hcju/isabelleps huggingface dataset:
llist_lift_admissible: fixes P :: "'b llist \<Rightarrow> bool"   and v :: "'b llist" assumes "ccpo.admissible lSup lprefix P"   and "\<And>u. \<lbrakk>u \<le> v; lfinite u\<rbrakk> \<Longrightarrow> P u" shows "P v"
eq_closure_of_def: fixes S :: "('d, 'e) eq_object_scheme"   and A :: "'d set" shows "closure_of\<^bsub>S\<^esub> A = {y \<in> carrier S. y .\<in>\<^bsub>S\<^esub> A}"
is_Exists_def: fixes formula :: "'a MFOTL.formula" shows "formula.is_Exists formula = (\<exists>x5. formula = formula.Exists x5)"
option_bindSomeI: fixes X :: "'c option"   and x :: "'c"   and F :: "'c \<Rightarrow> 'd option"   and y :: "'d" assumes "X = Some x"   and "F x = Some y" shows "X \<bind> F = Some y"
VsetI: fixes b :: "V"   and i :: "V" assumes "rank b \<sqsubset> i" shows "b \<in> elts (Vset i)"
distinct_6: shows "TernaryUnknown \<noteq> TernaryFalse"
swp: shows "sound (wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) (\<lambda>s. 1))"
treeLess_def: shows "treeLess \<equiv> {(T, T'). \<exists>inp binp i j. T' = Branch inp binp \<and> (inp i = Some T \<or> binp j = Some T)}"
G: fixes x :: "'a" assumes "x \<in> cbox a b" shows "g x \<in> G"
generate_into_G: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> additive.generate (R \<inter> H)" shows "a \<in> R"
defNode_1: fixes v :: "'val"   and g :: "'g" assumes "v \<in> allVars g" shows "defNode g v \<in> set (\<alpha>n g)"
matrix_equality: shows "mat_of_rows n fs = M (length fs) * mat_of_rows n (map gso [0..<length fs])"
elims_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list"   and y :: "bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x = y" obtains "xa__ = []" and "y"   | x :: "'b" and ys :: "'b list" where "xa__ = x # ys" and "y = (Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
simps_7: fixes c :: "int"   and e :: "MIR.num" shows "\<zeta> (Gt (CN 0 c e)) = c"
proots_fcompose_bij_eq: fixes q1 :: "'a poly"   and q2 :: "'a poly"   and A :: "'a set"   and B :: "'a set"   and p :: "'a poly" assumes "bij_betw (\<lambda>x. poly q1 x / poly q2 x) A B"   and "p \<noteq> 0"   and "\<forall>x\<in>A. poly q2 x \<noteq> (0::'a)"   and "max (degree q1) (degree q2) \<le> 1"   and "\<forall>c. q1 \<noteq> smult c q2"   and "infinite UNIV" shows "proots_count p B = proots_count (fcompose p q1 q2) A"
between: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "between (a, b) x = (dist a b = dist a x + dist x b)"
splitting_lemma_right: fixes C :: "'a monoid"   and B :: "'a monoid"   and A :: "'a monoid"   and g :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a" assumes "short_exact_sequence C B A g f"   and "g' \<in> hom C B"   and "\<And>z. z \<in> carrier C \<Longrightarrow> g (g' z) = z" obtains H :: "'a set" and K :: "'a set" where "H \<lhd> B" and "subgroup K B" and "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}" and "H <#>\<^bsub>B\<^esub> K = carrier B" and "f \<in> Group.iso A (subgroup_generated B H)" and "g \<in> Group.iso (subgroup_generated B K) C"
SUP_subset_mono: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd"   and g :: "'b \<Rightarrow> 'c \<times> 'd" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod (f x) (g x)" shows "less_eq_prod (Sup_prod (f ` A)) (Sup_prod (g ` B))"
lunit_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tensor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unity :: "'b"   and lunit :: "'b \<Rightarrow> 'b"   and runit :: "'b \<Rightarrow> 'b"   and assoc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "elementary_monoidal_category C tensor unity lunit runit assoc"   and "partial_magma.ide C a" shows "tensor unity (lunit a) = C (tensor (elementary_monoidal_category.\<iota> unity lunit) a) (category.inv C (elementary_monoidal_category.\<alpha> C tensor assoc (monoidal_category.unity C (elementary_monoidal_category.\<iota> unity lunit), monoidal_category.unity C (elementary_monoidal_category.\<iota> unity lunit), a)))"
