hcju/isabelleps huggingface dataset:
finite_span: fixes S :: "'c set" assumes "finite S"   and "S \<subseteq> carrier V" shows "span S = {lincomb a S |a. a \<in> S \<rightarrow> carrier K}"
simps_2: fixes i :: "'a" shows "deps (Delete (DeleteMessage i)) = {i}"
coeff_sum_monom: fixes n :: "nat"   and d :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "n \<le> d" shows "coeff (\<Sum>i\<le>d. monom (f i) i) n = f n"
akra_bazzi_real_recursion_axioms: shows "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"
divide_pos_neg: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) < x"   and "y < (0::'a)" shows "x / y < (0::'a)"
continuous_on_imp_closedin: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "closedin (top_of_set (f ` S)) T" shows "closedin (top_of_set S) (S \<inter> f -` T)"
alpha_Tree_eqvt': fixes t1 :: "('b, 'c, 'd, 'e) Tree"   and t2 :: "('b, 'c, 'd, 'e) Tree"   and p :: "perm" shows "(t1 =\<^sub>\<alpha> t2) = (p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2)"
freeVarsFL_cons: fixes A :: "formula"   and Gamma :: "formula list" shows "freeVarsFL (A # Gamma) = freeVarsF A \<union> freeVarsFL Gamma"
Bseq_iff: fixes X :: "nat \<Rightarrow> 'a" shows "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N))"
map2_map_map: fixes h :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and f :: "'e \<Rightarrow> 'c"   and xs :: "'e list"   and g :: "'e \<Rightarrow> 'd" shows "map2 h (map f xs) (map g xs) = map (\<lambda>x. h (f x) (g x)) xs"
hn_refine_fail: fixes \<Gamma> :: "assn"   and c :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn" shows "hn_refine \<Gamma> c \<Gamma>' R FAIL"
hoare_tripleD_4: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set"   and \<sigma> :: "heap option"   and r :: "'a" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P"   and "run c (Some h) \<sigma> r" shows "lim h \<le> lim (the_state \<sigma>)"
square_integrable_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> real \<Rightarrow> real"   and S :: "real set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i square_integrable S"   and "S \<in> sets lebesgue" shows "(\<lambda>x. \<Sum>i\<in>I. f i x) square_integrable S"
2: fixes x :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "vec_of x \<in> set (map vec_of bs)" shows "x \<in> set bs"
simps: fixes fs :: "face list"   and n :: "nat"   and f :: "face list list"   and h :: "nat list" shows "faces (Graph fs n f h) = fs"
R': shows "(m', n') \<in> \<langle>A\<rangle>mset_rel"
dynProcModifyReturn_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 init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and ModifAbr :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return' c Q,A"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<Turnstile>\<^bsub>/UNIV\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
vs: shows "vs = fst (splitAt ram1 vs) @ [ram1] @ s"
iMOD_0_card: fixes r :: "nat" shows "card [ r, mod 0 ] = Suc 0"
Unit_eq_dividesone: fixes u :: "'a" assumes "u \<in> carrier G" shows "(u \<in> Units G) = u divides \<one>"
