hcju/isabelleps huggingface dataset:
domintros: fixes S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and U :: "'x set" assumes "\<And>xa. \<lbrakk>0 \<notin> S; \<not> split_dom TYPE('b) (t, U - {SOME x'. x' \<in> U \<and> x' \<notin> max_subset U (\<lambda>V. S \<inter> .[V] = {})}, S); xa \<in> S; xa \<in> .[U]\<rbrakk> \<Longrightarrow> False"   and "\<And>ab ba xb ya xa. \<lbrakk>0 \<notin> S; (ab, ba) = split t (U - {SOME x'. x' \<in> U \<and> x' \<notin> max_subset U (\<lambda>V. S \<inter> .[V] = {})}) S; (xb, ya) = split t (U - {SOME x'. x' \<in> U \<and> x' \<notin> max_subset U (\<lambda>V. S \<inter> .[V] = {})}) S; \<not> split_dom TYPE('b) (monomial (Suc 0) (SOME x'. x' \<in> U \<and> x' \<notin> max_subset U (\<lambda>V. S \<inter> .[V] = {})) + t, U, (\<lambda>x. x - monomial (Suc 0) (SOME x'. x' \<in> U \<and> x' \<notin> max_subset U (\<lambda>V. S \<inter> .[V] = {}))) ` S); xa \<in> S; xa \<in> .[U]\<rbrakk> \<Longrightarrow> False" shows "split_dom TYPE('b) (t, U, S)"
fmeasurable_Diff_D: fixes T :: "'a set"   and S :: "'a set"   and M :: "'a measure" assumes "T - S \<in> fmeasurable M"   and "S \<in> fmeasurable M"   and "S \<subseteq> T" shows "T \<in> fmeasurable M"
V'_def: shows "V' = {sm, sn} \<union> Vs"
simps_3: fixes oper :: "pdf_operator"   and e :: "cexpr" shows "free_vars (oper $$\<^sub>c e) = free_vars e"
cases: fixes a1 :: "'a"   and a2 :: "'a" assumes "eq a1 a2" obtains x :: "'a" where "a1 = x" and "a2 = x"
tp_sum: fixes S :: "nat set"   and f :: "nat \<Rightarrow> real \<Rightarrow> real" assumes "finite S"   and "\<And>i. i \<in> S \<Longrightarrow> trigpoly (f i)" shows "trigpoly (\<lambda>x. \<Sum>i\<in>S. f i x)"
V'_V: fixes Z :: "('c \<Rightarrow> real) set" assumes "Z \<in> V'" shows "Z \<subseteq> V"
of_nat_neq_0: 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_1.of_nat one plus zero (Suc n) \<noteq> zero"
matchT_T_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.matchT_T tval aval indis c d \<equiv> \<forall>s t s'. (s, t) \<in> indis \<and> PL.transT tval aval (c, s) s' \<longrightarrow> (\<exists>t'. PL.transT tval aval (d, t) t' \<and> (s', t') \<in> indis)"
OR_def: fixes \<Phi> :: "'b pltl set" shows "OR \<Phi> \<equiv> SOME \<phi>. fold_graph Or_ltlp false\<^sub>p \<Phi> \<phi>"
eq: fixes y :: "'c option" shows "poly_subst (\<lambda>x. monomial (1::'d) (restrict_indets_subst x)) (dehomo_subst None y) = monomial (1::'d) (restrict_indets_subst y)"
assms: shows "vfind_match cs t = Some (env, pat, rhs)"
unary_wt_sym_0_imp_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and f :: "'b" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "arity_sym f = 1"   and "wt_sym f = 0" shows "of_nat \<delta> = of_nat \<epsilon>"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "Adj.C.ide a"   and "Adj.C.seq f a" shows "f \<cdot>\<^sub>C a = f"
islimpt_greaterThanLessThan2: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "b islimpt {a<..<b}"
plus_of_nat: fixes x :: "'a"   and n :: "nat" shows "f (x + of_nat n) = f x"
inext_singleton_cut_greater_not_empty_iff: fixes t0 :: "nat"   and I :: "nat set" shows "({inext t0 I} \<down>> t0 \<noteq> {}) = (I \<down>> t0 \<noteq> {} \<and> t0 \<in> I)"
