hcju/isabelleps huggingface dataset:
homeomorphism_on_extension_sequentially_precise: fixes S :: "'c set"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and u :: "nat \<Rightarrow> 'c"   and l :: "'c" assumes "\<And>u b. \<lbrakk>\<forall>n. u n \<in> S; b \<in> T; u \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> convergent (\<lambda>n. f (u n))"   and "\<And>u c. \<lbrakk>\<forall>n. u n \<in> S; c \<in> f ` T; (\<lambda>n. f (u n)) \<longlonglongrightarrow> c\<rbrakk> \<Longrightarrow> convergent u"   and "\<And>b. b \<in> T \<Longrightarrow> \<exists>u. (\<forall>n. u n \<in> S) \<and> u \<longlonglongrightarrow> b \<and> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b"   and "\<And>n. u n \<in> S \<union> T"   and "l \<in> T" shows "u \<longlonglongrightarrow> l = (\<lambda>n. f (u n)) \<longlonglongrightarrow> f l"
induct: fixes P :: "'a \<Rightarrow> bool"   and Qa :: "'b \<Rightarrow> bool"   and X :: "'c set"   and gpv :: "('a, 'b, 'c) gpv" assumes "pred_gpv' P Qa X gpv" obtains "\<And>x. x \<in> results_gpv (\<I>_uniform UNIV X) gpv \<Longrightarrow> P x" and "\<And>out. out \<in> outs_gpv (\<I>_uniform UNIV X) gpv \<Longrightarrow> Qa out"
set_restriction_finite: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set" assumes "set_restriction f"   and "finite A" shows "finite (f A)"
add_mset_eq_singleton_iff: fixes x :: "'d"   and M :: "'d multiset"   and y :: "'d" shows "(add_mset x M = {#y#}) = (M = {#} \<and> x = y)"
mult_oddprime_is_sum4sq: fixes p :: "int" assumes "prime (nat p)"   and "odd p" shows "\<exists>t>0. t < p \<and> is_sum4sq_int (p * t)"
remdups_mset_def: fixes S :: "'b multiset" shows "remdups_mset S = mset_set (set_mset S)"
ccspan_superset: fixes A :: "'a set" shows "A \<subseteq> space_as_set (ccspan A)"
PhiWhileMonotone: fixes b :: "BExpr" shows "Monotone (PhiWhile b)"
fun_type_arg_cong_1: fixes f :: "'fun"   and g :: "'fun"   and x :: "('fun, 'var) Term.term"   and X :: "('fun, 'var) Term.term list"   and y :: "('fun, 'var) Term.term"   and Y :: "('fun, 'var) Term.term list" assumes "f \<in> \<Union> (range \<Sigma>n) - \<C>"   and "g \<in> \<Union> (range \<Sigma>n) - \<C>"   and "\<Gamma> (Fun f (x # X)) = \<Gamma> (Fun g (y # Y))" shows "\<Gamma> x = \<Gamma> y"
unchanged: shows "\<forall>l\<le>k. \<forall>ts\<^sub>l \<S>\<^sub>l m\<^sub>l. c (n + l) = (ts\<^sub>l, m\<^sub>l, \<S>\<^sub>l) \<longrightarrow> ts\<^sub>l ! i = ts ! i"
fmeasurable_eq_sets: shows "fmeasurable (Pi\<^sub>M UNIV (\<lambda>i. M)) = P.events"
Lim_trivial_limit: fixes net :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "net = bot" shows "(f \<longlongrightarrow> l) net"
simps_6: fixes \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr" shows "(Read\<^sub>\<E> \<E>) [e] = Read (\<E> [e])"
cprank_max_Suc: fixes A :: "'b tensor"   and B :: "'b tensor"   and j :: "nat" assumes "dims A = dims B"   and "cprank_max1 A"   and "cprank_max j B" shows "cprank_max (Suc j) (A + B)"
my_iterate_rule_P: fixes S :: "('a, unit) HashMap.hashmap"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "my_invar S"   and "I (my_\<alpha> S) \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> my_\<alpha> S; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" shows "P (hs.iterate S f \<sigma>0)"
cblinfun_eq_on: fixes G :: "'a set"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and t :: "'a" assumes "\<And>x. x \<in> G \<Longrightarrow> A *\<^sub>V x = B *\<^sub>V x"   and "t \<in> closure (cspan G)" shows "A *\<^sub>V t = B *\<^sub>V t"