map_F_comp: fixes l1 :: "'ai \<Rightarrow> 'z"   and l2 :: "'aj \<Rightarrow> 'aa"   and l3 :: "'ak \<Rightarrow> 'ab"   and co1 :: "'al \<Rightarrow> 'ac"   and co2 :: "'am \<Rightarrow> 'ad"   and co3 :: "'an \<Rightarrow> 'ae"   and contra1 :: "'af \<Rightarrow> 'ao"   and contra2 :: "'ag \<Rightarrow> 'ap"   and contra3 :: "'ah \<Rightarrow> 'aq"   and l1' :: "'p \<Rightarrow> 'ai"   and l2' :: "'q \<Rightarrow> 'aj"   and l3' :: "'r \<Rightarrow> 'ak"   and co1' :: "'s \<Rightarrow> 'al"   and co2' :: "'t \<Rightarrow> 'am"   and co3' :: "'u \<Rightarrow> 'an"   and contra1' :: "'ao \<Rightarrow> 'v"   and contra2' :: "'ap \<Rightarrow> 'w"   and contra3' :: "'aq \<Rightarrow> 'x" shows "map_F l1 l2 l3 co1 co2 co3 contra1 contra2 contra3 \<circ> map_F l1' l2' l3' co1' co2' co3' contra1' contra2' contra3' = map_F (l1 \<circ> l1') (l2 \<circ> l2') (l3 \<circ> l3') (co1 \<circ> co1') (co2 \<circ> co2') (co3 \<circ> co3') (contra1' \<circ> contra1) (contra2' \<circ> contra2) (contra3' \<circ> contra3)"
Nml_Dom: fixes t :: "'c local.term" assumes "Nml t" shows "Nml (Dom t)"
elapsed_0D: fixes c :: "'c"   and u :: "'c \<Rightarrow> 'a"   and u' :: "'c \<Rightarrow> 'a" assumes "c \<in> \<X>"   and "elapsed u u' \<le> (0::'a)" shows "u' c - u c \<le> (0::'a)"
power_Suc2: fixes a :: "'a"   and n :: "nat" shows "a ^ Suc n = a ^ n \<cdot> a"
execute_nth_1: fixes i :: "nat"   and h :: "heap"   and a :: "'a array" assumes "i < Array.length h a" shows "execute (Array.nth a i) h = Some (Array.get h a ! i, h)"
GroupHomSetD_GroupHom: fixes T :: "'a \<Rightarrow> 'b"   and G :: "'a set"   and H :: "'b set" assumes "T \<in> GroupHomSet G H" shows "GroupHom G T"
sort'_def: fixes p :: "'a set list"   and l :: "('a, 'b) Combinators list" shows "sort' p l = FWNormalisationCore.sort l p"
mod_pCons_eq: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "pCons a p mod q = (if q = 0 then pCons a p else pCons a (p mod q) - smult (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) q)"
assms_5: fixes p :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and children' :: "(_) node_ptr list" assumes "h \<turnstile> get_child_nodes p \<rightarrow>\<^sub>r children"   and "h' \<turnstile> get_child_nodes p \<rightarrow>\<^sub>r children'" shows "set children' \<subseteq> set children"
coeff_eq: shows "local.coeff p1 = local.coeff p2"
perm_simps_1: fixes p :: "perm" shows "p \<bullet> Zero = Zero"
dynamically_complete_Liminf: fixes B :: "'a"   and Ns :: "'a set llist" assumes "B \<in> Bot"   and "chain (\<rhd>) Ns"   and "fair Ns"   and "lhd Ns \<Turnstile> {B}" shows "\<exists>B'\<in>Bot. B' \<in> Liminf_llist Ns"
strand_sem_eq_defs_2: fixes M :: "('fun, 'var) Term.term set"   and \<A> :: "('fun, 'var) strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "\<lbrakk>M; \<A>\<rbrakk>\<^sub>d' \<I> = \<lbrakk>M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>; \<A>\<rbrakk>\<^sub>d \<I>"
cross_ratio_reciprocal: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo"   and z :: "complex_homo" assumes "u \<noteq> v"   and "v \<noteq> w"   and "u \<noteq> w" shows "cross_ratio (reciprocal z) (reciprocal u) (reciprocal v) (reciprocal w) = cross_ratio z u v w"
analz_insertI: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<in> analz H" shows "X \<in> analz (insert Y H)"
simps_1: fixes tset1 :: "('e, 'f, 'g) Tree set['e]"   and tset2 :: "('e, 'f, 'g) Tree set['e]" shows "(tConj tset1 =\<^sub>\<alpha> tConj tset2) = rel_bset (=\<^sub>\<alpha>) tset1 tset2"
g: shows "(\<lambda>(x, y). g x y) \<in> M \<Otimes>\<^sub>M N \<rightarrow>\<^sub>M prob_algebra R"
append_child_heap_is_wellformed_preserved: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.append_child ptr child \<rightarrow>\<^sub>h h'" shows "Shadow_DOM.heap_is_wellformed h'"
genby_order2: fixes s :: "'b" assumes "s + s = (0::'b)" shows "\<langle>{s}\<rangle> = {0::'b, s}"
simps_3: fixes G :: "'a \<Rightarrow> 'b tm list \<Rightarrow> bool"   and H :: "('b, 'a) fm \<Rightarrow> bool"   and p :: "('b, 'a) fm"   and q :: "('b, 'a) fm" shows "boolean G H (p \<^bold>\<longrightarrow> q) = (boolean G H p \<longrightarrow> boolean G H q)"