insertI1: fixes a :: "'f"   and B :: "'f set" shows "a \<in> insert a B"
strict_order_iff: fixes a :: "'e"   and b :: "'e" shows "(a < b) = (a = inf a b \<and> a \<noteq> b)"
sel_1: fixes x1 :: "'a pmf"   and x2 :: "'a \<Rightarrow> 'a scheduler" shows "action_sch (Scheduler x1 x2) = x1"
perp_in_col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.Col Bet A B X \<and> Tarski_neutral_dimensionless.Col Bet C D X"
unintersect_coll2_def: fixes X :: "'d set" shows "unintersect_coll2 X = SPEC ((\<subseteq>) X)"
PDG_path_Nil: fixes n :: "'node" assumes "valid_node n" shows "n \<longrightarrow>\<^sub>d* n"
dims_smult: fixes \<alpha> :: "'b"   and A :: "'b tensor" shows "dims (\<alpha> \<cdot> A) = dims A"
induct: fixes P :: "('d, 'c) tpoly \<Rightarrow> bool"   and a0 :: "('d, 'c) tpoly" assumes "\<And>i. P (PNum i)"   and "\<And>x. P (PVar x)"   and "P (PSum [])"   and "\<And>p ps. \<lbrakk>P p; P (PSum ps)\<rbrakk> \<Longrightarrow> P (PSum (p # ps))"   and "P (PMult [])"   and "\<And>p ps. \<lbrakk>P p; P (PMult ps)\<rbrakk> \<Longrightarrow> P (PMult (p # ps))" shows "P a0"
f_sums_F: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a" shows "MC_syntax.f K x y sums MC_syntax.F K x y"
asymmetric_axioms: shows "asymmetric A (\<sqsubset>)"
him: shows "h ` (\<Union> \<G> - C) \<subseteq> rel_frontier T"
col_factor_def: fixes cis :: "nat list" shows "col_factor cis = prod_list (map2 (\<lambda>i B. lookup B [i]) cis (nths Bs (- I)))"
hypreal_of_real_isUb_iff: fixes Q :: "real set"   and Y :: "real" shows "isUb \<real> (hypreal_of_real ` Q) (hypreal_of_real Y) = isUb UNIV Q Y"
INF_insert: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (f ` insert a A) = inf (f a) (Inf (f ` A))"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (jmpl_instr instr s1)) \<and> t2 = snd (fst (jmpl_instr instr s2))"
ipurge_ref_last: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list"   and x :: "'b"   and X :: "'b set" shows "ipurge_ref I D u (xs @ [x]) X = (if (u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I) then ipurge_ref I D u xs {x' \<in> X. (D x, D x') \<notin> I} else ipurge_ref I D u xs X)"
op_elem_def: fixes oper :: "('d, 'c) operation" shows "op_elem oper \<equiv> case oper of Add i e \<Rightarrow> e | Rem is e \<Rightarrow> e"
left_residual: fixes a :: "'a"   and x :: "'a"   and b :: "'a" shows "(a * x \<le> b) = (x \<le> a r\<rightarrow> b)"
identify_blocksI: fixes k :: "nat"   and i :: "nat"   and j :: "nat"   and A :: "'c mat" assumes "k < n"   and "i \<le> j"   and "j < k"   and "\<And>l. \<lbrakk>i \<le> l; l < j\<rbrakk> \<Longrightarrow> A $$ (l, Suc l) = (1::'c)"   and "Suc j \<noteq> k \<Longrightarrow> A $$ (j, Suc j) \<noteq> (1::'c)"   and "0 < i \<Longrightarrow> A $$ (i - 1, i) \<noteq> (1::'c)" shows "(i, j) \<in> set (identify_blocks A k)"
minus_sub_one_diff_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and m :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "minus (uminus (neg_numeral.sub minus plus one m num.One)) one = uminus (numeral.numeral one plus m)"
assms_2: shows "\<exists>x\<in>X. c \<le> x"
assms_9: shows "\<forall>(k, \<gamma>)\<in>one_chain2. valid_path \<gamma>"
transpose_rowvector: fixes v :: "('a, 'c) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose (rowvector v) = columnvector v"
is_irred_map_scale: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and c :: "'b" assumes "\<not> punit.is_red F p" shows "\<not> punit.is_red F (c \<cdot> p)"
monoid_mult_ow_axioms: shows "monoid_mult_ow U 1\<^sub>o\<^sub>w (*\<^sub>o\<^sub>w)"
simps: fixes bs :: "atom list"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom list"   and y :: "'a" shows "(bs, x) \<approx>lst R f p (cs, y) = (f x - set bs = f y - set cs \<and> (f x - set bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
