hcju/isabelleps huggingface dataset:
div_field_poly_impl_def: fixes f :: "'b poly"   and g :: "'b poly" shows "div_field_poly_impl f g = (let cg = coeffs g in if cg = [] then 0 else let cf = coeffs f; ilc = inverse (last cg); ch = map ((*) ilc) cg; q = fst (divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg)) in poly_of_list (map ((*) ilc) q))"
addition_LE_def: shows "addition_LE \<equiv> inv_image Abacus.lex_triple addition_measure"
reachable_init: fixes s :: "'c"   and A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool" assumes "s \<in> init A" shows "reachablep A I s"
coeff_max_nonzeroTr: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "\<exists>j\<le>fst c. snd c j \<noteq> \<zero>\<^bsub>S\<^esub>" shows "snd c (c_max S c) \<noteq> \<zero>\<^bsub>S\<^esub>"
lead_coeff_prod: fixes f :: "'d \<Rightarrow> 'c poly"   and A :: "'d set" shows "lead_coeff (prod f A) = (\<Prod>x\<in>A. lead_coeff (f x))"
integral_bernoulli_pmf: fixes p :: "real"   and f :: "bool \<Rightarrow> real" assumes "0 \<le> p"   and "p \<le> 1" shows "measure_pmf.expectation (bernoulli_pmf p) f = f True * p + f False * (1 - p)"
not_Cons_self2: fixes x :: "'d"   and xs :: "'d list" shows "x # xs \<noteq> xs"
ht_hash_in_dom_bounded_hashcode_nat_in_dom: fixes l :: "('a \<times> 'b) list list"   and k :: "'a" assumes "ht_hash l"   and "1 < length l"   and "k \<in> dom (map_of (l ! bounded_hashcode_nat (length l) k))" shows "k \<in> dom (map_of (concat l))"
induct: fixes P :: "bool list \<Rightarrow> trie \<Rightarrow> bool"   and a0 :: "bool list"   and a1 :: "trie" assumes "\<And>ks. P ks Lf"   and "\<And>ks b lr. (\<And>x21 x22 x. ks = x21 # x22 \<Longrightarrow> P x22 x) \<Longrightarrow> P ks (Nd b lr)" shows "P a0 a1"
unless_def: fixes A :: "'c set"   and B :: "'c set" shows "A unless B \<equiv> A - B co A \<union> B"
f_3: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal r"
of_int_take_bit: fixes n :: "nat"   and k :: "int" shows "of_int (take_bit n k) = take_bit n (of_int k)"
Hyp: fixes c :: "class" assumes "class G C = Some c"   and "ws_prog G"   and "C \<noteq> Object" shows "wf_prog G \<longrightarrow> (\<forall>c m. class G (super c) = Some c \<longrightarrow> methd G (super c) sig = Some m \<longrightarrow> G\<turnstile>super c\<preceq>\<^sub>C declclass m)"
const_inject: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and name\<^sub>1 :: "name"   and name\<^sub>2 :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "const name\<^sub>1 = const name\<^sub>2" shows "name\<^sub>1 = name\<^sub>2"
cases: fixes a1 :: "'b set"   and a2 :: "'c"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" assumes "(a1, a2) \<in> foldSetD D f e" obtains "a1 = {}" and "a2 = e" and "e \<in> D"   | x :: "'b" and A :: "'b set" and y :: "'c" where "a1 = insert x A" and "a2 = f x y" and "x \<notin> A" and "f x y \<in> D" and "(A, y) \<in> foldSetD D f e"
f_expand_nth_mod_eq_0: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" assumes "n < length xs * k"   and "n mod k = 0" shows "xs \<odot> k ! n = xs ! (n div k)"
dist_real_def: fixes x :: "real"   and y :: "real" shows "dist x y = \<bar>x - y\<bar>"
dirichlet_inverse_gt_1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and i :: "'b" assumes "1 < n" shows "dirichlet_inverse f i n = - i * (\<Sum>d | d dvd n \<and> d < n. f (n div d) * dirichlet_inverse f i d)"