dim_unique: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set"   and V :: "'b set"   and n :: "nat" assumes "vector_space scale"   and "B \<subseteq> V"   and "V \<subseteq> module.span scale B"   and "\<not> module.dependent scale B"   and "card B = n" shows "vector_space.dim scale V = n"
intro: fixes c :: "nat \<times> nat \<Rightarrow> int"   and s :: "nat"   and t :: "nat"   and N :: "nat" assumes "Network_Impl c s t"   and "Edka_Impl_axioms c N" shows "Edka_Impl c s t N"
card_Diff_singleton_if: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
h2: shows "well_def_signs (length qs) (get_signs_R (reduce_system_R p (qs, matrix, subsets, signs)))"
CARD_1_UNIV: fixes x :: "'a" shows "UNIV = {x}"
CEProof_def: shows "CEProof = (([], [CEReq]), [(0, Cut CQ1Concl), (0, Cut CQ2Concl), (1, Rrule CohideRR 0), (Suc (Suc 0), CQ (Differential (Const 0)) (Const 0) CQRightSubst), (1, Rrule CohideRR 0), (1, CQ (Differential (Var vid1)) ($' vid1) CQLeftSubst), (0, Rrule EquivR 0), (0, Lrule EquivForwardL 1), (Suc (Suc 0), Lrule EquivForwardL 1), (Suc (Suc (Suc 0)), CloseId 0 0), (Suc (Suc 0), CloseId 0 0), (Suc 0, CloseId 0 0), (0, Lrule EquivBackwardL (Suc (Suc 0))), (0, CloseId 0 0), (0, Lrule EquivBackwardL (Suc 0)), (0, CloseId 0 0), (0, CloseId 0 0)])"
monotoneD: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "monotone orda ordb f"   and "orda x y" shows "ordb (f x) (f y)"
intros_5: fixes p :: "'a pfm"   and r :: "'a pfm" assumes "\<turnstile>\<^sub>! p" shows "\<turnstile>\<^sub>! [r]\<^sub>! p"
vcomp_in_hhom: fixes \<nu> :: "'d"   and \<mu> :: "'d"   and a :: "'d"   and b :: "'d" assumes "D.seq \<nu> \<mu>"   and "src\<^sub>D \<nu> = a"   and "trg\<^sub>D \<nu> = b" shows "\<guillemotleft>\<nu> \<cdot>\<^sub>D \<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>"
leads_to_refl: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(\<box>(\<lambda>s. P s \<longrightarrow> (\<diamond>P) s)) \<sigma>"
type_bexpr: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'BExp" shows "\<Gamma> \<turnstile>\<^sub>b e \<in> \<Union> (to_total \<Gamma> ` bexp_vars e) "
normal_form_disjunct_with_flatten_sigma_2_length_def: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set"   and N :: "'b ltln set" shows "normal_form_disjunct_with_flatten_sigma_2_length \<phi> M N \<equiv> size \<phi>[M]\<^sub>\<Sigma>\<^sub>2 + (\<Sum>\<psi>\<in>M. size (\<psi>[N]\<^sub>\<Sigma>\<^sub>1) + 2) + (\<Sum>\<psi>\<in>N. size (\<psi>[M]\<^sub>\<Pi>\<^sub>1) + 2)"
lebesgue_measurable_scaling: fixes x :: "real" shows "(*\<^sub>R) x \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "f' y x (b1 + b2) = f' y x b1 + f' y x b2"
ext_inject: fixes black :: "bool"   and more :: "'a"   and black' :: "bool"   and more' :: "'a" shows "(\<lparr>black = black, \<dots> = more\<rparr> = \<lparr>black = black', \<dots> = more'\<rparr>) = (black = black' \<and> more = more')"
valid_refs_def: shows "valid_refs = (\<lambda>s. \<forall>x. (\<exists>xa xb. xb \<in> roots (s (mutator xa)) \<and> (xb reaches x) s) \<longrightarrow> obj_at (\<lambda>s. True) x s)"
mopup_a_nth: fixes q :: "nat"   and n :: "nat"   and x :: "nat" assumes "q < n"   and "x < 4" shows "mopup_a n ! (4 * q + x) = mopup_a (Suc q) ! (4 * q + x)"
assms: fixes x :: "'a" assumes "x \<in># X" shows "count X x = 1"
dist_fun_lt_imp_dist_val_lt: fixes f :: "'c \<Rightarrow>\<^sub>C 'd"   and g :: "'c \<Rightarrow>\<^sub>C 'd"   and e :: "real"   and x :: "'c" assumes "dist f g < e" shows "dist (apply_bcontfun f x) (apply_bcontfun g x) < e"