sort_key_by_quicksort: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "sort_key f xs = sort_key f (filter (\<lambda>x. f x < f (xs ! (length xs div 2))) xs) @ filter (\<lambda>x. f x = f (xs ! (length xs div 2))) xs @ sort_key f (filter (\<lambda>x. f (xs ! (length xs div 2)) < f x) xs)"
carrier_is_subring: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "subring (carrier R) R"
in_sum: fixes N1 :: "'c set"   and N2 :: "'c set" assumes "LinearCombinations.submodule R N1 M"   and "LinearCombinations.submodule R N2 M" shows "N1 \<subseteq> submodule_sum N1 N2"
dfs_comp: fixes f :: "'f \<Rightarrow> 'c"   and g :: "('d, 'e) bintree \<Rightarrow> 'f"   and xs :: "('d, 'e) bintree option list" shows "dfs (f \<circ> g) xs = map f (dfs g xs)"
simps_2: fixes x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list" shows "wellformed_policy1_strong (x # xs) = (x = DenyAll \<and> DenyAll \<notin> set xs)"
axis_eq_axis: fixes i :: "'f"   and x :: "'e"   and j :: "'f"   and y :: "'e" shows "(axis i x = axis j y) = (x = y \<and> i = j \<or> x = (0::'e) \<and> y = (0::'e))"
B_def: shows "B = (\<Union>k. A k \<inter> {N k..})"
inf0: fixes a :: "'c" shows "a \<equiv> a \<sqinter> top"
takeWhile_eq_take: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "takeWhile P xs = take (length (takeWhile P xs)) xs"
mkArr_Fun: fixes f :: "'t" assumes "S'.arr f" shows "S'.mkArr (S'.set (S'.dom f)) (S'.set (S'.cod f)) (S'.Fun f) = f"
ordinal_plus_monoL: fixes x :: "ordinal"   and x' :: "ordinal"   and y :: "ordinal" assumes "x \<le> x'" shows "x + y \<le> x' + y"
MakeNTApp: fixes X :: "'g"   and f :: "('g, 'j, 'h, 'k, 'i, 'l) NatTrans" assumes "X \<in> obj\<^bsub>NTCatDom (MakeNT f)\<^esub>" shows "f $$ X = MakeNT f $$ X"
z3: fixes x :: "'b" shows "\<parallel>(A\<^sup>\<dagger>) x\<parallel> \<le> \<parallel>x\<parallel> * M"
mult_nonpos_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "Rings.class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq a zero"   and "less_eq zero b" shows "less_eq (times a b) zero"
the_adj_chamber_simplex: fixes C :: "'a set"   and z :: "'a set" assumes "chamber C"   and "z \<lhd> C" shows "the_adj_chamber C z \<in> X"
subcocycle_lim_birkhoff_2: fixes u :: "'a \<Rightarrow> real" assumes "integrable M u" shows "AE x in M. subcocycle_lim (birkhoff_sum u) x = real_cond_exp M Invariants u x"
add_commute: fixes a :: "'c"   and b :: "'c" shows "a \<squnion> b = b \<squnion> a"
rolling_rule_ltr: fixes g :: "'e \<rightarrow> 'd"   and f :: "'d \<rightarrow> 'e" shows "fix\<cdot>(g oo f) \<sqsubseteq> g\<cdot>(fix\<cdot>(f oo g))"
rBang: fixes P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> \<tau> \<prec> P'"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) P'"   and "guarded P" shows "Prop C \<Psi> (!P) P'"
fundchamber_next_WS_image_neq: fixes s :: "'a permutation"   and w :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "(w + s) `\<rightarrow> C0 \<noteq> w `\<rightarrow> C0"
division_split_left_inj: fixes \<D> :: "'a set set"   and S :: "'a set"   and K1 :: "'a set"   and k :: "'a"   and c :: "real"   and K2 :: "'a set" assumes "\<D> division_of S"   and "K1 \<inter> {x. x \<bullet> k \<le> c} = K2 \<inter> {x. x \<bullet> k \<le> c}"   and "K1 \<in> \<D>"   and "K2 \<in> \<D>"   and "K1 \<noteq> K2" shows "interior (K1 \<inter> {x. x \<bullet> k \<le> c}) = {}"
g: shows "\<guillemotleft>g : y' \<rightarrow>\<^sub>D y\<guillemotright>"
NotR_in_SNa: fixes M :: "trm"   and x :: "name"   and a :: "coname" assumes "SNa M" shows "SNa (NotR x.M a)"
reachable_conv: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>* v) = ((u, v) \<in> Restr ((arcs_ends G)\<^sup>*) (verts G))"
A3a: fixes x :: "i" assumes "AxiomsSet6" shows "x \<cdot> (dom x) \<cong> x"
