hcju/isabelleps huggingface dataset:
f: fixes \<Delta> :: "('k, 'l) network_scheme"   and f :: "('k \<times> 'k) \<times> 'k \<times> 'k \<Rightarrow> ennreal"   and S :: "('k \<times> 'k) set" assumes "web_flow (web_of_network \<Delta>) f"   and "orthogonal_current (web_of_network \<Delta>) f S"   and "separating (web_of_network \<Delta>) S"   and "\<And>e. e \<in> \<^bold>E\<^bsub>\<Delta>\<^esub> \<Longrightarrow> 0 < capacity \<Delta> e" shows "current (web_of_network \<Delta>) f"
rep_eq: fixes x :: "'a['b, 'c]"   and xa :: "'a['b, 'c]" shows "(x \<le> xa) = (mag (fromQ x) \<le> mag (fromQ xa))"
root_int_floor: fixes p :: "nat"   and x :: "int" shows "root_int_floor p x = \<lfloor>root p (real_of_int x)\<rfloor>"
elims_3: fixes x :: "cell list"   and xa :: "nat"   and xb :: "cell list \<times> cell list" assumes "\<not> wcode_backto_standard_pos_2 x xa xb" obtains ires :: "cell list" and rs :: "nat" and l :: "cell list" and r :: "cell list" where "x = ires" and "xa = rs" and "xb = (l, r)" and "\<not> (wcode_backto_standard_pos_2_O ires rs (l, r) \<or> wcode_backto_standard_pos_2_B ires rs (l, r))"
simps_1: fixes uu :: "'c" shows "del uu N0 = N0"
gcd_non_0_nat: fixes y :: "nat"   and x :: "nat" assumes "y \<noteq> 0" shows "gcd x y = gcd y (x mod y)"
subst_pres_fresh: fixes x :: "Vars.var"   and e :: "exp"   and y :: "Vars.var"   and z :: "Vars.var" assumes "atom x \<sharp> e \<or> x = y"   and "atom x \<sharp> z" shows "atom x \<sharp> e[y::=z]"
action_mult: fixes g :: "'a"   and h :: "'a"   and m :: "'c" assumes "g \<in> carrier G"   and "h \<in> carrier G"   and "m \<in> M" shows "\<phi> g (\<phi> h m) = \<phi> (g \<otimes> h) m"
Bonk_Schramm_extension_unfolded_wo_props'_1: fixes a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" shows "a \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b)"
last_ConsR: fixes xs :: "'d list"   and x :: "'d" assumes "xs \<noteq> []" shows "last (x # xs) = last xs"
orbit_list_def: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "orbit_list f x \<equiv> iterate 0 (funpow_dist1 f x x) f x"
vsv_axioms: shows "vsv (S\<lparr>ArrVal\<rparr>)"
ENR_imp_absolute_neighbourhood_retract: fixes S :: "'a set"   and S' :: "'b set"   and U :: "'b set" assumes "ENR S"   and "S homeomorphic S'"   and "S' \<subseteq> U" obtains V :: "'b set" where "openin (top_of_set U) V" and "S' retract_of V"
f: shows "simple_bochner_integrable (restrict_space M \<Omega>) f"
part1_su: fixes V :: "'a set" assumes "infinite V" shows "part1 V \<subseteq> V"
irreducible_M_PM: fixes p :: "'b poly" assumes "degree p = 0"   and "irreducible (coeff p 0)" shows "irreducible p"
weakenBisimCoinductAux: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox>\<^sub>w<(X \<union> weakenBisim)> Q \<and> \<Psi> \<rhd> P \<leadsto>\<^sub>w<(X \<union> weakenBisim)> Q \<and> (\<forall>\<Psi>'. (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X \<or> \<Psi> \<otimes> \<Psi>' \<rhd> P \<approx>\<^sub>w Q) \<and> ((\<Psi>, Q, P) \<in> X \<or> \<Psi> \<rhd> Q \<approx>\<^sub>w P)" shows "\<Psi> \<rhd> P \<approx>\<^sub>w Q"
Mp_rel_i_of_int_poly_i: fixes F :: "int poly" assumes "Mp F = F" shows "Mp_rel_i (of_int_poly_i ff_ops F) F"
Int_Collect: fixes x :: "'k"   and A :: "'k set"   and P :: "'k \<Rightarrow> bool" shows "(x \<in> A \<inter> {x. P x}) = (x \<in> A \<and> P x)"
f'_def: fixes x :: "real" shows "f' x = (1 + 1 / ln (b * x) * ln (1 + hb / b * ln x powr (- 1 - e))) powr (- e / 2)"
p_and_not_p: fixes P :: "'b \<Rightarrow> bool" shows "(P \<and> \<not> P) = false"
spref_rev_suf_iff: fixes s :: "'b list"   and w :: "'b list" shows "(s <p w) = (rev s <s rev w)"
