \<in> induction_axioms \<Longrightarrow> P H A"   and "\<And>H A B H'. \<lbrakk>H \<turnstile> A IMP B; P H (A IMP B); H' \<turnstile> A; P H' A\<rbrakk> \<Longrightarrow> P (H \<union> H') B"   and "\<And>H A B i. \<lbrakk>H \<turnstile> A IMP B; P H (A IMP B); atom i \<sharp> B; \<forall>C\<in>H. atom i \<sharp> C\<rbrakk> \<Longrightarrow> P H (SyntaxN.Ex i A IMP B)" shows "P x1 x2"
rel_state_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "(heap, 'e) state"   and g :: "'f Heap" shows "local.rel_state R f g \<equiv> \<forall>heap. P heap \<longrightarrow> (case run_state f heap of (v1, heap1) \<Rightarrow> case execute g heap of None \<Rightarrow> False | Some (v2, heap2) \<Rightarrow> R v1 v2 \<and> heap1 = heap2 \<and> P heap2)"
mbisim_simulation_silent1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s1' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set" assumes "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "\<tau>trsys.silent_move (multithreaded_base.redT final1 r1 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move1) s1 s1'"   and "FWbisimulation_base.mbisim bisim bisim_wait s1 s2" shows "\<exists>s2'. (\<tau>trsys.silent_move (multithreaded_base.redT final2 r2 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move2))\<^sup>*\<^sup>* s2 s2' \<and> FWbisimulation_base.mbisim bisim bisim_wait s1' s2'"
eventually_frequently: fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "F \<noteq> bot"   and "eventually P F" shows "frequently P F"
semiring_simprules_3: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
plus_absorb1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "\<Theta>[F](\<lambda>x. f x + g x) = \<Theta>[F](g)"
sum_bounded_above_strict: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and K :: "'c" assumes "\<And>i. i \<in> A \<Longrightarrow> f i < K"   and "0 < card A" shows "sum f A < of_nat (card A) * K"
assms_2: shows "0 \<le> b\<^sup>2 - 4 * a * c"
assms_2: shows "z \<noteq> 1 / cnj a"
act: shows "jAction (SPR.MCn (tLength t)) t a = [act]"
is_SNF_Smith_mxn: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m n" shows "is_SNF A (Smith_ED_mxn A)"
splus_left_commute: fixes s :: "'a"   and t :: "'a"   and v :: "'a" shows "local.punit.splus s (local.punit.splus t v) = local.punit.splus t (local.punit.splus s v)"
splus_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.splus pair_of_term term_of_pair t v = term_of_pair (t + term_powerprod.pp_of_term pair_of_term v, term_powerprod.component_of_term pair_of_term v)"
naturality: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "(fst (C.VV.cod f) \<star>\<^sub>C snd (C.VV.cod f)) \<cdot>\<^sub>C (fst f \<star>\<^sub>C snd f) = (fst f \<star>\<^sub>C snd f) \<cdot>\<^sub>C (fst (C.VV.dom f) \<star>\<^sub>C snd (C.VV.dom f))"
type_definition_action: shows "type_definition action.relation_of action_of {p. is_CSP_process p}"
card_fset_Suc_case: fixes S :: "'a fset"   and T :: "'a fset" assumes "Suc (fcard S) = fcard T"   and "P S" shows "P T"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c"   and list :: "'d list" shows "P (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (\<not> (list = [] \<and> \<not> P f1 \<or> (\<exists>x21 x22. list = x21 # x22 \<and> \<not> P (f2 x21 x22))))"