lexord_linear: fixes r :: "('a \<times> 'a) set"   and x :: "'a list"   and y :: "'a list" assumes "\<forall>a b. (a, b) \<in> r \<or> a = b \<or> (b, a) \<in> r" shows "(x, y) \<in> lexord r \<or> x = y \<or> (y, x) \<in> lexord r"
carrier_polynomial_shell: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)" shows "p \<in> carrier (poly_ring (poly_ring R))"
assms_1: fixes x :: "'a" assumes "\<sigma>' oid = \<lfloor>x\<rfloor>" shows "x = Object"
Cons: fixes xs :: "['b]"   and x :: "'b" assumes "distinct xs"   and "x \<notin> set xs" shows "distinct (x : xs)"
Inf_specs_Inf_spec: fixes d :: "nat" assumes "d = DIM('c)" shows "(Inf_specs d, Inf_spec) \<in> \<langle>lv_rel\<rangle>set_rel \<rightarrow> \<langle>lv_rel\<rangle>nres_rel"
pdevs_apply_Pdevs_prod_slist: fixes xs :: "(nat, 'd) slist"   and ys :: "(nat, 'e) slist"   and i :: "nat" shows "pdevs_apply (Pdevs (prod_slist xs ys)) i = (pdevs_apply (Pdevs xs) i, pdevs_apply (Pdevs ys) i)"
nat_filtration_AE_zero: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat" assumes "AE w in M. f w = 0"   and "f \<in> borel_measurable (nat_filtration n)"   and "0 < p"   and "p < 1" shows "\<forall>w. f w = 0"
terminates_fewer_guards_Normal: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c' :: "('e, 'd, 'f) com"   and s :: "'e"   and c :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile>c' \<down> Normal s"   and "c \<subseteq>\<^sub>g c'"   and "\<Gamma>\<turnstile> \<langle>c',Normal s\<rangle> \<Rightarrow>\<notin>range Fault" shows "\<Gamma>\<turnstile>c \<down> Normal s"
pos_unitr: fixes x :: "'a"   and y :: "'a"   and e :: "'a" assumes "D x y"   and "e \<in> E"   and "x \<oplus> y = e" shows "y = e"
q: shows "map_poly complex_of_real q \<noteq> 0"
Accessible_accessible: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.accessible (dfa.Accessible_dfa M) = dfa.accessible M"
add_diff_cancel_right': fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
isomorphic_transitive: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "L.isomorphic f g"   and "L.isomorphic g h" shows "L.isomorphic f h"
expandTrans_3: fixes R :: "pi"   and P :: "pi"   and Q :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "(R, expandSet P Q) \<in> sumComposeSet"   and "hnf P"   and "hnf Q" shows "P \<parallel> Q \<longmapsto> a<x> \<prec> P' = R \<longmapsto> a<x> \<prec> P'"
later: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> af A (\<lambda>y z. P y z \<or> P x y)" shows "af A P"
succ_rel_set_fst: fixes xs :: "'b list" shows "fst ` succ_rel xs = set xs"
mod_add_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (modulo a c) (modulo b c)) c = modulo (plus a b) c"
NT_widen: fixes P :: "'a prog"   and T :: "ty" shows "P \<turnstile> NT \<le> T = (T = NT \<or> (\<exists>C. T = Class C) \<or> (\<exists>U. T = U\<lfloor>\<rceil>))"
Arg_le_pi: fixes z :: "complex" shows "Arg z \<le> pi"
simps_2: fixes v :: "'c" shows "first_edge_of_walk [v] = undefined"
assms_1: shows "tree_take i\<cdot>t = tree_take i\<cdot>u"
sandwich_apply: fixes A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "sandwich A *\<^sub>V B = A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L A*"
leI: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" assumes "\<not> ko.lt ko x y" shows "ko.le ko y x"