ord_resolve: fixes CAs :: "'a literal multiset list"   and n :: "nat"   and Cs :: "'a literal multiset list"   and AAs :: "'a multiset list"   and As :: "'a list"   and D :: "'a literal multiset" assumes "length CAs = n"   and "length Cs = n"   and "length AAs = n"   and "length As = n"   and "n \<noteq> 0"   and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)"   and "\<forall>i<n. AAs ! i \<noteq> {#}"   and "\<forall>i<n. \<forall>A\<in>#AAs ! i. A = As ! i"   and "eligible As (D + negs (mset As))"   and "\<forall>i<n. strictly_maximal_wrt (As ! i) (Cs ! i)"   and "\<forall>i<n. S (CAs ! i) = {#}" shows "ord_resolve CAs (D + negs (mset As)) AAs As (\<Sum>\<^sub># (mset Cs) + D)"
InvariantVarsFAfterApplyLearn: fixes state :: "State"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "getConflictFlag state"   and "InvariantVarsF (getF state) F0 Vbl"   and "InvariantVarsM (getM state) F0 Vbl" shows "let state' = applyLearn state in InvariantVarsF (getF state') F0 Vbl"
checkpoint_nsc_code_def: shows "checkpoint_nsc_code = checkpoint_nsc checkpoint_s"
sat: shows "lst ! (db + i) \<lbrace> \<langle>X\<rangle> \<rbrace> = c $ i"
next_tame_def: fixes p :: "nat" shows "next_tame\<^bsub>p\<^esub> \<equiv> filter (\<lambda>g. \<not> final g \<or> is_tame g) \<circ> next_tame0\<^bsub>p\<^esub>"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "Fun_g.map f = C.null"
suffix_def: fixes xs :: "'a list"   and ys :: "'a list" shows "suffix xs ys = (\<exists>zs. ys = zs @ xs)"
wf_less_multiset: shows "wf {(M, N). M < N}"
add_0_left: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "mbot \<oplus> a = a"
truncate_down_zero: fixes prec :: "nat" shows "truncate_down prec 0 = 0"
dim_row: fixes n :: "nat"   and A :: "complex mat" assumes "gate n A" shows "dim_row A = 2 ^ n"
ipath_suffix: fixes E :: "('b \<times> 'b) set"   and r :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "ipath E r" shows "ipath E (suffix i r)"
simps_8: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and P :: "('f, 'd, 'e) psi" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.extractFrame SCompose SBottom (!P) = \<langle>\<epsilon>, SBottom\<rangle>"
assms: shows "\<forall>i<length xs. 0 < scf i"
simps: fixes instr :: "instr"   and i :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list" shows "iexec instr (i, s, stk) = (case instr of LOADI n \<Rightarrow> (i + 1, s, n # stk) | LOAD x \<Rightarrow> (i + 1, s, s x # stk) | ADD \<Rightarrow> (i + 1, s, (hd2 stk + hd stk) # tl2 stk) | STORE x \<Rightarrow> (i + 1, s(x := hd stk), tl stk) | JMP n \<Rightarrow> (i + 1 + n, s, stk) | JMPLESS n \<Rightarrow> (if hd2 stk < hd stk then i + 1 + n else i + 1, s, tl2 stk) | JMPGE n \<Rightarrow> (if hd stk \<le> hd2 stk then i + 1 + n else i + 1, s, tl2 stk))"
f8: shows "\<not> (\<exists>(a', b', c')\<in>set c. ((a' \<noteq> 0 \<or> b' \<noteq> 0) \<and> a' * r\<^sup>2 + b' * r + c' = 0) \<and> (\<forall>(d, e, f)\<in>set a. d * r\<^sup>2 + e * r + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. d * r\<^sup>2 + e * r + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. d * r\<^sup>2 + e * r + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. d * r\<^sup>2 + e * r + f \<noteq> 0))"
interleave_empty: fixes a :: "'d list"   and b :: "'d list" shows "([] \<in> a \<otimes> b) = (a = [] \<and> b = [])"
select_convs_2: fixes Obj :: "'h set"   and Mor :: "'g set"   and Dom :: "'g \<Rightarrow> 'h"   and Cod :: "'g \<Rightarrow> 'h"   and Id :: "'h \<Rightarrow> 'g"   and Comp :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and more :: "'i" shows "mor\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub> = Mor"