weak_greatest_unique: fixes x :: "'a"   and A :: "'a set"   and y :: "'a" assumes "greatest L x A"   and "greatest L y A" shows "x .= y"
set_restriction_mono: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set"   and B :: "'a set" assumes "set_restriction f"   and "A \<subseteq> B" shows "f A \<subseteq> f B"
reduced_GB_singleton: fixes f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "punit.reduced_GB {f} = (if f = 0 then {} else {punit.monic f})"
contour_integral_unique: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and g :: "real \<Rightarrow> complex" assumes "(f has_contour_integral i) g" shows "contour_integral g f = i"
hyps_5: shows "\<exists>DD\<subseteq>CC. finite DD \<and> D \<in> saturate DD"
2: shows "cmod \<tau> = cmod (gauss_sum 1) / \<bar>sqrt (real n)\<bar>"
find_base_vectors_5: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "row_echelon_form A"   and "A \<in> carrier_mat nr nc" shows "length (pivot_positions A) = card {i. i < nr \<and> row A i \<noteq> 0\<^sub>v nc}"
dim_row_polys_to_mat: fixes ts :: "'d list"   and ps :: "('d \<Rightarrow>\<^sub>0 'c) list" shows "dim_row (polys_to_mat ts ps) = length ps"
split: fixes P :: "'e \<Rightarrow> bool"   and f1 :: "'e"   and f2 :: "nat \<Rightarrow> 'e"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. nat = Suc x2 \<longrightarrow> P (f2 x2)))"
rel_gpv''_mono: fixes A :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and A' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and C :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and C' :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and R' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and R :: "'n \<Rightarrow> 'o \<Rightarrow> bool" assumes "A \<le> A'"   and "C \<le> C'"   and "R' \<le> R" shows "rel_gpv'' A C R \<le> rel_gpv'' A' C' R'"
of_mds_tyenv_wellformed: fixes m :: "Mode \<Rightarrow> 'Var set" assumes "mds_yields_stable_types m" shows "tyenv_wellformed m (\<Gamma>_of_mds m) (\<S>_of_mds m) {}"
axioms_1: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and a :: "'c"   and \<pi> :: "'b \<Rightarrow> 'c" assumes "product_cone J C D a \<pi>" shows "category J"
nn_integral_count_space_eq: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = 0"   and "\<And>x. x \<in> B - A \<Longrightarrow> f x = 0" shows "integral\<^sup>N (count_space A) f = integral\<^sup>N (count_space B) f"
terminatesD: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and s :: "'a" assumes "terminates g" shows "s \<in> terminates_on g"
select_convs_4: fixes types :: "tf list"   and funcs :: "nat list"   and tab :: "nat option"   and mem :: "nat option"   and globs :: "nat list"   and more :: "'a" shows "inst.mem \<lparr>types = types, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = mem"
fact_diff_Suc: fixes n :: "nat"   and m :: "nat" assumes "n < Suc m" shows "fact (Suc m - n) = of_nat (Suc m - n) * fact (m - n)"
second_isomorphism_grp_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and S :: "'c set" shows "second_isomorphism_grp_axioms G S \<equiv> subgroup S G"
smult_mult2: fixes a :: "'b"   and A :: "('b, 'c) sq_matrix"   and B :: "('b, 'c) sq_matrix" shows "a *\<^sub>S (A * B) = A * a *\<^sub>S B"