Ag_ind_aHom: fixes f :: "'a \<Rightarrow> 'c"   and D :: "'c set" assumes "f \<in> carrier A \<rightarrow> D"   and "bij_to f (carrier A) D" shows "Agii A f \<in> aHom A (Ag_ind A f)"
assms_1: shows "[A*\<^sub>vx]\<le>b"
simps_1: fixes rs :: "'rule stream"   and s :: "'state" shows "tree.root (mkTree rs s) = (s, shd (trim rs s))"
aform_err_leI: fixes y :: "real"   and e :: "nat \<Rightarrow> real"   and c :: "real \<times> real pdevs"   and d' :: "real"   and d :: "real" assumes "y \<in> aform_err e (c, d')"   and "d' \<le> d" shows "y \<in> aform_err e (c, d)"
FG_advice_monotone: fixes X :: "'b ltln set"   and Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "X \<subseteq> Y"   and "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<Sigma>\<^sub>1" shows "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<Sigma>\<^sub>1"
isometry_extend_closure: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "\<exists>g. isometry_on (closure X) g \<and> (\<forall>x\<in>X. g x = f x)"
simps_4: fixes x2 :: "'b"   and x1 :: "'b" shows "Lt x2 \<noteq> Le x1"
intros: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and s :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'d" assumes "f \<in> borel_measurable M"   and "\<And>i. simple_bochner_integrable M (s i)"   and "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0"   and "(\<lambda>i. simple_bochner_integral M (s i)) \<longlonglongrightarrow> x" shows "has_bochner_integral M f x"
writes_bind: fixes S :: "('a, 'b, 'c) prog set"   and f :: "('a, 'b, 'd) prog"   and h :: "'a"   and g :: "'d \<Rightarrow> ('a, 'b, 'e) prog"   and h' :: "'a" assumes "\<And>h2. writes S f h h2"   and "\<And>x h2. \<lbrakk>h \<turnstile> f \<rightarrow>\<^sub>r x; h \<turnstile> f \<rightarrow>\<^sub>h h2\<rbrakk> \<Longrightarrow> writes S (g x) h2 h'" shows "writes S (Heap_Error_Monad.bind f g) h h'"
non_empty_cycle_root_var_axioms_2: fixes r :: "'a"   and x :: "'a" shows "non_empty_cycle_root r x = (x ; 1 \<le> x\<^sup>+ ; r \<and> is_inj x \<and> is_p_fun x \<and> point r \<and> r \<le> x ; 1)"
inf_fres: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" shows "times_prod' y x = Inf_prod {z. less_eq_prod y (prod.fres z x)}"
assms_2: shows "{0..<length dims} \<subseteq> vars1 \<union> vars2"
Chinese_remTr1: fixes A :: "('d, 'e) Ring_scheme"   and n :: "nat"   and J :: "nat \<Rightarrow> 'd set"   and B :: "nat \<Rightarrow> 'd set Ring"   and S :: "nat \<Rightarrow> 'd \<Rightarrow> 'd set" assumes "Ring A"   and "\<forall>k\<le>n. ideal A (J k)"   and "\<forall>k\<le>n. B k = A /\<^sub>r J k"   and "\<forall>k\<le>n. S k = pj A (J k)" shows "ker\<^bsub>A,(r\<Pi>\<^bsub>{j. j \<le> n}\<^esub> B)\<^esub> A_to_prodag A {j. j \<le> n} S B = \<Inter> {I. \<exists>k\<in>{j. j \<le> n}. I = J k}"
vNM_utilityD_2: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes" shows "p \<succeq> q = (measure_pmf.expectation q u \<le> measure_pmf.expectation p u)"
assms: fixes i :: "nat" shows "X i \<in> M \<rightarrow>\<^sub>M N"
min_int_poly_irreducible: fixes x :: "'a" shows "irreducible (min_int_poly x)"
finitC: fixes C :: "nat set" assumes "C \<subseteq> PR" shows "finite C"
idiv_by_0_enat: fixes a :: "enat" shows "a div enat 0 = enat 0"
uminus_one_gbinomial: fixes k :: "nat" shows "- 1 gchoose k = (- 1) ^ k"
simps_3: fixes y :: "Syntax.var"   and x :: "Syntax.var"   and t' :: "Syntax.term"   and t :: "Syntax.term" assumes "atom y \<sharp> (x, t')" shows "Lam y t[t' / x] = Lam y (t[t' / x])"
UP_ring_subring_mult_closed: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "f \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "f \<otimes>\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