l_get_owner_document_wf_axioms_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_owner_document_wf_axioms heap_is_wellformed type_wf known_ptrs get_disconnected_nodes get_owner_document get_parent \<equiv> ((\<forall>h document_ptr disc_nodes node_ptr. heap_is_wellformed h \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes \<longrightarrow> node_ptr \<in> set disc_nodes \<longrightarrow> h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r document_ptr) \<and> (\<forall>h node_ptr owner_document disc_nodes. heap_is_wellformed h \<longrightarrow> h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None \<longrightarrow> h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r owner_document \<longrightarrow> h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> node_ptr \<in> set disc_nodes)) \<and> (\<forall>h ptr owner_document. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document \<longrightarrow> owner_document |\<in>| document_ptr_kinds h) \<and> (\<forall>h ptr. heap_is_wellformed h \<longrightarrow> known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> ptr |\<in>| object_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_owner_document ptr)"
list_of_LCons: fixes xs :: "'c llist"   and x :: "'c" assumes "lfinite xs" shows "list_of (LCons x xs) = x # list_of xs"
ack_less_mono1: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j" shows "ack i k < ack j k"
sg_unit_closed: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "\<one>\<^bsub>G\<^esub> \<in> H"
norm_axis_1: fixes m :: "'a" shows "\<parallel>\<e> m\<parallel> = 1"
PPs_closed_plus: fixes s :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t \<in> .[X]" shows "s + t \<in> .[X]"
edge_end: fixes p :: "'a"   and q :: "'a" assumes "is_point p"   and "is_point q"   and "p \<noteq> q" shows "end_points (p ; q\<^sup>T) = q"
extremum_strict: fixes a :: "'a" shows "\<not> \<top> < a"
eq_zero_iff: fixes x :: "nat" shows "(principal_dchar n x = 0) = (\<not> coprime x n)"
marked_not_white: fixes r :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "white r s" shows "\<not> marked r s"
star_of_Suc_lessI: fixes n :: "nat"   and N :: "nat star" assumes "hypnat_of_nat n < N"   and "hypnat_of_nat (Suc n) \<noteq> N" shows "hypnat_of_nat (Suc n) < N"
ord_le_eq_trans: fixes a :: "'b Word.word"   and b :: "'b Word.word"   and c :: "'b Word.word" assumes "a \<le>s b"   and "b = c" shows "a \<le>s c"
geom_rand_walk_strictly_positive: assumes "0 < init"   and "0 < d"   and "d < u" shows "\<forall>n w. 0 < geom_proc n w"
fct_gen_subalgebra_space: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c" shows "space (fct_gen_subalgebra M N X) = space M"
max_0_1_4: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq (numeral.numeral one plus x) zero = numeral.numeral one plus x"
updS_def: fixes s :: "'a bdd_scheme"   and x :: "nat \<times> nat \<times> nat"   and r :: "nat" shows "updS s x r = dcl_update (\<lambda>m. m(x \<mapsto> r)) s"
ANR_delete: fixes S :: "'a set"   and a :: "'a" assumes "ANR S" shows "ANR (S - {a})"
Osum_antisym: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "antisym r"   and "antisym r'" shows "antisym (r \<union>o r')"
r_div2_def: shows "r_div2 \<equiv> r_shrink (Pr 1 Z (Cn 3 r_ifle [Cn 3 r_mul [r_constn 2 2, Cn 3 S [recf.Id 3 0]], recf.Id 3 2, Cn 3 S [recf.Id 3 1], recf.Id 3 1]))"
existence_ivl_empty2: fixes x0 :: "'a"   and t0 :: "real" assumes "x0 \<notin> X" shows "existence_ivl t0 x0 = {}"