simps_4: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int" shows "aval (Times a\<^sub>1 a\<^sub>2) s = aval a\<^sub>1 s * aval a\<^sub>2 s"
elims: fixes x :: "('c, 'd) t"   and y :: "'c \<Rightarrow> 'd option" assumes "get x = y" obtains xs :: "('c \<times> 'd) list" where "x = T xs" and "y = map_of xs"
H: shows "H = (\<^bold>g [^] w, g' [^] w)"
le_def: shows "le = (\<Lambda> x y. case compare\<cdot>x\<cdot>y of LT \<Rightarrow> TT | EQ \<Rightarrow> TT | GT \<Rightarrow> FF)"
pred_inject_3: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" shows "pred_behaviour P (Goes_wrong a) = P a"
Max_less_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less (linorder.Max less_eq A) x = (\<forall>a\<in>A. less a x)"
simps: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma>' :: "'b \<Rightarrow> Sec option"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_types.closed_glob_helper aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>' ((c\<^sub>1, mds), mem\<^sub>1) ((c\<^sub>2, mds\<^sub>2), mem\<^sub>2) = (\<forall>x. (dma x = High \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v\<^sub>1 v\<^sub>2. (((c\<^sub>1, mds), mem\<^sub>1(x := v\<^sub>1)), (c\<^sub>2, mds), mem\<^sub>2(x := v\<^sub>2)) \<in> {(x, y). sifum_types.\<R>\<^sub>3_aux aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>' x y})) \<and> (dma x = Low \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v. (((c\<^sub>1, mds), mem\<^sub>1(x := v)), (c\<^sub>2, mds), mem\<^sub>2(x := v)) \<in> {(x, y). sifum_types.\<R>\<^sub>3_aux aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>' x y})))"
the_inv_into_map_impl_def: fixes V :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "the_inv_into_map_impl V f = FOREACH V (\<lambda>x m. RETURN (m(f x \<mapsto> x))) Map.empty"
arr_Rel_ArrVal_vdomain: shows "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrDom\<rparr>"
ocnet_cast: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and R :: "nat set"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), R:*cast(m), \<sigma>', s') \<in> S"   and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j" shows "((\<sigma>, s), \<tau>, \<sigma>', s') \<in> ocnet_sos S"
ring_distribs_2: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a + b) * c = a * c + b * c"
goal_def: fixes pad :: "nat \<Rightarrow> 'a vec \<Rightarrow> 'a vec"   and m :: "nat" shows "goal pad m \<equiv> pad m ` N.span U = LinearCombinations.module.span class_ring (M (n + m)) (pad m ` U)"
insert_body_delete_commute: fixes i2 :: "'c"   and e :: "'c \<times> 'd \<times> bool"   and xs :: "('c \<times> 'd \<times> bool) list"   and x :: "'c \<times> 'd \<times> bool" assumes "i2 \<noteq> fst e" shows "delete (insert_body xs e) i2 \<bind> (\<lambda>t. Some (x # t)) = delete xs i2 \<bind> (\<lambda>y. Some (x # insert_body y e))"
conv_radius_cmult_left: fixes c :: "'b"   and f :: "nat \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "conv_radius (\<lambda>n. c * f n) = conv_radius f"
fac_body_final_fac_body': shows "fac_body_final = fac_body'"
distinct_1: shows "X5.Xd1 \<noteq> X5.Xd1'"
c_graph_def: shows "c_graph = (\<lambda>f. {c_pair x (f x) |x. x \<in> UNIV})"
v11_vcomp_vconverse': shows "T\<lparr>ArrVal\<rparr> \<circ>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>)\<inverse>\<^sub>\<circ> = vid_on (\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>))"