distinct_94: fixes x121 :: "expr"   and x122 :: "expr"   and x4 :: "val" shows "x121;; x122 \<noteq> Val x4"
Ord_LeastI_ex: fixes P :: "V \<Rightarrow> bool" assumes "\<exists>i. Ord i \<and> P i" shows "P (LEAST i. Ord i \<and> P i)"
transpI: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x z" shows "transp r"
sum_upto_primepows: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and x :: "real" assumes "\<And>n. \<not> primepow n \<Longrightarrow> f n = (0::'a)"   and "\<And>p i. \<lbrakk>prime p; 0 < i\<rbrakk> \<Longrightarrow> f (p ^ i) = g p i" shows "sum_upto f x = (\<Sum>(p, i)\<in>{(p, i). prime p \<and> 0 < i \<and> real (p ^ i) \<le> x}. g p i)"
punit_splus: shows "MPoly_Type_Class.punit.splus = (+)"
stoch_proc_filt_def: fixes M :: "'b measure"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and N :: "'c measure"   and n :: "nat" shows "stoch_proc_filt M X N n = gen_subalgebra M (sigma_sets (space M) (\<Union>i\<in>{m. m \<le> n}. {X i -` A \<inter> space M |A. A \<in> sets N}))"
pre2: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "I l s"   and "\<not> bval b s" shows "Q l s \<and> 1 \<le> E s \<and> S s = s"
wf_asym: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and x :: "'a" assumes "wf r"   and "(a, x) \<in> r" obtains "(x, a) \<notin> r"
completeness_countable: fixes Cs :: "fterm literal set set" assumes "infinite UNIV"   and "finite Cs"   and "\<forall>C\<in>Cs. finite C"   and "\<forall>F G. \<not> eval\<^sub>c\<^sub>s F G Cs" shows "\<exists>Cs'. resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
subseqs_subformulas\<^sub>\<mu>_restricted_list: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln" shows "(X \<subseteq> subformulas\<^sub>\<mu> \<phi> \<inter> restricted_subformulas \<phi>) = (\<exists>xs. X = set xs \<and> xs \<in> set (subseqs (filter (\<lambda>x. x \<in> restricted_subformulas \<phi>) (subformulas\<^sub>\<mu>_list \<phi>))))"
matrix_seq_def: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat" shows "matrix_seq dim f \<equiv> (\<forall>n. f n \<in> carrier_mat dim dim) \<and> (\<forall>n. partial_density_operator (f n)) \<and> (\<forall>n. f n \<le>\<^sub>L f (Suc n))"
expl_cond_expect_def: fixes M :: "'c measure"   and Y :: "'c \<Rightarrow> 'd"   and X :: "'c \<Rightarrow> real" shows "expl_cond_expect M Y X = img_dce M Y X \<circ> Y"
int_shiftl_numeral_1: fixes w :: "num"   and w' :: "num" shows "numeral w << numeral w' = numeral (num.Bit0 w) << pred_numeral w'"
zip_append1: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'e list" shows "zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
integer_of_nat_numeral: fixes n :: "num" shows "integer_of_nat (numeral n) = numeral n"
Sup_measure_def: shows "Sup = Sup_lexord space (Sup_lexord sets Sup_measure' (\<lambda>U. sigma (\<Union> (space ` U)) (\<Union> (sets ` U)))) (\<lambda>U. sigma (\<Union> (space ` U)) {})"
continuous_transform_within: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and d :: "real"   and g :: "'b \<Rightarrow> 'c" assumes "continuous (at x within s) f"   and "0 < d"   and "x \<in> s"   and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "continuous (at x within s) g"
divides_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a divides\<^bsub>add_monoid R\<^esub> b"   and "b divides\<^bsub>add_monoid R\<^esub> c"   and "a \<in> carrier R" shows "a divides\<^bsub>add_monoid R\<^esub> c"
runErrorT_liftET: fixes m :: "'e\<cdot>'g" shows "runErrorT\<cdot>(liftET\<cdot>m) = fmap\<cdot>Ok\<cdot>m"
lemma2_6_1_diff: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and S :: "'a set" shows "r \<down>m M - r \<down>s S \<subseteq> r \<down>m (M -s S)"