is_Guard_basic_simps_21: fixes F :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "is_Guard (guaranteeStrip F g c) = True"
RUV_split_le: fixes u :: "'a"   and p :: "'b"   and u' :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and v :: "'a"   and M :: "'d set"   and P :: "'b multiset"   and q :: "'b"   and Ml :: "'d set"   and Me :: "'d set"   and h :: "'d \<Rightarrow> 'd set"   and U :: "'a set"   and Ml' :: "'d set"   and Me' :: "'d set"   and h' :: "'d \<Rightarrow> 'd set"   and V :: "'a set" assumes "(u, Call p, u') \<in> edges fg"   and "proc_of fg v = p"   and "(v, M, P) \<in> S_cs fg 1"   and "q \<in># P"   and "(v, Ml, Me, h) \<in> RU_cs fg U"   and "(entry fg q, Ml', Me', h') \<in> RU_cs fg V"   and "(mon_n fg u \<union> mon fg p) \<inter> (Me \<union> Ml' \<union> Me') = {}"   and "h [*] h'" shows "(u, mon fg p \<union> M \<union> Ml, Me \<union> Ml' \<union> Me') \<in> RUV_cs fg U V"
subst_typ_combine_single: fixes fresh_idn :: "variable"   and \<tau> :: "typ"   and S :: "String.literal set"   and \<tau>2 :: "typ"   and idn :: "variable" assumes "fresh_idn \<notin> fst ` tvsT \<tau>" shows "subst_typ [((fresh_idn, S), \<tau>2)] (subst_typ [((idn, S), Tv fresh_idn S)] \<tau>) = subst_typ [((idn, S), \<tau>2)] \<tau>"
square_free_factorizationD'_4: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, bs)"   and "p = 0" shows "c = (0::'a) \<and> bs = []"
CAS_\<tau>red1r_xt3: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and v :: "'addr val"   and D :: "String.literal"   and F :: "String.literal"   and v' :: "'addr val" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e), xs) (Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e'), xs')"
disjoint_implies_distinct: fixes x :: "i"   and y :: "i" assumes "\<not> O x y" shows "x \<noteq> y"
f1_l0: fixes l :: "real" assumes "real l0 \<le> l / c" shows "f1 * (l / c) \<le> f1 * l - 1"
ord0_order_trans: fixes x :: "ord0"   and y :: "ord0"   and z :: "ord0" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
less_eq_def: fixes s :: "'a"   and s' :: "'a" shows "s \<preceq> s' \<equiv> \<exists>rs. s' = s \<star> rs"
funcset_vimage_diff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and C :: "'c set" assumes "f \<in> A \<rightarrow> B" shows "A - f -` (B - C) = A \<inter> f -` C"
in_closureI: fixes x :: "'c"   and X :: "'c set" assumes "x \<in> X" shows "x \<in> closure X"
that_3: shows "domain c1 \<inter> domain c2 \<subseteq> carrier"
assms_3: shows "x $ 1 \<noteq> y $ 1 \<or> x $ 2 \<noteq> y $ 2 \<or> x $ 3 \<noteq> y $ 3 \<or> x $ 4 \<noteq> y $ 4 \<or> x $ 5 \<noteq> y $ 5 \<or> x $ 6 \<noteq> y $ 6 \<or> x $ 7 \<noteq> y $ 7"
is_expansion_aux_imp_smallo'': fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and e' :: "real" assumes "is_expansion_aux xs f basis"   and "ms_exp_gt e' (ms_aux_hd_exp xs)" obtains e :: "real" where "e < e'" and "f \<in> o(\<lambda>x. hd basis x powr e)"
preserves_ide: fixes a :: "'c" assumes "Cop.ide a" shows "S.ide (F a)"
of_nat_ge_zero: fixes n :: "nat" shows "(0::'a) \<le> of_nat n"
that: fixes p' :: "('a, 'b, 'c, 'd) seqp" assumes "p = {l}send(fmsg) . p'"   and "l' \<in> labels \<Gamma> qq" shows "thesis"
inversions_append: fixes xs :: "'a list"   and ys :: "'a list" shows "inversions (xs @ ys) = inversions xs \<union> map_prod ((+) (length xs)) ((+) (length xs)) ` inversions ys \<union> map_prod id ((+) (length xs)) ` inversions_between xs ys"