linear_compose_scale_right: fixes f :: "('a \<Rightarrow> real) \<Rightarrow> real"   and c :: "real" assumes "\<forall>x. x \<in> diff_fun_space \<longrightarrow> f x \<in> UNIV"   and "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) f" shows "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) (\<lambda>x. c *\<^sub>R f x)"
vec_setE: fixes a :: "'b"   and v :: "'b Matrix.vec" assumes "a \<in> set\<^sub>v v" obtains i :: "nat" where "v $v i = a" and "i < dim_vec v"
W2a: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) r\<rightarrow> b = (b l\<rightarrow> a) r\<rightarrow> a"
x_zero_prop: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (0::'a) \<parallel> y = d (x \<cdot> (0::'a)) \<cdot> y"
wary_raw_db_subst: fixes i :: "nat"   and x :: "'v" shows "wary_subst (raw_db_subst i x)"
absorb1: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "inf a b = a"
hom: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "continuous_on U f"   and "f ` U \<subseteq> s"   and "P f"   and "continuous_on U g"   and "g ` U \<subseteq> s"   and "P g" shows "homotopic_with_canon P U s f g"
subdg_antisym: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA>" shows "\<AA> = \<BB>"
pre_splitFace_edges_f_in_g: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and f :: "face"   and vs :: "nat list" assumes "pre_splitFace g ram1 ram2 f vs" shows "\<E> f \<subseteq> \<E> g"
that: assumes "r = a"   and "h' = update a i v h"   and "i < Array.length h a" shows "thesis"
diff_cancel_imp_eq: fixes n :: "nat"   and i :: "nat"   and j :: "nat" assumes "0 < n"   and "n + i - j = n" shows "i = j"
fv\<^sub>s\<^sub>s\<^sub>t_is_subterm_trms\<^sub>s\<^sub>s\<^sub>t: fixes x :: "'c"   and A :: "('d, 'c) stateful_strand_step list" assumes "x \<in> fv\<^sub>s\<^sub>s\<^sub>t A" shows "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t A"
pointermap_empty_sane: shows "pointermap_sane empty_pointermap"
parallel2_wiring_def: shows "(|\<^sub>w) = (\<lambda>(f, g) (f', g'). (map_sum f f', map_sum g g'))"
2: shows "insertion_fun f (\<lambda>m. a when Poly_Mapping.single v n = m) = a * (\<Prod>va. f va ^ lookup (Poly_Mapping.single v n) va)"
prob_space_imp_sigma_finite: fixes M :: "'b measure" assumes "prob_space M" shows "sigma_finite_measure M"
prems_1: shows "is \<lhd> dims (tensors_from_net (witness_l rs) $ j)"
ServiceCenter_L1: fixes x :: "nat"   and i :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and t :: "nat" assumes "\<forall>t2<x. i t2 = []"   and "ServiceCenter i a"   and "t \<le> x" shows "a t = []"
diff_diff_right: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> b" shows "c - (b - a) = c + a - b"
ZFTSnd: fixes a :: "ZF"   and b :: "ZF"   and c :: "ZF" shows "ZFTSnd (ZFTriple a b c) = b"
det3_self12: fixes b :: "real \<times> real"   and a :: "real \<times> real" shows "coll b b a"
iIN_sub1: fixes n :: "nat"   and d :: "nat"   and k :: "nat" assumes "n + d \<le> k" shows "k \<ominus> [n\<dots>,d] = [k - (n + d)\<dots>,d]"
assms: shows "((\<lambda>(s, n). P (s, n) \<and> vars b \<subseteq> dom s) \<and>* (\<lambda>(s, c). s = 0 \<and> c = Suc 0)) (ps, n)"
trace_core_closure_start: fixes core1 :: "('a, 'b, 'c, 'd, 'e, 'f) core"   and core2 :: "('g, 'b, 'c, 'd, 'e, 'f) core"   and E :: "'b set"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'g spmf" shows "trace_core_closure core1 core2 E IA IU p q p q"