lucas: fixes n :: "nat"   and a :: "nat" assumes "2 \<le> n"   and "[a ^ (n - 1) = 1] (mod n)"   and "\<forall>p. prime p \<and> p dvd n - 1 \<longrightarrow> [a ^ ((n - 1) div p) \<noteq> 1] (mod n)" shows "prime n"
res_iso: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "x \<le> y" shows "f x \<le> f y"
run_alt_def_snth: fixes r :: "'c stream"   and p :: "'a" shows "run r p = (\<forall>k. r !! k \<in> en (target (stake k r) p))"
simps_30: fixes f1 :: "String.literal \<Rightarrow> typ \<Rightarrow> 'a"   and f2 :: "variable \<Rightarrow> typ \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'a"   and f4 :: "typ \<Rightarrow> Core.term \<Rightarrow> 'a"   and f5 :: "Core.term \<Rightarrow> Core.term \<Rightarrow> 'a"   and x51 :: "Core.term"   and x52 :: "Core.term" shows "(case x51 $ x52 of Ct x xa \<Rightarrow> f1 x xa | Fv x xa \<Rightarrow> f2 x xa | Bv x \<Rightarrow> f3 x | Abs x xa \<Rightarrow> f4 x xa | x $ xa \<Rightarrow> f5 x xa) = f5 x51 x52"
older_seniors_card: fixes x :: "nat"   and n :: "nat"   and y :: "nat" shows "(card (older_seniors x n) = card (older_seniors y n)) = (older_seniors x n = older_seniors y n)"
det_mult_row: fixes A :: "(('d, 'e) vec, 'e) vec"   and a :: "'e"   and k :: "'d" shows "det (mult_row A a k) = k * det A"
divide_cancel_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide c a = divide c b) = (c = zero \<or> a = b)"
nth_find_first: fixes a :: "'a"   and xs :: "'a list" assumes "a \<in> set xs" shows "xs ! find_first a xs = a"
cases: fixes a1 :: "('test, 'atom, 'choice) cmd \<times> 'state"   and a2 :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" assumes "enabled a1 a2" obtains \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" and s :: "('test, 'atom, 'choice) cmd \<times> 'state" where "a1 = s" and "a2 = \<omega>" and "enabled (shd \<omega>) (stl \<omega>)" and "shd \<omega> \<in> G s"
surj_def: fixes f :: "'d \<Rightarrow> 'c" shows "top_pres ((`) f) = (\<forall>y. \<exists>x. y = f x)"
iMODb_mod_0_card: fixes r :: "nat"   and c :: "nat" shows "card [ r, mod 0, c ] = Suc 0"
map_cong: fixes x :: "'b pmf"   and ya :: "'b pmf"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in> set_pmf ya \<Longrightarrow> f z = g z" shows "map_pmf f x = map_pmf g ya"
int_cases: fixes z :: "int" obtains n :: "nat" where "z = int na__"   | n :: "nat" where "z = - int (Suc n)"
card_set_of_intersecting_sets_by_card: fixes A :: "'a set"   and I :: "'a set"   and k :: "nat"   and n :: "nat" assumes "A \<subseteq> I"   and "finite I"   and "k \<le> n"   and "n \<le> card I"   and "k \<le> card A" shows "card {B. B \<subseteq> I \<and> card B = n \<and> card (A \<inter> B) = k} = (card A choose k) * (card I - card A choose (n - k))"
ExprListTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and exprs :: "'a expr list"   and \<pi> :: "'a OCL_Types.type list" assumes "\<Gamma> \<turnstile>\<^sub>L exprs : \<pi>" obtains "exprs = []" and "\<pi> = []"   | expr :: "'a expr" and \<tau> :: "'a OCL_Types.type" and exprsa :: "'a expr list" and \<pi>' :: "'a OCL_Types.type list" where "exprs = expr # exprsa" and "\<pi> = \<tau> # \<pi>'" and "\<Gamma> \<turnstile>\<^sub>E expr : \<tau>" and "\<Gamma> \<turnstile>\<^sub>L exprsa : \<pi>'"
isomorphic_as_rings_def: fixes \<R> :: "'c set \<times> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<times> 'c \<times> 'c"   and \<R>' :: "'d set \<times> ('d \<Rightarrow> 'd \<Rightarrow> 'd) \<times> ('d \<Rightarrow> 'd \<Rightarrow> 'd) \<times> 'd \<times> 'd" shows "(\<R> \<cong>\<^sub>R \<R>') = (let (R, addition, multiplication, zero, unit) = \<R>; (R', addition', multiplication', zero', unit') = \<R>' in \<exists>\<eta>. ring_isomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit')"
length_annos_annotate: fixes f :: "nat \<Rightarrow> 'a"   and c :: "com" shows "length (annos (annotate f c)) = asize c"