ipassmt_disjoint_nonempty_inj: fixes ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and ifce :: "iface"   and i_ips :: "('b word \<times> nat) list"   and k :: "iface" assumes "ipassmt_sanity_disjoint ipassmt"   and "ipassmt ifce = Some i_ips"   and "ipcidr_union_set (set i_ips) \<noteq> {}"   and "ipassmt k = Some i_ips" shows "k = ifce"
right_neutral: fixes a :: "'k" shows "a + (0::'k) = a"
distinct_set_remove_mid: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list" assumes "distinct (xs @ [x] @ ys)" shows "set (xs @ ys) = set (xs @ [x] @ ys) - {x}"
tau_ideal1: fixes x :: "'a"   and y :: "'a" assumes "\<tau> x = x"   and "y \<le> x" shows "\<tau> y = y"
5: fixes x :: "'a"   and y :: "'a" assumes "fst (rep_nat_term x) \<noteq> fst (rep_nat_term y)"   and "\<not> lex_pp (fst (rep_nat_term x)) (fst (rep_nat_term y))"   and "x < y" shows "False"
distinct_rev: fixes xs :: "'b list" shows "distinct (rev xs) = distinct xs"
l2_inv3D_aux: fixes x :: "l2_state"   and c :: "msg" assumes "x \<in> l2_inv3"   and "c \<in> synth (analz (ik x))" shows "c \<in> synth (analz (generators x))"
loc_reachp_loc_reach_eq: fixes lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "loc_reachp lc = (\<lambda>x xa xb. \<langle>x, xa, xb\<rangle> \<in> loc_reach lc)"
var_prm_action: fixes a :: "'b"   and b :: "'b" shows "[a \<leftrightarrow> b] \<cdot> Var a = Var b"
eq_commute: fixes a :: "'k"   and b :: "'k" shows "(a = b) = (b = a)"
cmp_rule_eq: fixes s :: "bdd"   and ni :: "nat"   and i :: "nat ifex"   and ni' :: "nat"   and i' :: "nat ifex" assumes "bdd_sane s"   and "(ni, i) \<in> Rmi s"   and "(ni', i') \<in> Rmi s" shows "(ni = ni') = (i = i')"
suffix_nth: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "suffix k x n = x (k + n)"
BIJ: shows "bij_betw (f(B\<^sub>1 := u)) (P\<^sub>1 \<union> wrap B\<^sub>1) (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2 \<union> wrap {u}))"
strong_induct: fixes P :: "'e \<Rightarrow> 'd prefix \<Rightarrow> bool"   and z :: "'e"   and prefix :: "'d prefix" assumes "\<And>a1 list a2 z. P z (pInput a1 list a2)"   and "\<And>a1 a2 z. P z (pOutput a1 a2)"   and "\<And>z. P z pTau" shows "P z prefix"
WhileTrue: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile>c \<down> Normal s"   and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>While b c \<down> s'" shows "\<Gamma>\<turnstile>While b c \<down> Normal s"
min_height_size1: fixes t :: "'b tree" shows "2 ^ min_height t \<le> size1 t"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "Right.arr f"   and "Right.dom f = a" shows "Right.comp f a = f"
pairwise_mono: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a set" assumes "pairwise P A"   and "\<And>x y. P x y \<Longrightarrow> Q x y"   and "B \<subseteq> A" shows "pairwise Q B"
prems_1: shows "(a # u) @ v =\<^sub>F v @ a # u"
strict_trans1: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "b \<preceq> a"   and "c \<prec> b" shows "c \<prec> a"
bin_lcp_def: shows "i.\<alpha> = x \<cdot> y \<and>\<^sub>p y \<cdot> x"
abs_eq_0_iff: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "(abs a = zero) = (a = zero)"
all_not_cin_conv: fixes A :: "'d cset" shows "(\<forall>x. \<not> cin x A) = (A = cempty)"
singletonD: fixes b :: "'c"   and a :: "'c" assumes "b \<in> {a}" shows "b = a"
chine_assoc_def: shows "\<nu>\<pi>\<rho>.chine_assoc \<equiv> \<langle>\<nu>\<pi>\<rho>.Prj\<^sub>1\<^sub>1 \<lbrakk>\<nu>.leg0, \<pi>.leg1 \<cdot> \<pi>\<rho>.prj\<^sub>1\<rbrakk> \<langle>\<nu>\<pi>\<rho>.Prj\<^sub>0\<^sub>1 \<lbrakk>\<pi>.leg0, \<rho>.leg1\<rbrakk> \<nu>\<pi>\<rho>.Prj\<^sub>0\<rangle>\<rangle>"
