hcju/isabelleps huggingface dataset:
list_of_dlist_inject: fixes x :: "'c set_dlist"   and y :: "'c set_dlist" shows "(list_of_dlist x = list_of_dlist y) = (x = y)"
char_inv: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and a :: "'c" assumes "character G \<chi>"   and "a \<in> carrier G" shows "\<chi> (inv\<^bsub>G\<^esub> a) = cnj (\<chi> a)"
arbitrary_union_of_inc: fixes P :: "'d set \<Rightarrow> bool"   and S :: "'d set" assumes "P S" shows "(arbitrary union_of P) S"
simps: fixes s :: "'a + 'b"   and a :: "'b" shows "setrp s a = (\<exists>x. a = x \<and> s = Inr x)"
inj_of_nat: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_char_0 one times plus zero" shows "inj (semiring_1.of_nat one plus zero)"
length_gt_zero: fixes xs :: "'b list" assumes "0 < length xs" shows "xs \<noteq> []"
min_wpl_is_optimal: fixes t :: "int tree"   and i :: "int"   and j :: "int"   and w :: "int \<Rightarrow> int \<Rightarrow> nat" assumes "inorder t = [i..j]" shows "Optimal_BST.min_wpl w i j \<le> wpl.wpl w i j t"
sorted_wrt_drop: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and n :: "nat" assumes "sorted_wrt f xs" shows "sorted_wrt f (drop n xs)"
PostConjI_sound: 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 P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and R :: "'e set"   and B :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c R,B" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c (Q \<inter> R),(A \<inter> B)"
nsqn_invalidate_other: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> kD rt"   and "dip \<notin> dom dests" shows "nsqn (invalidate rt dests) dip = nsqn rt dip"
set_concat: fixes xs :: "'a list list" shows "list.set (concat xs) = \<Union> (list.set ` list.set xs)"
eq_iff: fixes x :: "'b"   and y :: "'b"   and b :: "'b" shows "(x = y) = (flat_ord b x y \<and> flat_ord b y x)"
all_nth_imp_all_set: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>i<length xs. P (xs ! i)"   and "x \<in> set xs" shows "P x"
map_of_zip_upto2_length_eq_nth: fixes B :: "'b list"   and i :: "nat" assumes "distinct B"   and "i < length B" shows "map_of (zip B [0..<length B]) (B ! i) = Some i"
dual_dual_ord: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "less_eq_prod (dual_prod x) (dual_prod y) = less_eq_prod y x"
quietFrame: fixes P :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "quiet P" shows "insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F \<langle>\<epsilon>, \<Psi>\<rangle>"
zero_is_solution_homogeneous_system: fixes A :: "(('b, 'c) vec, 'd) vec" shows "0 \<in> solution_set A 0"
CompDefinedI: fixes f :: "'d"   and C :: "('e, 'd, 'f) Category_scheme"   and g :: "'d" assumes "f \<in> mor\<^bsub>C\<^esub>"   and "g \<in> mor\<^bsub>C\<^esub>"   and "cod\<^bsub>C\<^esub> f = dom\<^bsub>C\<^esub> g" shows "f \<approx>>\<^bsub>C\<^esub> g"
o_eq_dest_lhs: fixes a :: "'d \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'd"   and c :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "a \<circ> b = c" shows "a (b v) = c v"
test_bit_conj_lt: fixes x :: "'b word"   and m :: "nat" shows "(x !! m \<and> m < LENGTH('b)) = x !! m"
finite_vars_of_a_eq: fixes e1 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and e2 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" assumes "finite (Aexp.vars e1)"   and "finite (Aexp.vars e2)" shows "finite (Bexp.vars (\<lambda>\<sigma>. e1 \<sigma> = e2 \<sigma>))"