subc_psubst: fixes p :: "('a, 'b) form"   and c :: "'a"   and f :: "'a \<Rightarrow> 'c"   and s :: "'a FOL_Fitting.term" assumes "\<forall>x\<in>params p. x \<noteq> c \<longrightarrow> f x \<noteq> f c" shows "psubst f (subc c s p) = subc (f c) (psubstt f s) (psubst f p)"
code: shows "0 = Octo 0 0 0 0 0 0 0 0"
power_le_mono: fixes n :: "nat"   and m :: "nat" assumes "2 ^ n \<le> 2 ^ m"   and "n < LENGTH('b)"   and "m < LENGTH('b)" shows "n \<le> m"
induct: fixes P1 :: "('b \<Rightarrow> ('c, 'd, 'e) formula) \<Rightarrow> ('c, 'd + 'b, 'e) hp \<Rightarrow> bool"   and P2 :: "('b \<Rightarrow> ('c, 'd, 'e) formula) \<Rightarrow> ('c, 'd + 'b, 'e) formula \<Rightarrow> bool"   and x1 :: "'b \<Rightarrow> ('c, 'd, 'e) formula"   and x2 :: "('c, 'd + 'b, 'e) hp"   and x3 :: "'b \<Rightarrow> ('c, 'd, 'e) formula"   and x4 :: "('c, 'd + 'b, 'e) formula" assumes "\<And>\<sigma> a. P1 \<sigma> ($\<alpha> a)"   and "\<And>\<sigma> a b. \<lbrakk>PPadmit \<sigma> a; P1 \<sigma> a; PPadmit \<sigma> b; P1 \<sigma> b; PPUadmit \<sigma> b (BVP (PPsubst a \<sigma>)); hpsafe (PPsubst a \<sigma>)\<rbrakk> \<Longrightarrow> P1 \<sigma> (a ;; b)"   and "\<And>\<sigma> a. \<lbrakk>PPadmit \<sigma> a; P1 \<sigma> a; PPUadmit \<sigma> a (BVP (PPsubst a \<sigma>)); hpsafe (PPsubst a \<sigma>)\<rbrakk> \<Longrightarrow> P1 \<sigma> a**"   and "\<And>\<sigma> \<phi> ODE. \<lbrakk>PFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; PFUadmit \<sigma> \<phi> (BVO ODE)\<rbrakk> \<Longrightarrow> P1 \<sigma> (EvolveODE ODE \<phi>)"   and "\<And>\<sigma> a b. \<lbrakk>PPadmit \<sigma> a; P1 \<sigma> a; PPadmit \<sigma> b; P1 \<sigma> b\<rbrakk> \<Longrightarrow> P1 \<sigma> (a \<union>\<union> b)"   and "\<And>\<sigma> x \<theta>. P1 \<sigma> (x := \<theta>)"   and "\<And>\<sigma> x \<theta>. P1 \<sigma> (DiffAssign x \<theta>)"   and "\<And>\<sigma> \<phi>. \<lbrakk>PFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>\<rbrakk> \<Longrightarrow> P1 \<sigma> (? \<phi>)"   and "\<And>\<sigma> \<theta>1 \<theta>2. P2 \<sigma> (Geq \<theta>1 \<theta>2)"   and "\<And>\<sigma> f args. P2 \<sigma> ($\<phi> f args)"   and "\<And>\<sigma> \<phi>. \<lbrakk>PFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>\<rbrakk> \<Longrightarrow> P2 \<sigma> (! \<phi>)"   and "\<And>\<sigma> \<phi> \<psi>. \<lbrakk>PFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; PFadmit \<sigma> \<psi>; P2 \<sigma> \<psi>\<rbrakk> \<Longrightarrow> P2 \<sigma> (\<phi> && \<psi>)"   and "\<And>\<sigma> \<phi> x. \<lbrakk>PFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; PFUadmit \<sigma> \<phi> {Inl x}\<rbrakk> \<Longrightarrow> P2 \<sigma> (Exists x \<phi>)"   and "\<And>\<sigma> \<phi> a. \<lbrakk>PFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; PPadmit \<sigma> a; P1 \<sigma> a; PFUadmit \<sigma> \<phi> (BVP (PPsubst a \<sigma>))\<rbrakk> \<Longrightarrow> P2 \<sigma> (\<langle> a \<rangle> \<phi>)"   and "\<And>\<sigma> \<phi> C. \<lbrakk>PFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; PFUadmit \<sigma> \<phi> UNIV\<rbrakk> \<Longrightarrow> P2 \<sigma> (InContext C \<phi>)" shows "(PPadmit x1 x2 \<longrightarrow> P1 x1 x2) \<and> (PFadmit x3 x4 \<longrightarrow> P2 x3 x4)"