dghm_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> = [\<FF>\<lparr>ObjMap\<rparr>, \<FF>\<lparr>ArrMap\<rparr>, \<FF>\<lparr>HomDom\<rparr>, \<FF>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
AbsV_aux: fixes f :: "ValD \<rightarrow> ValD"   and f' :: "(('o ValK \<rightarrow> 'o) \<rightarrow> 'o) \<rightarrow> ('o ValK \<rightarrow> 'o) \<rightarrow> 'o" assumes "\<eta>: ValF\<cdot>f \<mapsto> unitK\<cdot>(ValKF\<cdot>f')" shows "\<eta>: ValF\<cdot>(strictify\<cdot>f) \<mapsto> unitK\<cdot>(ValKF\<cdot>(\<Lambda> x c. x\<cdot>(\<Lambda> x'. f'\<cdot>(unitK\<cdot>x')\<cdot>c)))"
submanifold_def: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set" shows "submanifold charts k S \<equiv> c_manifold' charts k \<and> submanifold_axioms S"
not_prime_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "\<not> normalization_semidom.prime zero times one normalize zero"
cases: fixes a :: "('a, 'b, 'c) hp" assumes "hpsafe a" obtains x :: "'c" where "aa__ = $\<alpha> x__"   | e :: "('a, 'c) trm" and x :: "'c" where "aa__ = (xa__ := e__)" and "dsafe e__"   | e :: "('a, 'c) trm" and x :: "'c" where "aa__ = DiffAssign x e" and "dsafe e"   | P :: "('a, 'b, 'c) formula" where "aa__ = ? P__" and "fsafe P__"   | ODE :: "('a, 'c) ODE" and P :: "('a, 'b, 'c) formula" where "aa__ = EvolveODE ODE P" and "osafe ODE" and "fsafe P"   | a :: "('a, 'b, 'c) hp" and b :: "('a, 'b, 'c) hp" where "aa__ = (ab__ \<union>\<union> b__)" and "hpsafe ab__" and "hpsafe b__"   | a :: "('a, 'b, 'c) hp" and b :: "('a, 'b, 'c) hp" where "aa__ = (ac__ ;; b)" and "hpsafe ac__" and "hpsafe b"   | a :: "('a, 'b, 'c) hp" where "aa__ = a**" and "hpsafe a"
I_1: shows "MultiSequents.mset C \<equiv> \<LM> Compound F Fs  \<RM>"
ZusammenziehungA: fixes \<DD> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<DD> # \<DD> # \<Gamma> \<Rightarrow> \<Theta>" shows "\<DD> # \<Gamma> \<Rightarrow> \<Theta>"
comp_final: fixes e :: "char list exp"   and Vs :: "char list list" assumes "final e" shows "compE\<^sub>1 Vs e = fin\<^sub>1 e"
assms_4: fixes e\<^sub>1' :: "Syntax.term" assumes "\<lless>\<pi>, e\<^sub>1\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>1'\<ggreater> \<and> e' = Syntax.Let e\<^sub>1' x e\<^sub>2 \<and> \<not> value e\<^sub>1" shows "Q"
antisym: fixes a :: "'d list"   and b :: "'d list" assumes "prefix a b"   and "prefix b a" shows "a = b"
basis_null_space_iarrays_eq: fixes A :: "(('b, 'c) vec, 'd) vec" shows "basis_null_space_iarrays (matrix_to_iarray A) = set (map (\<lambda>i. row_iarray i (fst (Gauss_Jordan_iarrays_PA (transpose_iarray (matrix_to_iarray A))))) [rank_iarray (matrix_to_iarray A)..<ncols_iarray (matrix_to_iarray A)])"
2: shows "II' = \<Union> {I \<in> P. I \<subseteq> II'}"
that: fixes n :: "nat" assumes "x = local.of_nat n" shows "thesis"
vars_def: fixes e :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b" shows "vars e = {v. \<exists>\<sigma> val. e (\<sigma>(v := val)) \<noteq> e \<sigma>}"
f_class: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and CDef :: "classDef"   and D :: "nat"   and Cf :: "varDef list"   and Dg :: "varDef list"   and DgCf :: "varDef list" assumes "CT C = Some CDef"   and "cSuper CDef = D"   and "cFields CDef = Cf"   and "fields(CT,D) = Dg"   and "DgCf = Dg @ Cf" shows "fields(CT,C) = DgCf"