wf_subst_subst_idem: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "subst_idem \<theta>"
maintainedI: fixes A :: "('d, 'b) labeled_graph"   and G :: "('d, 'c) labeled_graph"   and B :: "('d, 'b) labeled_graph" assumes "\<And>f. graph_homomorphism A G f \<Longrightarrow> extensible (A, B) G f" shows "maintained (A, B) G"
is_pole_cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "\<forall>\<^sub>F x in at a. f x = g x"   and "a = b" shows "is_pole f a = is_pole g b"
keys_def: shows "keys \<equiv> map_fun lookup id (\<lambda>f. {k. f k \<noteq> (0::'d)})"
skip_elim: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Skip, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "c' = Stop \<and> mds = mds' \<and> mem = mem'"
oalist_inv_raw_ConsD3: fixes k :: "'a"   and v :: "'e"   and xs :: "('a \<times> 'e) list"   and k' :: "'a" assumes "oalist_inv_raw ((k, v) # xs)"   and "k' \<in> fst ` set xs" shows "lt k k'"
elims: fixes x :: "opinl"   and y :: "op" assumes "deinl x = y" obtains "x = OpAddNum" and "y = OpAdd"   | "x = OpMulNum" and "y = OpMul"
inheritedUnEq: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and A :: "(nat \<times> 'b) set"   and B :: "(nat \<times> 'b) set" assumes "inherited subs P" shows "P (A \<union> B) = (P A \<and> P B)"
Euclid_induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a :: "nat"   and b :: "nat" assumes "\<And>a b. P a b = P b a"   and "\<And>a. P a 0"   and "\<And>a b. P a b \<longrightarrow> P a (a + b)" shows "P a b"
mk_mat_def: fixes f :: "'g \<Rightarrow> 'f \<Rightarrow> 'e" shows "HMA_Connect.mk_mat f = (\<chi>i j. f i j)"
dagger_trans_eq: fixes x :: "'a list set" shows "p_prod (p_star x) (p_star x) = p_star x"
RDirProd_carrier: fixes R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" shows "carrier (RDirProd R S) = carrier R \<times> carrier S"
mul_closed_set_tOp_closed: fixes S :: "'a set"   and s :: "'a"   and t :: "'a" assumes "mul_closed_set R S"   and "s \<in> S"   and "t \<in> S" shows "s \<cdot>\<^sub>r t \<in> S"
indRelLPO_def: shows "indRelLPO \<equiv> _Collect (x, y) (indRelLPOp x y)"
rfn_tool20: fixes m :: "nat"   and a :: "nat"   and b :: "nat"   and c :: "nat" assumes "0 < m"   and "a = b * m + c"   and "c < m" shows "a mod m = c"
diamond_y_gen_valid: fixes a :: "real"   and b :: "real" assumes "a \<le> 0"   and "0 \<le> b" shows "diamond_y_gen piecewise_C1_differentiable_on {a..b}"
base_2: shows "f = (\<lambda>t. \<Sum>i\<in>S. x i * \<chi> (A i) t)"
upd_const_same: fixes b :: "'b"   and g :: "('a, 'c) finfun"   and a :: "'a" shows "finfun_update (finfun_comp (Pair b) g) a (b, finfun_apply g a) = finfun_comp (Pair b) g"
chain_tranclp_imp_exists_chain: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "chain R\<^sup>+\<^sup>+ xs" shows "\<exists>ys. chain R ys \<and> emb xs ys \<and> lhd ys = lhd xs \<and> llast ys = llast xs"
measurable_const: fixes c :: "'a"   and M' :: "'a measure"   and M :: "'b measure" assumes "c \<in> space M'" shows "(\<lambda>x. c) \<in> M \<rightarrow>\<^sub>M M'"
abs_real_of_ereal: fixes x :: "ereal" shows "\<bar>real_of_ereal x\<bar> = real_of_ereal \<bar>x\<bar>"
char_of_char: fixes c :: "char" shows "char_of (of_char c) = c"
inv_is_inverse: fixes f :: "'a" assumes "S.iso f" shows "S.inverse_arrows f (S.inv f)"
monoD: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "f x < f y" shows "x < y"