assms: shows "[\<lparr>p\<rparr> in v]"
sym_def: fixes hd :: "('a, 'b) hd" shows "hd.sym hd = (case hd of Sym x2 \<Rightarrow> x2)"
insert: fixes A :: "'d set"   and x :: "'d"   and g :: "'d \<Rightarrow> 'a" assumes "finite A"   and "x \<notin> A" shows "sum_0 g (insert x A) = g x + sum_0 g A"
generated_ringI_Basic: fixes A :: "'a set" assumes "A \<in> generated_ring" shows "A \<in> G.generated_ring"
push_forward_def: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real" shows "inclusion.push_forward X = restrict0 diff_fun_space (\<lambda>g. X (restrict0 sub.carrier (g \<circ> (\<lambda>x. x))))"
\<R>\<^sub>3_closed_glob_consistent: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" shows "closed_glob_consistent (\<R>\<^sub>3 \<Gamma>' \<S>' P')"
1: shows "xd \<in> X0 - trap \<times> UNIV"
openin_Int_closure_of_subset: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S" shows "S \<inter> X closure_of T \<subseteq> X closure_of (S \<inter> T)"
WellDefinedVarE: fixes S :: "('a, 'b) Signature"   and A :: "'a" assumes "Sig S \<triangleright> Vx : A \<turnstile> Vx : A" obtains "Sig S \<triangleright> \<turnstile> A Type"
Tuple: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and u :: "trm"   and U :: "ty" assumes "\<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> u : U" shows "\<Gamma> \<turnstile> \<langle>t, u\<rangle> : T \<otimes> U"
cross_ratio_swap_34: fixes p :: "proj2"   and q :: "proj2"   and s :: "proj2"   and r :: "proj2" shows "cross_ratio p q s r = 1 / cross_ratio p q r s"
subst_pp_id: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and f :: "'b \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>x. x \<in> keys t \<Longrightarrow> f x = monomial (1::'c) (monomial 1 x)" shows "subst_pp f t = monomial (1::'c) t"
Sup_subdistl: fixes x :: "'a"   and Y :: "'a set" shows "\<Squnion> ((\<cdot>) x ` Y) \<le> x \<cdot> \<Squnion> Y"
isnonconstant_def: fixes p :: "poly" shows "isnonconstant p = (\<not> isconstant p)"
simps_2: fixes n :: "nat"   and l :: "'b tree23"   and p :: "'b"   and r :: "'b tree23" shows "full\<^sub>i n (OF l p r) = (full n l \<and> full n r)"
pre_imp_pre_set: fixes s :: "'d"   and c :: "('d, 'e, 'f) ann"   and a :: "'d set" assumes "s \<in> pre c"   and "a \<in> pre_set c" shows "s \<in> a"
S_finite: fixes os :: "'c list" shows "finite {as. length as = length os \<and> set as \<subseteq> \<S>}"
assms: shows "eval cid t t' n \<gamma> \<or> eval cid t t' n \<gamma>'"
Rep_cfun_strict1: fixes x :: "'b" shows "\<bottom>\<cdot>x = \<bottom>"
2: fixes x :: "nat"   and xs :: "nat list"   and s :: "nat"   and x' :: "nat"   and s' :: "nat" assumes "cond2 a b (x # xs) s"   and "x' \<le> x"   and "s' \<le> s" shows "cond2 a b (x' # xs) s'"
Fs_dim_3: shows "mat_of_rows n (map gs1.gso [0..<length (map of_int_hom.vec_hom fs)]) \<in> carrier_mat (length (map of_int_hom.vec_hom fs)) n"
scast_distrib: fixes M :: "'b word \<Rightarrow> 'b word \<Rightarrow> 'b word"   and L :: "int \<Rightarrow> int \<Rightarrow> int"   and M' :: "'c word \<Rightarrow> 'c word \<Rightarrow> 'c word"   and a :: "'b word"   and b :: "'b word" assumes "\<And>x y. uint (M x y) = L (uint x) (uint y) mod 2 ^ LENGTH('b)"   and "\<And>x y. uint (M' x y) = L (uint x) (uint y) mod 2 ^ LENGTH('c)"   and "\<And>x y. L (x mod 2 ^ LENGTH('c)) (y mod 2 ^ LENGTH('c)) mod 2 ^ LENGTH('c) = L x y mod 2 ^ LENGTH('c)"   and "is_down SCAST('b \<rightarrow> 'c)" shows "SCAST('b \<rightarrow> 'c) (M a b) = M' (SCAST('b \<rightarrow> 'c) a) (SCAST('b \<rightarrow> 'c) b)"