Inf_F_to_Inf_FL: fixes \<iota>\<^sub>F :: "'f inference"   and Ll :: "'l list" assumes "\<iota>\<^sub>F \<in> Inf_F"   and "length Ll = length (prems_of \<iota>\<^sub>F)" shows "\<exists>L0. Infer (zip (prems_of \<iota>\<^sub>F) Ll) (concl_of \<iota>\<^sub>F, L0) \<in> Inf_FL"
echelon_form_of_det_def: fixes A :: "(('b, 'c) vec, 'c) vec"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "echelon_form_of_det A bezout = echelon_form_of_upt_k_det bezout (1::'b, A) (ncols A - 1)"
assms_3: assumes "x \<le> 0"   and "0 \<le> y" shows "P (gcd (- x) y)"
lookup_lift_poly_syz_alt: fixes n :: "nat"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and i :: "nat"   and t :: "'a"   and j :: "nat" shows "lookup (lift_poly_syz n b i) (term_of_pair (t, j)) = (if (t, j) = (0::'a, i) then 1::'c else if n \<le> j then lookup b (term_of_pair (t, j - n)) else (0::'c))"
1: shows "j < length M2 \<longrightarrow> (v # M \<otimes> M2) ! j = vec_mat_Tensor v M2 ! j"
WT_converterD_1: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and q :: "'h" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "q \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g run_converter conv q \<surd>"
divides_mult: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b c"   and "algebraic_semidom.coprime times one a b" shows "dvd.dvd times (times a b) c"
Suc_n_minus_m_eq: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n"   and "1 < m" shows "Suc (n - m) = n - (m - 1)"
ncs_2: fixes Iv :: "'a set \<times> (nat \<Rightarrow> QDelta)" shows "Iv \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set ncs = Iv \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set cs"
root_subterm: fixes t :: "'b trm" shows "t \<in> subterms_of t"
funion_strictE: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "c |\<in>| A |\<union>| B" obtains "c |\<in>| A"   | "c |\<notin>| A" and "c |\<in>| B"
monom_def: fixes a :: "'a"   and n :: "nat" shows "local.monom a n = a # replicate n \<zero>"
e: shows "nprv (insert \<phi> F) (exu x \<phi>)"
sets_Collect_countable_All': fixes I :: "'c set"   and P :: "'c \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> {x \<in> \<Omega>. P i x} \<in> M"   and "countable I" shows "{x \<in> \<Omega>. \<forall>i\<in>I. P i x} \<in> M"
distinct_1: fixes x2 :: "'a" shows "ERR \<noteq> OK x2"
ideal_points_moebius_circline: fixes M :: "moebius"   and H :: "circline" assumes "unit_circle_fix M"   and "is_poincare_line H" shows "ideal_points (moebius_circline M H) = moebius_pt M ` ideal_points H"
simps_3: fixes c1 :: "'a acom"   and c2 :: "'a acom"   and c1' :: "'a acom"   and c2' :: "'a acom" shows "(c1;; c2 \<le> c1';; c2') = (c1 \<le> c1' \<and> c2 \<le> c2')"
arrI: fixes f :: "'a" assumes "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>" shows "hom.arr f"
scc_non_empty: fixes E :: "('b \<times> 'b) set" shows "\<not> is_scc E {}"
Diff_subset_conv: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
ladder_stepdown_\<alpha>_0_altdef: fixes \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list"   and a1 :: "symbol list"   and A :: "symbol"   and a2 :: "symbol list" assumes "LeftDerivationLadder \<alpha> D L \<gamma>"   and "1 < length L"   and "splits_at (ladder_\<alpha> \<alpha> D L 1) (ladder_i L 1) a1 A a2" shows "ladder_stepdown_\<alpha>_0 \<alpha> D L = a1 @ snd (snd (D ! ladder_n L 0)) @ a2"
sq_mtx_mult_diag_diag: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" shows "sq_mtx_diag f * sq_mtx_diag g = (\<d>\<i>\<a>\<g> i. f i * g i)"
Invariants_Tinv: assumes "invertible_qmpt" shows "qmpt.Invariants M Tinv = Invariants"
H_def: fixes x :: "nat"   and y :: "nat" shows "H x y = lim (\<lambda>n. h' (k n) (x, y))"