recurrent_subset_infty_series_infinite: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> recurrent_subset_infty A" shows "(\<Sum>n. indicator A ((T ^^ n) x)) = \<infinity>"
parent_def: fixes i :: "nat" shows "local.parent i = (case (left_spec xs i, right_spec xs i) of (None, None) \<Rightarrow> None | (None, Some y) \<Rightarrow> Some y | (Some x, None) \<Rightarrow> Some x | (Some x, Some y) \<Rightarrow> Some (max x y))"
affine_independent_insert: fixes S :: "'b set"   and a :: "'b" assumes "\<not> affine_dependent S"   and "a \<notin> affine hull S" shows "\<not> affine_dependent (insert a S)"
remdups_mset_singleton: fixes a :: "'c" shows "remdups_mset {#a#} = {#a#}"
refl: fixes a :: "'a" shows "domain_semigroup.ds_ord d (\<cdot>) a a"
transitionI: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "(l, g, \<mu>) \<in> PTA.trans_of A"   and "(X, l') \<in> set_pmf \<mu>" shows "A \<turnstile> l \<longrightarrow>\<^bsup>g,\<mu>,X\<^esup> l'"
proper_def: fixes P :: "assn_raw" shows "proper P = ((\<forall>h as. aseval P (pHeap h as) \<longrightarrow> in_range (h, as)) \<and> (\<forall>h h' as. aseval P (pHeap h as) \<longrightarrow> relH as h h' \<longrightarrow> in_range (h', as) \<longrightarrow> aseval P (pHeap h' as)))"
hyps_3: shows "u \<cdot> \<sigma> \<rhd> t \<cdot> \<sigma>"
amd1_update_refine: fixes dflt :: "'d" shows "(uncurry2 (\<lambda>k v f. f(k := v)), uncurry2 op_map_update) \<in> [\<lambda>((k, v), m). v \<noteq> dflt]\<^sub>f (Id \<times>\<^sub>r Id) \<times>\<^sub>r (Id \<rightarrow> dflt_option_rel_aux dflt) \<rightarrow> Id \<rightarrow> dflt_option_rel_aux dflt"
simps_1: shows "ternary_to_bool TernaryTrue = Some True"
comparator_double_def: fixes x :: "double"   and y :: "double" shows "comparator_double x y = (let c = lcompare_double x y in if c = 0 then order.Eq else if c < 0 then order.Lt else order.Gt)"
coerce_eq_ID: shows "COERCE('a, 'a) = ID"
is_expansion_minus: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a"   and G :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "is_expansion F basis"   and "is_expansion G basis" shows "is_expansion (minus F G) basis"
reductive_inference_system_axioms: shows "reductive_inference_system gr.ord_\<Gamma>"
typed: shows "\<turnstile> \<Gamma> {c} \<Gamma>'"
frontier_of_closures: fixes X :: "'c topology"   and S :: "'c set" shows "X frontier_of S = X closure_of S \<inter> X closure_of (topspace X - S)"
has_as_equalizer_def: fixes f0 :: "'j"   and f1 :: "'j"   and e :: "'j" shows "J.has_as_equalizer f0 f1 e \<equiv> J.par f0 f1 \<and> limit_cone (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) (\<cdot>\<^sub>J) (parallel_pair_diagram.map (\<cdot>\<^sub>J) f0 f1) (J.dom e) (parallel_pair_diagram.mkCone (\<cdot>\<^sub>J) f0 e)"
msubstneq: fixes a :: "poly"   and r :: "tm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and c :: "poly"   and t :: "tm"   and d :: "poly"   and s :: "tm" assumes "islin (Eq (CNP 0 a r))" shows "Ifm vs (x # bs) (msubstneq c t d s a r) = Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'b) # bs) (NEq (CNP 0 a r))"
split_list_last_propE: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'a list" and x :: "'a" and zs :: "'a list" where "xs = ys @ x # zs" and "P x" and "\<forall>z\<in>set zs. \<not> P z"