l8_13_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and X :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "C \<noteq> D"   and "Tarski_neutral_dimensionless.Col Bet X A B"   and "Tarski_neutral_dimensionless.Col Bet X C D"   and "\<exists>U V. Tarski_neutral_dimensionless.Col Bet U A B \<and> Tarski_neutral_dimensionless.Col Bet V C D \<and> U \<noteq> X \<and> V \<noteq> X \<and> Tarski_neutral_dimensionless.Per Bet Cong U X V" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D"
assms: shows "\<langle>M; 0\<rangle> \<turnstile>\<^sub>n t"
multiple_orig_sub_system: fixes n :: "nat" assumes "0 < n" shows "sub_incidence_system \<V> \<B> \<V> (multiple_blocks n)"
app_vimageE: fixes b :: "V"   and r :: "V"   and A :: "V" assumes "b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A" obtains a :: "V" where "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" and "a \<in>\<^sub>\<circ> A"
trancl_to_strict_subtuple': fixes ym :: "'e \<rightharpoonup>\<^sub>f 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xm :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "acyclicP_on (fmran' ym) R"   and "(strict_subtuple (\<lambda>x y. R x y \<or> x = y))\<^sup>+\<^sup>+ xm ym" shows "strict_subtuple (\<lambda>x y. R x y \<or> x = y)\<^sup>*\<^sup>* xm ym"
abs_eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(abs_fset x = abs_fset y) = list_eq x y"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Cod \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = Cod t \<^bold>\<star> Cod u \<^bold>\<star> Cod v"
cases: fixes a :: "'c"   and r :: "('d \<times> 'c) set" assumes "a \<in> Range r" obtains aa :: "'d" and b :: "'c" where "a = b" and "(aa, b) \<in> r"
ConjunctiveD: fixes a :: "'c \<Rightarrow> 'c"   and A :: "('c \<Rightarrow> 'c) set" assumes "a \<in> Comp.Conjunctive" shows "a \<circ> Inf A = Inf ((\<circ>) a ` A)"
maximal_exists: fixes P :: "'g set \<Rightarrow> bool"   and N :: "nat"   and B :: "'g set" assumes "\<And>A. P A \<Longrightarrow> finite A \<and> card A \<le> N"   and "P B" shows "\<exists>A. finite A \<and> maximal A P"
ltln_weak_strong_stable_words_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = w \<Turnstile>\<^sub>n (\<phi> or\<^sub>n G\<^sub>n \<psi>[\<G>\<F> \<psi> w]\<^sub>\<Pi>\<^sub>1) M\<^sub>n \<psi>"
oa_facts_3: fixes v :: "i"   and x :: "\<nu>" shows "[\<^bold>\<diamond>\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<lparr>O!,x\<^sup>P\<rparr> in v]"
reduced_word_for_sum_list: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and A :: "'a set"   and a :: "'a"   and as :: "'a list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A a as" shows "monoid_add.sum_list plus zero as = a"
eucl_rel_poly_exists: fixes x :: "'b poly"   and y :: "'b poly" shows "\<exists>q r. eucl_rel_poly x y (q, r)"
except_id_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" shows "(except p S = p) = (keys p \<inter> S = {})"
mean_def: fixes s :: "real list" shows "mean s = sum_list s / real (length s)"
tendsto_vec_lambda: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c"   and a :: "'a \<Rightarrow> 'c"   and net :: "'b filter" assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net" shows "((\<lambda>x. \<chi>i. f x i) \<longlongrightarrow> (\<chi>i. a i)) net"