kuhn_simplex_lemma: fixes p :: "nat"   and n :: "nat"   and rl :: "(nat \<Rightarrow> nat) \<Rightarrow> nat" assumes "\<forall>s. ksimplex p (Suc n) s \<longrightarrow> rl ` s \<subseteq> {..Suc n}"   and "odd (card {f. \<exists>s a. ksimplex p (Suc n) s \<and> a \<in> s \<and> f = s - {a} \<and> rl ` f = {..n} \<and> ((\<exists>j\<le>n. \<forall>x\<in>f. x j = 0) \<or> (\<exists>j\<le>n. \<forall>x\<in>f. x j = p))})" shows "odd (card {s. ksimplex p (Suc n) s \<and> rl ` s = {..Suc n}})"
vars_add_monom: fixes p2 :: "'b mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and p1 :: "'b mpoly" assumes "p2 = MPoly_Type.monom m a"   and "m \<notin> keys (mapping_of p1)" shows "vars (p1 + p2) = vars p1 \<union> vars p2"
smn: fixes n :: "nat"   and p :: "nat"   and cs :: "nat list" assumes "0 < n" shows "smn n p cs = encode (Cn n (r_universal (n + length cs)) (r_constn (n - 1) p # map (r_constn (n - 1)) cs @ map (recf.Id n) [0..<n]))"
universal_interpretation_validate: fixes L :: "'at Literal" shows "UNIV \<Turnstile> L = (\<exists>A. L = Pos A)"
lmember_code_1: fixes x :: "'c" shows "lmember x LNil = False"
comm_monoid_mult_hom_axioms: shows "comm_monoid_mult_hom poly_x_mult_y"
H0': fixes x12 :: "nat"   and r :: "int mpoly"   and Aa :: "nat multiset" assumes "2 * the (fmlookup A x12) - r \<in> More_Modules.ideal polynomial_bool"   and "r - spec \<in> More_Modules.ideal polynomial_bool" shows "spec \<in> pac_ideal (insert (the (fmlookup A x12)) ((\<lambda>x. the (fmlookup A x)) ` set_mset Aa))"
arrE: fixes f :: "'a" assumes "Left_a.arr f" shows "left a f"
cloWW0: shows "closedin (top_of_set W) W0"
degrees_leI: fixes X :: "'a pdevs list"   and K :: "nat" assumes "\<And>x. x \<in> set X \<Longrightarrow> degree x \<le> K" shows "degrees X \<le> K"
periodic_arg_plus_0: fixes x :: "'b" shows "x + (0::'b) * ((2::'b) * of_real pi) = x"
of_nat_word_less_iff: fixes m :: "nat"   and n :: "nat" shows "(word_of_nat m < word_of_nat n) = (take_bit LENGTH('b) m < take_bit LENGTH('b) n)"
fact_Suc: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero (Suc n) = times (semiring_1.of_nat one plus zero (Suc n)) (semiring_char_0.fact one plus zero n)"
clock_trans_common: fixes t :: "'s Trace"   and ec :: "'s odlist \<times> 's odlist"   and a :: "'a" assumes "t \<in> Clock.jkbpC"   and "clock_simAbs ec = Clock.sim_equiv_class a t" shows "toSet (local.clock_trans (fst ec) (fst ec)) = rel_ext (\<lambda>uu_. \<exists>t' s. uu_ = s \<and> t' \<leadsto> s \<in> Clock.jkbpC \<and> tLength t' = tLength t)"
coplanar_trans_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and R :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet P Q R"   and "Tarski_neutral_dimensionless.Coplanar Bet P Q R A"   and "Tarski_neutral_dimensionless.Coplanar Bet P Q R B" shows "Tarski_neutral_dimensionless.Coplanar Bet Q R A B"
ENF_offending_set: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_all_edges_normal_form P" shows "set_offending_flows G nP = (if sinvar G nP then {} else {{(e1, e2). (e1, e2) \<in> edges G \<and> \<not> P (nP e1) (nP e2)}})"
abs_eq: fixes x :: "int" shows "Word.cast (Word.Word x) = Word.Word (take_bit LENGTH('c) x)"
timpl_closure_setI: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" assumes "t \<in> M" shows "t \<in> timpl_closure_set M TI"
domain_eq: shows "Domainp (pcr_commute (=) (=)) = (\<lambda>f. \<forall>x y. f x y = f y x)"
tt_lower: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" assumes "punit.tt p \<prec> v" shows "punit.tt (punit.lower p v) = punit.tt p"