cross7_add_left: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec"   and z :: "(real, 7) vec" shows "(x + y) \<times>\<^sub>7 z = x \<times>\<^sub>7 z + y \<times>\<^sub>7 z"
insert: fixes x :: "'d"   and xs :: "'d list"   and y :: "'d" assumes "ListMem x xs" shows "ListMem x (y # xs)"
bc: shows "pmf (sds R34) b = pmf (sds R24) c"
insert_not_elem: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A"   and "A \<noteq> {}" shows "set.F (insert x A) = x \<^bold>* set.F A"
no_head_sub': fixes b :: "'a list" assumes "b \<in> \<C>" shows "no_head_gen \<C> b \<subseteq> no_head \<C> b"
check_poly_weak_mono: fixes p :: "('b monom \<times> 'c) list"   and v :: "'b" assumes "check_poly_weak_mono p v" shows "poly_weak_mono p v"
ring_irreducibleI': fixes r :: "'a" assumes "r \<in> carrier R - {\<zero>}"   and "irreducible (Ring_Divisibility.mult_of R) r" shows "ring_irreducible r"
psimps: fixes s :: "char list \<Rightarrow> int"   and b :: "bexp"   and C :: "acom" assumes "times_dom (s, b, C)" shows "Nielson_VCGi.times s b C = (if bval b s then Suc (Nielson_VCGi.times (postQ C s) b C) else 0)"
sum_list_append: fixes xs :: "'b list"   and ys :: "'b list" shows "sum_list (xs @ ys) = sum_list xs + sum_list ys"
head_add_vert: fixes u :: "'a" shows "head (add_vert u) = head G"
BoundOutputEq: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" shows "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P' = P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'"
ide_hcomp: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.ide V \<nu>"   and "partial_magma.ide V \<mu>"   and "src \<nu> = trg \<mu>" shows "partial_magma.ide V (H \<nu> \<mu>)"
ParCL: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and c1' :: "('test, 'atom) com"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>c (c1', s')" shows "(c1 | c2, s) \<rightarrow>c (c1' | c2, s')"
IH: assumes "\<forall>v. v \<in> T (\<sigma> \<rightarrow> \<tau>) \<eta> \<longrightarrow> Ee (v # \<rho>) \<subseteq> T (\<sigma> \<rightarrow> \<tau>) \<eta>" shows "f \<in> T (\<sigma> \<rightarrow> \<tau>) \<eta>"
simps_1: fixes valuation :: "Literal list" shows "clauseTrue [] valuation = False"
tendsto_add_zero: fixes f :: "'d \<Rightarrow> 'e"   and F :: "'d filter"   and g :: "'d \<Rightarrow> 'e" assumes "(f \<longlongrightarrow> (0::'e)) F"   and "(g \<longlongrightarrow> (0::'e)) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> (0::'e)) F"
le_rexp_Zero: fixes r :: "'a rexp" assumes "local.le_rexp r Zero" shows "r = Zero"
append_eq_mapE: fixes fl :: "'c list"   and fl' :: "'c list"   and f :: "'d \<Rightarrow> 'c"   and ls :: "'d list" assumes "fl @ fl' = map f ls" obtains l :: "'d list" and l' :: "'d list" where "ls = l @ l'" and "map f l = fl" and "map f l' = fl'"
pheap_merge: fixes h1 :: "'b tree"   and h2 :: "'b tree" assumes "is_root h1"   and "is_root h2"   and "pheap h1"   and "pheap h2" shows "pheap (merge h1 h2)"
higher_deriv_ident: fixes n :: "nat"   and z :: "'a" shows "(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1::'a else (0::'a))"
top_eq: fixes L :: "('a, 'b) gorder_scheme"   and t :: "'a" assumes "bounded_lattice L"   and "t \<in> carrier L"   and "\<And>x. x \<in> carrier L \<Longrightarrow> le L x t" shows "t = \<top>\<^bsub>L\<^esub>"
inorder_joinR: fixes l :: "('b \<times> color) tree"   and x :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.inorder (joinR l x r) = Tree2.inorder l @ x # Tree2.inorder r"
M1: shows "T_measure1 (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u)) < T_measure1 (A Q \<pi>)"