Derive_Derive: fixes \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and \<gamma> :: "symbol list" assumes "Derivation \<alpha> (D @ E) \<gamma>" shows "Derive (Derive \<alpha> D) E = Derive \<alpha> (D @ E)"
p_times_monomial_nonzero: fixes p :: "'a"   and j :: "nat" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "X^\<^bsup>R j\<^esup> \<cdot>\<^sub>r p \<noteq> \<zero>"
partial_connectTE: fixes s :: "'c net_state"   and i :: "nat"   and i' :: "nat"   and s' :: "'c net_state"   and S :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and T :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set" assumes "(s, connect(i, i'), s') \<in> pnet_sos S T" obtains sa :: "'c net_state" and s'a :: "'c net_state" and t :: "'c net_state" and t' :: "'c net_state" where "s = SubnetS sa t" and "s' = SubnetS s'a t'" and "(sa, connect(i, i'), s'a) \<in> S" and "(t, connect(i, i'), t') \<in> T"
remainder_expansion: fixes K :: "'a set"   and A :: "'a set"   and \<psi> :: "'a"   and B :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K \<turnstile> \<psi>"   and "\<not> B \<turnstile> \<psi>"   and "B \<in> K .\<bottom>. \<phi>" shows "B \<oplus> \<psi> = K"
invertible_commutes_with_inverse: fixes a :: "'a" assumes "f.source.additive.invertible a"   and "a \<in> A" shows "(g \<circ> f \<down> A) (f.source.additive.inverse a) = g.target.additive.inverse ((g \<circ> f \<down> A) a)"
nscd: shows "\<forall>i<length ds. P (cs ! i) (ds ! i)"
dist_in_closed_segment: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> closed_segment a b" shows "dist x a \<le> dist a b \<and> dist x b \<le> dist a b"
analz_image_freshK: fixes evs :: "event list"   and KK :: "nat set"   and K :: "nat" assumes "evs \<in> bankerb_gets"   and "KK \<subseteq> - range shrK" shows "(Key K \<in> analz (Key ` KK \<union> knows Spy evs)) = (K \<in> KK \<or> Key K \<in> analz (knows Spy evs))"
openE: fixes S :: "'b set"   and x :: "'b" assumes "open S"   and "x \<in> S" obtains e :: "real" where "0 < e" and "ball x e \<subseteq> S"
inverse_linear_lower: fixes a :: "real"   and x :: "real"   and b :: "real"   and alpha :: "real" assumes "0 < a"   and "x \<in> {a..b}"   and "- inverse (b * b) \<le> alpha" shows "inverse b + alpha * (x - b) \<le> inverse x"
inject_1: fixes a :: "'c"   and a' :: "'c" shows "(Inl a = Inl a') = (a = a')"
l6: fixes p :: "'node"   and R :: "'node set" shows "{q. not_blocked p R q} \<inter> R \<inter> W = {}"
Cons: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "x < y" shows "ord_class.lexordp (x # xs) (y # ys)"
Res: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and x :: "name" assumes "P \<longmapsto>\<alpha> \<prec> P'"   and "x \<sharp> \<alpha>" shows "\<lparr>\<nu>x\<rparr>P \<longmapsto>\<alpha> \<prec> \<lparr>\<nu>x\<rparr>P'"
rv_0: fixes A :: "((complex, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "(real, 'b) Finite_Cartesian_Product.vec" assumes "real_non_neg_mat A" shows "((\<chi>i. complex_of_real (v $h i)) = 0) = (v = 0)"
sigma_algebra_axioms: fixes M :: "'b measure" shows "sigma_algebra (space M) (sets M)"
varsf: shows "free_vars f \<subseteq> shift_var_set (shift_var_set V')"
ldrop_all: fixes xs :: "'c llist"   and m :: "enat" assumes "llength xs \<le> m" shows "ldrop m xs = LNil"
assms_14: shows "sparse_row_matrix r1 \<le> x"