f: fixes n :: "nat" assumes "1 \<le> n" shows "cmod (f n) \<le> C"
fold_mset_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b"   and M :: "'c multiset" shows "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)"
nprv_neg_fls: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F (neg \<phi>)"   and "nprv F \<phi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F fls"
c_R: fixes n :: "nat" assumes "N \<le> n" shows "real_of_int (c (Suc n)) / real_of_int (a n) = R n"
iLast_iEx_iff: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLast t0 I P = (\<diamond> t [\<dots>0] \<oplus> iprev t0 I. P t)"
inv_equality: fixes y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "y \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub>"   and "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"   and "y \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x = y"
A4: shows "\<forall>(P, Q)\<in>RelB. CondB (P, Q)"
f1: shows "\<langle>col_fst (U * |u\<rangle>)|col_fst (U * |v\<rangle>)\<rangle> = (\<langle>|u\<rangle>| * U\<^sup>\<dagger> * U * |v\<rangle>) $$ (0, 0)"
fd_iso2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows " |x\<rangle> z \<le>  |y\<rangle> z"
3: fixes a :: "'a"   and b :: "'a"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'a" assumes "P a b I f"   and "finite I"   and "i \<notin> I" shows "P a b (insert i I) f"
override_on_emptyset: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "override_on f g {} = f"
arr_cod: fixes f :: "'a" assumes "hom.arr f" shows "hom.arr (hom.cod f)"
abcd_dcba_only_22: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c d]]" shows "\<not> [[d c a b]]"
iT_Plus_neg_Min: fixes I :: "nat set"   and k :: "nat" assumes "I \<oplus>- k \<noteq> {}" shows "iMin (I \<oplus>- k) = iMin (I \<down>\<ge> k) - k"
conf_clinit_Cons: fixes P :: "jvm_prog"   and sh :: "sheap"   and f :: "frame"   and frs :: "frame list" assumes "conf_clinit P sh (f # frs)" shows "conf_clinit P sh frs"
set: fixes x1 :: "'c match_expr"   and x2 :: "action" shows "set_rule (Rule x1 x2) = set_match_expr x1"
deduct_normalize: fixes M :: "('fun, 'var) Term.term set"   and P :: "'fun \<Rightarrow> ('fun, 'var) Term.term list \<Rightarrow> bool"   and k :: "nat"   and t :: "('fun, 'var) Term.term"   and f :: "'fun"   and T :: "('fun, 'var) Term.term list" assumes "\<forall>m\<in>M. \<forall>f T. Fun f T \<sqsubseteq> m \<longrightarrow> P f T"   and "\<langle>M; k\<rangle> \<turnstile>\<^sub>n t"   and "Fun f T \<sqsubseteq> t"   and "\<not> P f T" shows "\<exists>l\<le>k. \<langle>M; l\<rangle> \<turnstile>\<^sub>n Fun f T \<and> (\<forall>t\<in>set T. \<exists>j<l. \<langle>M; j\<rangle> \<turnstile>\<^sub>n t)"
supp_Dummy: fixes v :: "Vars.var" shows "supp (Dummy v) = supp v"
proj_ik\<^sub>s\<^sub>t_is_proj_rcv_set: fixes n :: "'f"   and A :: "('f strand_label \<times> ('d, 'e) strand_step) list" shows "ik\<^sub>s\<^sub>t (proj_unl n A) = {t. (ln n, receive\<langle>t\<rangle>\<^sub>s\<^sub>t) \<in> set A \<or> (\<star>, receive\<langle>t\<rangle>\<^sub>s\<^sub>t) \<in> set A}"