inext_nth_atLeast: fixes n :: "nat"   and a :: "nat" shows "{n..} \<rightarrow> a = n + a"
s4: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.antidomain_left_monoid antidomain_op one times" shows "times (antidomain_op x) (antidomain_op (times (antidomain_op x) y)) = times (antidomain_op x) (antidomain_op y)"
ideal_ring_multiple1: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and x :: "'e"   and r :: "'e" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "r \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> r \<in> I"
I_def: shows "I = (\<lambda>t. integral {0..t} f)"
append_eq_append_conv_if: fixes xs\<^sub>1 :: "'a list"   and xs\<^sub>2 :: "'a list"   and ys\<^sub>1 :: "'a list"   and ys\<^sub>2 :: "'a list" shows "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) = (if |xs\<^sub>1| \<le> |ys\<^sub>1| then xs\<^sub>1 = take |xs\<^sub>1| ys\<^sub>1 \<and> xs\<^sub>2 = drop |xs\<^sub>1| ys\<^sub>1 @ ys\<^sub>2 else take |ys\<^sub>1| xs\<^sub>1 = ys\<^sub>1 \<and> drop |ys\<^sub>1| xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)"
simps_4: fixes b :: "bool"   and e :: "expr"   and r :: "ref_ty" shows "assigns_if b (e InstOf r) = assignsE e"
jw_Lam_inv: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and x :: "Syntax.var"   and e :: "Syntax.term"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile>\<^sub>W Lam x e : \<tau>"   and "atom x \<sharp> \<Gamma>" obtains \<tau>\<^sub>1 :: "ty" and \<tau>\<^sub>2 :: "ty" where "\<tau> = Fun \<tau>\<^sub>1 \<tau>\<^sub>2" and "\<Gamma>(x $$:= \<tau>\<^sub>1) \<turnstile>\<^sub>W e : \<tau>\<^sub>2"
foundation5: fixes \<tau> :: "'d state \<times> 'd state"   and P :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> P and Q" shows "\<tau> \<Turnstile> P \<and> \<tau> \<Turnstile> Q"
mon_c_single: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and s :: "'b list" shows "mon_c fg {#s#} = mon_s fg s"
Zp_nat_mult_closed: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp" shows "[n] \<cdot> x \<in> carrier Zp"
flow_isCont_state_space: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> existence_ivl0 x0" shows "isCont (\<lambda>(x, t). flow0 x (t - t0)) (x0, t)"
HR: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "rel_aka.H p x q = (x \<subseteq> rel_R p q)"
sq_norm_fs_mu_g_bound: fixes fs :: "int vec list"   and i :: "nat"   and bnd :: "rat" assumes "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"   and "length fs = m"   and "i < m"   and "\<mu>_bound_row fs bnd i"   and "g_bound fs" shows "rat_of_int \<parallel>fs ! i\<parallel>\<^sup>2 \<le> rat_of_nat (Suc i * N) * bnd"
tendsto_minus_cancel_left: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c"   and F :: "'b filter" shows "(f \<longlongrightarrow> - y) F = ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
prod_relE: fixes p :: "'d \<times> 'e"   and p' :: "'f \<times> 'g"   and R1 :: "('d \<times> 'f) set"   and R2 :: "('e \<times> 'g) set" assumes "(p, p') \<in> R1 \<times>\<^sub>r R2" obtains a :: "'d" and b :: "'e" and a' :: "'f" and b' :: "'g" where "p = (a, b)" and "p' = (a', b')" and "(a, a') \<in> R1" and "(b, b') \<in> R2"
lspasl_c_der: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a" assumes "h1,h2\<triangleright>h0"   and "h1,h3\<triangleright>h0" shows "(h1,h2\<triangleright>h0) \<and> h2 = h3"
fls_subdegree_ge0I: fixes f :: "'a fls" assumes "\<And>k. k < 0 \<Longrightarrow> fls_nth f k = (0::'a)" shows "0 \<le> fls_subdegree f"