nMTcharn: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and p :: "'d list" shows "none_MT_rules C p = (\<forall>r\<in>set p. dom (C r) \<noteq> {})"
QCongAnFlat_def: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "QCongAnFlat a \<equiv> QCongA a \<and> (\<forall>A B C. a A B C \<longrightarrow> \<not> Bet A B C)"
totient_prime: fixes p :: "nat" assumes "prime p" shows "totient p = p - 1"
fpower_supdistl: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'b" assumes "mono f" shows "f \<circ> fstar g \<le> (\<Sqinter>i. f \<circ> fpower g i)"
Quotient_alt_def5: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Quotient R Abs Rep T = (T \<le> BNF_Def.Grp UNIV Abs \<and> BNF_Def.Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>)"
integrable_periodic_offset: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and c :: "real" assumes "integrable (lebesgue_on {a..b}) f"   and "\<And>x. f (x + (b - a)) = f x" shows "integrable (lebesgue_on {a..b}) (\<lambda>x. f (x + c))"
in_sstart: fixes s :: "'b stream"   and S :: "'b set"   and x :: "'b"   and xs :: "'b list" shows "(s \<in> sstart S (x # xs)) = (shd s = x \<and> stl s \<in> sstart S xs)"
baE: fixes t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'BB"   and n :: "nat"   and \<phi> :: "'BB \<Rightarrow> bool" assumes "pb.eval the_bb t t' n [\<phi>]\<^sub>b" shows "\<phi> (\<sigma>\<^bsub>the_bb\<^esub>t n)"
vdisjnt_vinsert_left: fixes a :: "V"   and X :: "V"   and Y :: "V" shows "vdisjnt (vinsert a X) Y = (a \<notin>\<^sub>\<circ> Y \<and> vdisjnt X Y)"
lang_of_append_rexp: fixes r :: "'b rexp"   and trm :: "'b trm" shows "lang_trm (Append_rexp r trm) = lang_trm trm \<cdot> lang r"
cod_in_codomains: fixes f :: "'b" assumes "C'.codomains f \<noteq> {}" shows "C'.cod f \<in> C'.codomains f"
matchC_C_matchC_LC: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.matchC_C aval tval cval indis theta c d" shows "PL_Indis.matchC_LC aval tval cval indis theta c d"
d_ipurge_unwinding: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b" assumes "deterministic P" shows "secure P I D = d_weakly_future_consistent P I D (rel_ipurge P I D)"
finite_ReZ_segments_congE: fixes p1 :: "real \<Rightarrow> complex"   and z1 :: "complex"   and p2 :: "real \<Rightarrow> complex"   and z2 :: "complex" assumes "finite_ReZ_segments p1 z1"   and "\<And>t. \<lbrakk>0 < t; t < 1\<rbrakk> \<Longrightarrow> Re (p1 t - z1) = Re (p2 t - z2)" shows "finite_ReZ_segments p2 z2"
assms_8: shows "moebius_pt M2 z2 = w2"
Full_fairness: shows "\<turnstile> \<box>[FullNxt]_vars \<and> WF(Deq mid q2 out)_vars \<and> \<box>WF(Deq inp q1 mid)_vars \<longrightarrow> WF(Deq inp qc out)_(inp, qc, out)"
A0: shows "A *v (\<chi>i. complex_of_real (v $h i)) = 0"
segment_to_closest_point: fixes S :: "'b set"   and a :: "'b" assumes "closed S"   and "S \<noteq> {}" shows "open_segment a (closest_point S a) \<inter> S = {}"
simps: fixes zero :: "'c"   and A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" shows "pivot_positions_main_gen zero A nr nc i j = (if i < nr then if j < nc then if A $$ (i, j) = zero then pivot_positions_main_gen zero A nr nc i (Suc j) else (i, j) # pivot_positions_main_gen zero A nr nc (Suc i) (Suc j) else [] else [])"
map_filter_simps_2: fixes f :: "'e \<Rightarrow> 'd option" shows "List.map_filter f [] = []"