partial_fraction_decomposition: fixes ys :: "('a poly \<times> nat) list"   and x :: "'a poly"   and a :: "'a poly"   and zs :: "'a poly list list" assumes "\<And>y. y \<in> fst ` set ys \<Longrightarrow> is_unit (fps_of_poly y)"   and "pairwise coprime (set (map (\<lambda>(x, n). x ^ Suc n) ys))"   and "distinct (map (\<lambda>(x, n). x ^ Suc n) ys)"   and "partial_fraction_decomposition x ys = (a, zs)" shows "fps_of_poly a + (\<Sum>i<length ys. \<Sum>j\<le>snd (ys ! i). from_decomp (zs ! i ! j) (fst (ys ! i)) (snd (ys ! i) + 1 - j)) = fps_of_poly x / fps_of_poly (\<Prod>(x, n)\<leftarrow>ys. x ^ Suc n)"
distinct_disc_16: fixes event :: "('d, 'e, 'f) event" assumes "isSnapshot event" shows "\<not> isRecv event"
ih: fixes M :: "'a multiset" assumes "k = size M" shows "P M"
c_iop_welldef: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and a :: "'e" assumes "Group G"   and "G \<triangleright> N"   and "a \<in> carrier G" shows "c_iop G N (N \<bullet>\<^bsub>G\<^esub> a) = N \<bullet>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> a"
absn_beta_equiv: fixes n :: "nat"   and s :: "dB"   and t :: "dB" shows "(Abs ^^ Suc n) s \<degree> t \<leftrightarrow> (Abs ^^ n) (s[liftn n t 0/n])"
is_sum2sq_int_nat_eq: fixes n :: "nat" shows "is_sum2sq_nat n = is_sum2sq_int (int n)"
cons_delete_list_eq2: fixes a :: "'b"   and A :: "'b list" shows "a # removeAll a A \<approx>2 if List.member A a then A else a # A"
subgroup_of_restricted_group: fixes U :: "'a set" assumes "subgroup U (G\<lparr>carrier := H\<rparr>)" shows "U \<subseteq> H"
compact_cball: fixes x :: "'b"   and e :: "real" shows "compact (cball x e)"
Map_ide_1: fixes a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.ide a" shows "Cop_S.Map a = Cop_S.Dom a"
rel_pred_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_pred R = R ===> (=)"
split_n_conv_app: fixes es :: "v list"   and n :: "nat"   and es1 :: "v list"   and es2 :: "v list" assumes "split_n es n = (es1, es2)" shows "es = es1 @ es2"
dom_map_option: fixes f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and m :: "'c \<Rightarrow> 'e option" shows "dom (\<lambda>k. map_option (f k) (m k)) = dom m"
"2_4": shows "(poincare_map p2 (poincare_map p1 x0), D2 o\<^sub>L (D1 o\<^sub>L d0)) \<in> P2"
inter_closed: fixes S :: "'a set set" assumes "\<And>x. x \<in> S \<Longrightarrow> x closed" shows "\<Inter> S closed"
zip_with_prev_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "zip_with_prev f x xs = map2 f (x # xs) xs"
runit_in_hom_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.arr V (prebicategory_with_homs_and_units.runit V H \<a> \<i> src f) \<and> src f \<in> weak_composition.sources V H (prebicategory_with_homs_and_units.runit V H \<a> \<i> src f) \<and> trg f \<in> weak_composition.targets V H (prebicategory_with_homs_and_units.runit V H \<a> \<i> src f)"
sup_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "sup f g = (\<lambda>x. sup (f x) (g x))"
Inf_pres_downset_var: fixes X :: "'a set" shows "\<Inter> (ord.downset (\<lambda>x y. y \<le> x) ` X) = ord.downset (\<lambda>x y. y \<le> x) (\<Squnion> X)"
choose_strategy: fixes v :: "'a" assumes "v \<in> S - W" shows "strategy p (choose v)"
get_ancestors_si_never_empty: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors" shows "ancestors \<noteq> []"