vimage2p_def: fixes f :: "'b \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'f"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> 'd" shows "BNF_Def.vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
cauchy_def: fixes X :: "nat \<Rightarrow> rat" shows "cauchy X = (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
in_rtrancl_UnI: fixes x :: "'b \<times> 'b"   and R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "x \<in> R\<^sup>* \<or> x \<in> S\<^sup>*" shows "x \<in> (R \<union> S)\<^sup>*"
except_single: fixes u :: "'b"   and c :: "'c"   and S :: "'b set" shows "except (monomial c u) S = (monomial c u when u \<notin> S)"
group_incl_imp_subgroup: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "Group.group (G\<lparr>carrier := H\<rparr>)" shows "subgroup H G"
\<mu>: shows "local.iso \<mu> \<and> \<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>"
hom_add: fixes x :: "'a poly poly"   and y :: "'a poly poly"   and xa :: "'a"   and ya :: "'a" shows "poly2 (x + y) xa ya = poly2 x xa ya + poly2 y xa ya"
CFG_wf_axioms: shows "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"
not_squarefreeI: fixes x :: "'b"   and n :: "'b" assumes "x\<^sup>2 dvd n"   and "\<not> x dvd (1::'b)" shows "\<not> squarefree n"
assms: shows "\<tau>exec_meth ci P ins xt t h s s'"
add_mem_right_cancel: fixes x :: "V"   and y :: "V"   and z :: "V" shows "(x + y \<in> elts (x + z)) = (y \<in> elts z)"
a_preserves_SNa: fixes M :: "trm"   and M' :: "trm" assumes "SNa M"   and "M \<longrightarrow>\<^sub>a M'" shows "SNa M'"
return_spmf_None_parametric: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_spmf A (return_pmf None) (return_pmf None)"
cobounded1: fixes a :: "'g"   and b :: "'g" shows "min a b \<le> a"
insert_insert_body: fixes e1 :: "'c \<times> 'd \<times> bool"   and e2 :: "'c \<times> 'd \<times> bool"   and i2 :: "'c option"   and xs :: "('c \<times> 'd \<times> bool) list" assumes "fst e1 \<noteq> fst e2"   and "i2 \<noteq> Some (fst e1)" shows "insert (insert_body xs e1) e2 i2 = insert xs e2 i2 \<bind> (\<lambda>ys. Some (insert_body ys e1))"
covie: assumes "ov\<inverse> \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "ov\<inverse> O e = ov\<inverse>"
assms_1: shows "(s1, s2) \<in> trm_ord \<or> (s1, t2) \<in> trm_ord"
mod_ring_mult32: shows "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (mult_p32 pp) (*)"
partitions_remove1: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat" assumes "p partitions n"   and "0 < p k" shows "p(k := p k - 1) partitions n - k"
a_acq: shows "a \<in> acquired pending_write (x # sb) (A \<union> B)"
forward: assumes "csa \<in> consistent_sign_vectors fs UNIV \<and> has_no_zeros csa" shows "csa \<in> set (characterize_consistent_signs_at_roots (coprime_r (cast_rat_list fs)) (cast_rat_list fs))"
no_infinite_subset_chain: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "finite UNIV"   and "mono \<tau>"   and "\<forall>i. (\<tau> ^^ i) {} \<subset> (\<tau> ^^ (i + 1)) {}" shows "False"
simps_6: fixes v :: "'b rexp" shows "toplevel_summands (Star v) = {Star v}"
Red_I_eq_Red_I_diff_Red_F: fixes N :: "'f set" shows "Red_I N = Red_I (N - Red_F N)"
diagram_def: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah" shows "diagram J C D \<equiv> category C \<and> category J \<and> functor J C D"
indefinite_integral_continuous_1': fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real" assumes "f integrable_on {a..b}" shows "continuous_on {a..b} (\<lambda>x. integral {x..b} f)"