\<gamma>: fixes p :: "('a \<times> 'a set) set" assumes "p tagged_division_of cbox a b"   and "\<gamma> fine p" shows "norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) - i) < norm i - B * content (cbox a b)"
Pow_bottom: fixes B :: "'b set" shows "{} \<in> Pow B"
Hist_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "Hist rmhist = (\<forall>p. HistP rmhist p)"
set_plus_rearrange2: fixes a :: "'b"   and b :: "'b"   and C :: "'b set" shows "a +o (b +o C) = (a + b) +o C"
degree_prod_sum_le: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "finite S" shows "degree (prod f S) \<le> sum (degree \<circ> f) S"
runit_coherence: fixes f :: "'c" assumes "C.ide f" shows "\<r>\<^sub>D[F f] = F \<r>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f))"
SPEC_rule_conjI: fixes A :: "'a nres"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A \<le> SPEC P"   and "A \<le> SPEC Q" shows "A \<le> SPEC (\<lambda>v. P v \<and> Q v)"
leading_term_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" shows "leading_term R f = monom (UP R) (f (deg R f)) (deg R f)"
logic_actual_nec_3: fixes \<phi> :: "'b \<Rightarrow> \<o>" shows "[[\<^bold>\<A>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<^bold>\<A>\<phi> \<alpha>)]]"
If: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com" assumes "\<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s} c\<^sub>1 {Q}"   and "\<turnstile>\<^sub>t {\<lambda>s. P s \<and> \<not> bval b s} c\<^sub>2 {Q}" shows "\<turnstile>\<^sub>t {P} IF b THEN c\<^sub>1 ELSE c\<^sub>2 {Q}"
empty: shows "(\<lambda>x. \<epsilon> (''1'' @ x)) \<cdot> (\<lambda>x. \<epsilon> (tl x)) = \<epsilon>"
exists_no_Fvars: shows "\<exists>\<phi>. \<phi> \<in> fmla \<and> prv \<phi> \<and> Fvars \<phi> = {}"
sub_add: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "h \<in> carrier P" shows "Cring_Poly.compose R (f \<oplus>\<^bsub>P\<^esub> h) g = Cring_Poly.compose R f g \<oplus>\<^bsub>P\<^esub> Cring_Poly.compose R h g"
norm_uminus_minus: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus (uminus x) y) = norm (plus x y)"
assms_2: fixes M :: "'a"   and xvec :: "name list"   and N :: "'a" assumes "\<alpha> = M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>" shows "Prop"
multrow_mat_inv: fixes k :: "nat"   and n :: "nat"   and a :: "'a" assumes "k < n"   and "a \<noteq> (0::'a)" shows "multrow_mat n k a * multrow_mat n k (inverse a) = 1\<^sub>m n"
lead_coeff_pderiv: fixes p :: "'a poly" shows "lead_coeff (pderiv p) = of_nat (degree p) * lead_coeff p"
q: shows "q = monomial (1::'b) (term_of_pair (0::'a, component_of_term u))"
has_vector_derivative_transform_within_open: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and S :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "(f has_vector_derivative f') (at x)"   and "open S"   and "x \<in> S"   and "\<And>y. y \<in> S \<Longrightarrow> f y = g y" shows "(g has_vector_derivative f') (at x)"
\<phi>_simps_1: shows "arr t\<^sub>0u\<^sub>1.\<phi>"
typedef_right_total: fixes Rep :: "'b \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and T :: "'d \<Rightarrow> 'b \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "right_total T"
p2: fixes y :: "xml" shows "monotone sum_bot.le_fun sum_bot_ord (p2 y)"