simps: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and a :: "'e" shows "reachablep A I a = ((\<exists>s. a = s \<and> s \<in> init A) \<or> (\<exists>s a s'. a = s' \<and> reachablep A I s \<and> (s, a, s') \<in> automaton.trans A \<and> I a))"
homology_excision_axiom: fixes X :: "'b topology"   and U :: "'b set"   and T :: "'b set"   and S :: "'b set"   and p :: "int" assumes "X closure_of U \<subseteq> X interior_of T"   and "T \<subseteq> S" shows "hom_induced p (subtopology X (S - U)) (T - U) (subtopology X S) T id \<in> Group.iso (relative_homology_group p (subtopology X (S - U)) (T - U)) (relative_homology_group p (subtopology X S) T)"
cases: fixes a1 :: "pi"   and a2 :: "pi" assumes "a1 \<approx> a2" obtains P :: "pi" and Q :: "pi" where "a1 = P" and "a2 = Q" and "P \<leadsto><weakBisim> Q" and "Q \<approx> P"
dependent_finite: fixes S :: "'c set" assumes "finite S" shows "m2.dependent S = (\<exists>u. (\<exists>v\<in>S. u v \<noteq> (0::'a)) \<and> (\<Sum>v\<in>S. u v *b v) = (0::'c))"
per_mult: fixes w :: "'b list"   and n :: "nat"   and m :: "nat" assumes "periodN w n"   and "m \<noteq> 0" shows "periodN w (m * n)"
2: fixes x :: "'a formula" assumes "P x" shows "P (- x)"
empty_nth: fixes i :: "nat"   and x :: "'a list" assumes "\<not> i < length x" shows "x ! i = [] ! (i - length x)"
liftAll2_def: fixes phi :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and inp1 :: "'j \<Rightarrow> 'h option"   and inp2 :: "'j \<Rightarrow> 'i option" shows "liftAll2 phi inp1 inp2 \<equiv> \<forall>i v1 v2. inp1 i = Some v1 \<and> inp2 i = Some v2 \<longrightarrow> phi v1 v2"
s1'_def: shows "s1' \<equiv> activate_cond_actions1 (ls, (ts1, m1), ws, is) (ls, (ts2, m2), ws, is) cts"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> UNIV; y \<in> UNIV\<rbrakk> \<Longrightarrow> x * y = y * x" shows "Group.comm_monoid \<Z>"
solution_VAR: fixes sys :: "'a \<Rightarrow> ('b, 'a) process_term"   and X :: "'a" assumes "guarded sys" shows "solution sys (VAR X) = solution sys (sys X)"
convergence: fixes xs :: "('a \<times> ('a, 'b) operation) list"   and ys :: "('a \<times> ('a, 'b) operation) list" assumes "set xs = set ys"   and "hb.concurrent_ops_commute xs"   and "hb.concurrent_ops_commute ys"   and "distinct xs"   and "distinct ys"   and "hb.hb_consistent xs"   and "hb.hb_consistent ys" shows "hb.apply_operations xs = hb.apply_operations ys"
new: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and v :: "('r, 'l, 'v) val"   and l :: "'l" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"   and "l \<notin> \<Union> {doms ls |ls. ls \<in> ran s}" shows "revision_step_relaxed r s (s(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (Loc l)])))"
IN_g: fixes x :: "'v" shows "d_IN g x = d_IN f x + j (None, \<langle>x\<rangle>)"
n'_def: shows "n' = 2 * n mod (p - 1)"
subcls1_confluent: fixes a :: "char list"   and b :: "char list"   and P :: "'b prog"   and c :: "char list" assumes "P \<turnstile> a \<prec>\<^sup>1 b"   and "P \<turnstile> a \<preceq>\<^sup>* c"   and "a \<noteq> c" shows "P \<turnstile> b \<preceq>\<^sup>* c"
lepoll_def: fixes A :: "'b set"   and B :: "'c set" shows "A \<lesssim> B \<equiv> \<exists>f. inj_on f A \<and> f ` A \<subseteq> B"