subst_comp_upd2: fixes v :: "'a"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" assumes "v \<notin> subst_domain s"   and "v \<notin> range_vars s" shows "s(v := t) = s \<circ>\<^sub>s Var(v := t)"
intros_18: fixes P :: "pi" shows "!P \<equiv>\<^sub>s P \<parallel> !P"
strictify_conv_if: fixes f :: "'b \<rightarrow> 'a"   and x :: "'b" shows "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
put_option_def: fixes T :: "nat"   and K :: "real" shows "put_option T K = (\<lambda>w. max (K - prices Mkt stk T w) 0)"
less_eq_set_def: fixes A :: "'g set"   and B :: "'g set" shows "(A \<subseteq> B) = ((\<lambda>x. x \<in> A) \<le> (\<lambda>x. x \<in> B))"
nlists_empty_gt0: fixes n :: "nat" assumes "0 < n" shows "nlists {} n = {}"
uniformly_convergent_uniform_limit_iff: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" shows "uniformly_convergent_on X f = uniform_limit X f (\<lambda>x. lim (\<lambda>n. f n x)) sequentially"
semiring_norm_73: fixes m :: "num"   and n :: "num" shows "(num.Bit1 m \<le> num.Bit1 n) = (m \<le> n)"
conG: fixes n :: "nat" shows "continuous_on UNIV (G n)"
simps: fixes i :: "nat"   and j :: "nat" shows "f2.\<mu> i j = (if j < i then f2 ! i \<bullet> f2.gso j / \<parallel>f2.gso j\<parallel>\<^sup>2 else if i = j then 1::'a else (0::'a))"
strategyproofness: shows "pmf (sds R38) c + pmf (sds R38) a \<le> pmf (sds R2) c + pmf (sds R2) a"
suma_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C D E F SumA G H I" shows "A \<noteq> B \<and> B \<noteq> C \<and> D \<noteq> E \<and> E \<noteq> F \<and> G \<noteq> H \<and> H \<noteq> I"
a'Z: shows "emb a A \<in> B - A \<rightarrow> {\<zero>\<^bsub>K\<^esub>}"
open_core_eq: fixes a :: "'a set" assumes "a open" shows "core a = a"
polyn_ring_S_nonzero: shows "1\<^sub>r\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>"
less_vec_def: fixes v :: "'a vec"   and w :: "'a vec" shows "(v < w) = (v \<le> w \<and> \<not> w \<le> v)"
length_zip: fixes xs :: "'d list"   and ys :: "'e list" shows "length (zip xs ys) = min (length xs) (length ys)"
rpr_complete: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c" assumes "rpr A r"   and "x \<in> A"   and "y \<in> A" shows "x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x"
relative_to_imp_subset: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and T :: "'c set" assumes "(P relative_to S) T" shows "T \<subseteq> S"
that: fixes t' :: "'a list" assumes "t \<sim> t'"   and "u \<le>p t' \<^sup>@ k" shows "thesis"
equiv_order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<simeq> y"   and "z \<sqsubseteq> y"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sqsubseteq> x"
flat_ord_antisym: fixes a :: "'g"   and x :: "'g"   and y :: "'g" assumes "flat_ord a x y"   and "flat_ord a y x" shows "x = y"
qFreshAll_simps_1: fixes xs :: "'a"   and x :: "'b"   and ys :: "'a"   and y :: "'b" shows "qFresh xs x (qVar ys y) = (xs \<noteq> ys \<or> x \<noteq> y)"
remap_polys_change_all_def: fixes spec :: "int mpoly" shows "remap_polys_change_all spec = (\<lambda>\<V> A. SPEC (\<lambda>(st, \<V>', A'). (\<not> is_failed st \<longrightarrow> pac_ideal (set_mset (ran_m A)) = pac_ideal (set_mset (ran_m A')) \<and> \<Union> (vars ` set_mset (ran_m A)) \<subseteq> \<V>' \<and> \<Union> (vars ` set_mset (ran_m A')) \<subseteq> \<V>') \<and> (st = FOUND \<longrightarrow> spec \<in># ran_m A')))"
pell_mul_commutes: fixes z1 :: "'a \<times> 'a"   and z2 :: "'a \<times> 'a" shows "pell_mul z1 z2 = pell_mul z2 z1"
scheme_def: fixes u :: "(nat \<times> nat) set" shows "scheme u \<equiv> fst ` u \<union> snd ` u"