vec_mat_Tensor_elements: fixes v :: "'a list"   and M :: "'a list list" shows "\<forall>i j. (i < length v * row_length M \<and> j < length M) \<and> mat (row_length M) (length M) M \<longrightarrow> vec_mat_Tensor v M ! j ! i = v ! (i div row_length M) * M ! j ! (i mod row_length M)"
cases: fixes a :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and \<sigma> :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ((status \<times> 'x) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" assumes "a \<in> h.mthr.if.\<E> \<sigma>" obtains E' :: "('thread_id \<times> ('addr, 'thread_id, status \<times> 'x, 'heap, 'addr, ('addr, 'thread_id) obs_event action) thread_action) llist" where "a = lconcat (lmap (\<lambda>(t, ta). llist_of (map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) E')" and "h.mthr.if.mthr.Runs \<sigma> E'"
hyps: fixes xs :: "'a llist" assumes "enat n < llength xs" shows "lnth (lappend xs ys) n = lnth xs n"
finite_set_mset_mset_set: fixes A :: "'k set" assumes "finite A" shows "set_mset (mset_set A) = A"
sum_comp_morphism: fixes h :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "h (0::'b) = (0::'a)"   and "\<And>x y. h (x + y) = h x + h y" shows "sum (h \<circ> g) A = h (sum g A)"
IH: fixes p :: "('a, 'out, 'b) gpv" shows "interaction_bound' (p \<bind> f) \<le> local.interaction_bound p + (\<Squnion>x\<in>results'_gpv p. interaction_bound' (f x))"
assms_3: shows "(u, v) \<notin> cfE_of f"
fls_times_both_shifted_simp: fixes m :: "int"   and f :: "'a fls"   and n :: "int"   and g :: "'a fls" shows "fls_shift m f * fls_shift n g = fls_shift (m + n) (f * g)"
UN_extend_simps_1: fixes a :: "'b"   and B :: "'c \<Rightarrow> 'b set"   and C :: "'c set" shows "insert a (\<Union> (B ` C)) = (if C = {} then {a} else \<Union>x\<in>C. insert a (B x))"
blinfun_apply_inject: fixes x :: "'a \<Rightarrow>\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>L 'b" shows "((*\<^sub>v) x = (*\<^sub>v) y) = (x = y)"
hyp1: shows "(\<Sum>s\<leftarrow>signs. z I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s})) = (\<Sum>s\<in>set signs. z I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s}))"
simps_2: fixes x :: "'b" shows "Missing_Lemmas.max_list [x] = x"
insert_spec_none: fixes xs :: "'a list"   and oid :: "'a" shows "set (insert_spec xs (oid, None)) = set xs \<union> {oid}"
abort_2: assumes "\<And>x a\<^sub>1 a\<^sub>2. \<lbrakk>x \<in> S; f x a\<^sub>1 a\<^sub>2\<rbrakk> \<Longrightarrow> r a\<^sub>1 a\<^sub>2" shows "r a\<^sub>0 a\<^sub>1"
setdist_subset_right: fixes T :: "'c set"   and u :: "'c set"   and S :: "'c set" assumes "T \<noteq> {}"   and "T \<subseteq> u" shows "setdist S u \<le> setdist S T"
emp_in_no_head: fixes C :: "'b list set"   and b :: "'b list" shows "\<epsilon> \<in> no_head C b"
atc_io_reduction_on_sets_reduction: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and TS :: "'d list set"   and \<Omega> :: "('d, 'e) ATC set" assumes "M1 \<preceq> M2"   and "observable M1"   and "observable M2" shows "M1 \<preceq>\<lbrakk>TS.\<Omega>\<rbrakk> M2"
dist_def: fixes v :: "nat"   and d :: "nat"   and v' :: "nat" shows "dist v d v' \<equiv> \<exists>p. isPath v p v' \<and> length p = d"
cl7: shows "1\<^sub>\<sigma> \<parallel> 1\<^sub>\<sigma> = 1\<^sub>\<sigma>"
i_aggregate_Cons: fixes k :: "nat"   and xs :: "'b list"   and f :: "nat \<Rightarrow> 'b"   and ag :: "'b list \<Rightarrow> 'b" assumes "0 < k"   and "length xs = k" shows "i_aggregate (xs \<frown> f) k ag = [ag xs] \<frown> i_aggregate f k ag"
Ch_f_range: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and h :: "'f" assumes "Contracts Xd Xh Pd Ch" shows "f_range (Ch h)"