poly_div_minus_left: fixes x :: "'a poly"   and y :: "'a poly" shows "- x div y = - (x div y)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>i j. \<lbrakk>i < j \<Longrightarrow> P (i + 1) j; i < j \<Longrightarrow> P i (j - 1)\<rbrakk> \<Longrightarrow> P i j" shows "P a0 a1"
ipoly_mult_rat_main: fixes d :: "int"   and n :: "int"   and p :: "int poly"   and x :: "'a" assumes "d \<noteq> 0"   and "n \<noteq> 0" shows "ipoly (poly_mult_rat_main n d p) x = of_int n ^ degree p * ipoly p (x * of_int d / of_int n)"
Int: fixes a :: "'d set"   and M :: "'d measure"   and b :: "'d set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a \<inter> b \<in> sets M"
assms: shows "the (rt1 ip) \<sqsubseteq> the (rt2 ip)"
ex_head_tail: fixes t :: "lam" shows "\<exists>ts h. t = h \<degree>\<degree> ts \<and> ((\<exists>n. h = Var n) \<or> (\<exists>x u. h = (Lam [x].u)))"
indep_Y: shows "indep_vars (\<lambda>i. borel) Y I"
r_neg1: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> (x \<oplus>\<^bsub>G\<^esub> y) = y"
WhileTRedsThrow: fixes b :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 c :: "char list exp"   and e :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>b,s\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>c,s\<^sub>1\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>while (b) c,s\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2\<rangle>"
subst_atm_mset_empty: fixes \<sigma> :: "'s" shows "{#} \<cdot>am \<sigma> = {#}"
awalk_verts_append: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and q :: "('a \<times> 'a) list"   and v :: "'a" assumes "G.awalk u (p @ q) v" shows "pawalk_verts u (p @ q) = pawalk_verts u p @ tl (pawalk_verts (last (pawalk_verts u p)) q)"
list_all2_map1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and as :: "'e list"   and bs :: "'d list" shows "list_all2 P (map f as) bs = list_all2 (\<lambda>x. P (f x)) as bs"
intros_2: fixes f :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "Sem (Basic f) (Some s) (Some (f s))"
scale_minus_left: fixes a :: "'a"   and x :: "'b" shows "- a *a x = - (a *a x)"
y_1: shows "\<forall>x. 0 < x \<and> x < y \<longrightarrow> \<psi> j x \<down>= 0 \<or> \<psi> j x \<down>= 1"
assms_2: fixes x :: "'a" assumes "x \<in> I" shows "M x = N x"
column_eq_fst_QR_decomposition: fixes A :: "((real, 'a) vec, 'b) vec"   and i :: "'a"   and ia :: "'a" assumes "rank A = ncols A"   and "column i (fst (QR_decomposition A)) = column ia (fst (QR_decomposition A))" shows "i = ia"
fun_pow_iterates_above: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and k :: "nat"   and a :: "'b" assumes "bourbaki_witt_fixpoint lub leq f" shows "(f ^^ k) a \<in> bourbaki_witt_fixpoint.iterates_above lub leq f a"
p_and_not_p: fixes P :: "'d upred" shows "(P \<and> \<not> P) = false"
asymptotics3: shows "\<forall>\<^sub>F x in at_top. (1 + ln x powr (- e / 2)) / 2 \<le> 1"
lf: shows "locally_finite_in X ((\<lambda>i. T i \<inter> f i -` U) ` I)"
g': fixes y :: "'a" assumes "y \<in> V" shows "g' y = inv (blinfun_apply (invf o\<^sub>L f' (g y + x0)))"
countable'': fixes y :: "'a" shows "countable {x. (x, y) \<in> support_flow (\<Squnion> Y)}"
poly_as_sum_of_monoms': fixes p :: "'b poly"   and n :: "nat" assumes "degree p \<le> n" shows "(\<Sum>i\<le>n. X ^ i * C (coeff p i)) = p"
iffb_inv1: shows "fb\<^sup>-\<^sub>\<F> \<circ> fb\<^sub>\<F> = id"
abs_eq: fixes xa :: "'b ltl set"   and x :: "'b ltl" shows "\<up>eval\<^sub>G xa (Abs x) = Abs (eval\<^sub>G xa x)"