keys_to_list_def: fixes p :: "('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "pp_pm.extended_ord.punit.keys_to_list p = pp_pm.extended_ord.punit.pps_to_list (keys p)"
U1_seqr: fixes P :: "('d, 'i) urel"   and Q :: "('i, 'g) urel" shows "\<lceil>P ;; Q\<rceil>\<^sub>1 = P ;; \<lceil>Q\<rceil>\<^sub>1"
that: fixes n :: "nat" assumes "f\<cdot>C\<^bsup>n\<^esup> \<noteq> \<bottom>" shows "thesis"
cm_preserves_inv_implications_nonneg: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('a, 'c) configuration"   and c1 :: "('a, 'c) configuration"   and loc :: "'a"   and t :: "'c"   and n :: "int" assumes "dataflow_topology summary results_in"   and "dataflow_topology.next_change_multiplicity' c0 c1 loc t n"   and "inv_implications_nonneg c0" shows "inv_implications_nonneg c1"
exhaust: fixes y :: "('a, 'b) out_port" obtains x1 :: "'a" where "y = Reg x1"   | x21 :: "'a" and x22 :: "('a, 'b) antecedent" where "y = Hyp x21 x22"
rep_eq: fixes x :: "nat"   and xa :: "'b poly" shows "lookup (mpoly_of_poly_aux x xa) = (\<lambda>m. if keys m \<subseteq> {x} then poly.coeff xa (lookup m x) else (0::'b))"
cobounded2: fixes b :: "'d multiset"   and a :: "'d multiset" shows "b \<subseteq># a \<union># b"
productF_path_reverse: fixes w :: "('d \<times> 'e) list"   and r1 :: "'f list"   and r2 :: "'g list"   and A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and p1 :: "'f"   and p2 :: "'g" assumes "length w = length r1"   and "length r1 = length r2"   and "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "path AB (w || r1 || r2) (p1, p2)"   and "p1 \<in> nodes A"   and "p2 \<in> nodes B" shows "path A (w || r1) p1 \<and> path B (w || r2) p2 \<or> target (w || r1 || r2) (p1, p2) = FAIL \<and> 0 < length w \<and> path A (butlast (w || r1)) p1 \<and> path B (butlast (w || r2)) p2 \<and> succ A (last w) (target (butlast (w || r1)) p1) = {} \<and> succ B (last w) (target (butlast (w || r2)) p2) \<noteq> {}"
add_spmat_commute: fixes a :: "(nat \<times> (nat \<times> 'a) list) list"   and b :: "(nat \<times> (nat \<times> 'a) list) list" shows "add_spmat a b = add_spmat b a"
piecewise_continuous_on_cong: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a"   and I :: "'a set"   and J :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "a = c"   and "b = d"   and "I = J"   and "\<And>x. \<lbrakk>c \<le> x; x \<le> d; x \<notin> J\<rbrakk> \<Longrightarrow> f x = g x" shows "piecewise_continuous_on a b I f = piecewise_continuous_on c d J g"
fv_snd_rcv_union: fixes x :: "('d, 'c) strand_step" shows "fv\<^sub>s\<^sub>t\<^sub>p x = fv\<^sub>s\<^sub>n\<^sub>d x \<union> fv\<^sub>r\<^sub>c\<^sub>v x \<union> fv\<^sub>e\<^sub>q assign x \<union> fv\<^sub>e\<^sub>q check x \<union> fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q x"
dependent_live_vars_ddep_empty_fst: fixes n'' :: "'node"   and V :: "'var"   and as :: "'edge list"   and n' :: "'node"   and V' :: "'var" assumes "n'' -{V}as\<rightarrow>\<^sub>d\<^sub>d n'"   and "V' \<in> Use n''" shows "(V', [], as) \<in> dependent_live_vars n'"
homeomorphic_empty_space: fixes X :: "'c topology"   and Y :: "'d topology" assumes "X homeomorphic_space Y" shows "(topspace X = {}) = (topspace Y = {})"
index_smult_mat_1: fixes i :: "nat"   and A :: "'d mat"   and j :: "nat"   and a :: "'d" assumes "i < dim_row A"   and "j < dim_col A" shows "(a \<cdot>\<^sub>m A) $$ (i, j) = a * A $$ (i, j)"
disj_def: fixes p :: "fm"   and q :: "fm" shows "Ferrack.disj p q = (if p = T \<or> q = T then T else if p = F then q else if q = F then p else if p = q then p else Or p q)"