def_sel: shows "\<tau> \<Turnstile> \<delta> select_object_any0\<^sub>S\<^sub>e\<^sub>t f s_set"
trans_mult: fixes R :: "('a \<times> 'a) set" shows "trans (mult R)"
intro: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>y x. f y \<circ> f x = f x \<circ> f y" shows "comp_fun_commute f"
cring_simprules_9: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "cring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
assms_1: shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"
to_fun_diff: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "a \<in> carrier R" shows "to_fun (p \<ominus>\<^bsub>P\<^esub> q) a = to_fun p a \<ominus> to_fun q a"
Zp_defs_2: shows "\<zero> = padic_zero p"
alg: fixes x :: "complex" assumes "x \<in> X" shows "algebraic x"
diagonalization: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {xx}" shows "prv (eql (tJ \<psi>) \<langle>inst \<psi> (tJ \<psi>)\<rangle>) \<and> prv (eqv (\<phi>J \<psi>) (inst \<psi> \<langle>\<phi>J \<psi>\<rangle>))"
that: fixes f :: "nat set cset \<Rightarrow> nat set" assumes "bij_betw f {A. A \<noteq> {}\<^sub>c} UNIV" shows "thesis"
vp_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as n' \<equiv> CFG.path sourcenode targetnode valid_edge n as n' \<and> CFG.valid_path kind get_return_edges as"
subfield: shows "subfield (poly_of_const ` K) (K [X])"
0: shows "stfx.zero_stalk V \<in> stfx.carrier_stalk"
typeof_OutOfMemory: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt OutOfMemory)) = \<lfloor>Class OutOfMemory\<rfloor>"
cd_le_log: shows "cd \<le> 2 powr (b / c) - 1"
split_n_conv_take_drop: fixes es :: "v list"   and n :: "nat" shows "split_n es n = (take n es, drop n es)"
dests_bigger_than_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:15..PAodv-:19} \<union> {PPkt-:7..PPkt-:11} \<union> {PRreq-:11..PRreq-:15} \<union> {PRreq-:24..PRreq-:28} \<union> {PRrep-:10..PRrep-:14} \<union> {PRerr-:1..PRerr-:5} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> kD (rt \<xi>) \<and> sqn (rt \<xi>) ip \<le> the (dests \<xi> ip)))"
chine_assoc_def: shows "trg_trg_f.chine_assoc \<equiv> \<langle>trg_trg_f.Prj\<^sub>1\<^sub>1 \<lbrakk>trg.leg0, trg.leg1 \<cdot> trg_f.prj\<^sub>1\<rbrakk> \<langle>trg_trg_f.Prj\<^sub>0\<^sub>1 \<lbrakk>trg.leg0, f.leg1\<rbrakk> trg_trg_f.Prj\<^sub>0\<rangle>\<rangle>"
acyclic_5a_def: fixes x :: "'a" shows "acyclic_5a x \<equiv> \<forall>y\<le>x. y\<^sup>\<star> \<sqinter> (x \<sqinter> - y)\<^sup>\<star> = (1::'a)"
assoc: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b"   and c :: "'a \<times> 'b" shows "times_prod' (times_prod' a b) c = times_prod' a (times_prod' b c)"
coeff_isolate_variable_sparse: fixes p :: "'a mpoly"   and var :: "nat"   and deg :: "nat"   and x :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (isolate_variable_sparse p var deg) x = (if lookup x var = 0 then MPoly_Type.coeff p (Poly_Mapping.update var deg x) else (0::'a))"
map_simp: fixes f :: "'a" assumes "arr f" shows "\<ll>' f = local.inv (\<ll> (cod f)) \<cdot> local.map f"
hyps_5: shows "\<exists>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
obj_simps_2: fixes a :: "'a" assumes "H.obj a" shows "local.dom a = a"
inv_f_eq: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" assumes "inj f"   and "f x = y" shows "inv f y = x"
fresh_cnj: fixes a :: "atom"   and A :: "fmla"   and B :: "fmla" shows "a \<sharp> A AND B = (a \<sharp> A \<and> a \<sharp> B)"
assms: shows "bidirected_digraph (with_proj G) (swap_in (parcs G))"
countable_UN': fixes X :: "'d set"   and A :: "'d \<Rightarrow> 'a set" assumes "countable X"   and "A ` X \<subseteq> M" shows "\<Union> (A ` X) \<in> M"