le_ASSERT_iff: fixes S :: "'a nres"   and \<Phi> :: "bool"   and S' :: "'a nres" shows "(S \<le> ASSERT \<Phi> \<bind> (\<lambda>_. S')) = (\<Phi> \<longrightarrow> S \<le> S')"
euclidean_ring_ops_poly: shows "euclidean_ring_ops (poly_ops ops) poly_rel"
interpret_floatarith_inner_eq: fixes xs :: "floatarith list"   and ys :: "floatarith list"   and vs :: "real list" assumes "length xs = length ys" shows "interpret_floatarith (inner_floatariths xs ys) vs = (\<Sum>i<length ys. interpret_floatariths xs vs ! i * interpret_floatariths ys vs ! i)"
hterms_of_fterms_fterms_of_hterms: fixes ts :: "hterm list" shows "hterms_of_fterms (fterms_of_hterms ts) = ts"
split_head_repeat_3: fixes j :: "nat"   and list2 :: "cell list"   and list1 :: "cell list" shows "(Bk \<up> j @ list2 = Oc # list1) = (j = 0 \<and> Oc # list1 = list2)"
sifum_refinement_eg1_eg2_def: shows "sifum_refinement_eg1_eg2 \<equiv> sifum_example \<and> sifum_example2"
absorb_same: fixes j :: "jump" shows "absorb j (Some (Jump j)) = None"
top_le: fixes a :: "'b \<times> 'c" assumes "less_eq_prod top_prod a" shows "a = top_prod"
ceiling_diff_one: fixes x :: "'a" shows "\<lceil>x - (1::'a)\<rceil> = \<lceil>x\<rceil> - 1"
cmd: shows "m O d \<subseteq> s \<union> ov \<union> d"
neutral_ow_def: fixes U :: "'b set"   and z :: "'b" shows "neutral_ow U z \<equiv> z \<in> U"
fps_lr_inverse_unique_divring_2: fixes f :: "'a fps"   and g :: "'a fps" assumes "f * g = 1" shows "fps_right_inverse f (fps_nth g 0) = g"
path_Obj: fixes x :: "'obj" assumes "x \<in> Obj" shows "G.path x x []"
coincides_cls_val_process: fixes Mkt :: "('c, 'd) discrete_market"   and Mkt2 :: "('c, 'd) discrete_market"   and A :: "'d set"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "coincides_on Mkt Mkt2 A"   and "support_set pf \<subseteq> A" shows "\<forall>n w. cls_val_process Mkt pf n w = cls_val_process Mkt2 pf n w"
pred_option_conj: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "pred_option (\<lambda>x. P x \<and> Q x) = (\<lambda>x. pred_option P x \<and> pred_option Q x)"
execute_ref: fixes v :: "'a"   and h :: "heap" shows "execute (ref v) h = Some (Ref.alloc v h)"
nmp_eq: shows "PP x y = (P x y \<and> \<not> P y x)"
closure_of_Union: fixes \<F> :: "'c set set"   and X :: "'c topology" assumes "finite \<F>" shows "X closure_of \<Union> \<F> = \<Union> ((closure_of) X ` \<F>)"
line_comm: fixes P :: "Points"   and Q :: "Points" assumes "P \<noteq> Q" shows "line P Q = line Q P"
strictly_subsumes_def: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "strictly_subsumes C D = (subsumes C D \<and> \<not> subsumes D C)"
vs: shows "mset vs = mset (remove1 w vs) + {#w#}"
c_meet_a_eq_d: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "d_aux a b c \<le> e_aux a b c" shows "c_aux a b c \<sqinter> a_aux a b c = d_aux a b c"
un_non_empty: shows "j \<squnion> k \<noteq> bot"
cycle_Cons: fixes x :: "'c"   and xs :: "'c list" shows "cycle (x # xs) = x ## cycle (xs @ [x])"
assms: shows "\<sigma> x = None \<or> \<sigma> x = Some t"
D: shows "KM_map\<cdot>ValK_copy \<sqsubseteq> fst (fix\<cdot>ValK_copy_rec)"
splitAt_in_snd: fixes v :: "'c"   and ram :: "'c"   and vs :: "'c list" assumes "v \<in> set (snd (splitAt ram vs))" shows "v \<in> set vs"
sum_eq: fixes a :: "ordinal"   and b :: "ordinal" assumes "additive_principal a"   and "b < a" shows "b + a = a"
HRP_HPair: fixes x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{HRP x x', HRP y y'} \<turnstile> HRP (HPair x y) (Q_HPair x' y')"