simps_20: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm"   and x2 :: "'b" shows "PApp x31 x32 \<noteq> PVar x2"
assms_4: assumes "j < i" shows "f1 ! i = f2 ! i"
chine_composite_2: shows "Chn (local.dom \<mu> \<star> local.dom \<nu> \<star> local.dom \<pi>) = doms.\<mu>.leg0 \<down>\<down> doms.\<nu>.leg1 \<cdot> doms.\<nu>\<pi>.prj\<^sub>1"
ballf: fixes z :: "complex" assumes "z \<in> ball 0 r" shows "f z \<in> ball 0 r"
half2_st_def: shows "half2_st = Abs_freelist (if gcd_nat.greater_eq (add_order (s + t)) 2 then pair_relator_halflist s t else pair_relator_halflist t s)"
guessed_frame_Resp_xsk: fixes R :: "fid_t" assumes "role (guessed_runs R) = Resp" shows "guessed_frame R xsk = Some (NonceF (R $ sk))"
locally_path_connected_imp_locally_connected: fixes S :: "'b set" assumes "locally path_connected S" shows "locally connected S"
symrun_interp_decreases_add_head: fixes \<Gamma> :: "'a constr list"   and \<Gamma>' :: "'a constr list"   and \<gamma> :: "'a constr" assumes "set \<Gamma> \<subseteq> set \<Gamma>'" shows "\<lbrakk>\<lbrakk> \<gamma> # \<Gamma>' \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<subseteq> \<lbrakk>\<lbrakk> \<gamma> # \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
conv_radius_cong_weak: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<And>n. f n = g n" shows "conv_radius f = conv_radius g"
iso_iff_mon_epi: fixes f :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" shows "(f \<in> Group.iso G H) = (f \<in> mon G H \<and> f \<in> epi G H)"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
Rel2Mon_Id: fixes x :: "'a \<Rightarrow> (unit \<times> 'a) option" shows "(Rel2Mon \<circ> Mon2Rel) x = x"
that: fixes k :: "nat" assumes "k \<noteq> 0"   and "r \<^sup>@ k = x" shows "thesis"
heap'_def: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'a \<Rightarrow> htype option"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'d prog" shows "heap' addr2thread_id thread_id2addr allocate typeof_addr heap_write P \<equiv> heap addr2thread_id thread_id2addr allocate (\<lambda>_. typeof_addr) heap_write P"
dot_cross7_self_3: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "x \<times>\<^sub>7 y \<bullet> y = 0"
reachable_in_oreachable: fixes s :: "'d \<times> 'e"   and A :: "('d \<times> 'e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "s \<in> reachable A I" shows "s \<in> oreachable A (act I) U"
fbox_demodalisation3: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(d y \<le>  |x] d z) = (d y \<cdot> x \<le> x \<cdot> d z)"
prems_3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
m: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> m) = meets p q"
chain_singleton: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "chain R (LCons x LNil)"
Atm_WbisT: fixes atm :: "'atom" assumes "compatAtm atm" shows "Atm atm \<approx>wT Atm atm"
deg_q1: fixes t :: "'x \<Rightarrow>\<^sub>0 'a" assumes "t \<in> keys q1" shows "deg_pm t = poly_deg q"
basic_trans_rules_31: fixes A :: "'f set"   and B :: "'f set"   and c :: "'f" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
r_badblock_def: shows "r_badblock \<equiv> let f = Cn 1 r_singleton_encode [r_not] in Let (Cn 3 r_cons [r_constn 2 0, recf.Id 3 1]) (Pr 1 f)"
currentD_outside_OUT: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f"   and "x \<notin> \<^bold>V\<^bsub>\<Gamma>\<^esub>" shows "d_OUT f x = 0"
assms: shows "edges G \<subseteq> {(l, v1, v2). v1 \<in> vertices G \<and> v2 \<in> vertices G}"
ins_root_order: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "root_order k t" shows "root_order_up\<^sub>i k (ins k x t)"
inj_on_diff_nat: fixes N :: "nat set"   and k :: "nat" assumes "\<And>n. n \<in> N \<Longrightarrow> k \<le> n" shows "inj_on (\<lambda>n. n - k) N"
simps_9: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl" shows "w \<Turnstile> F \<phi> = (\<exists>k. suffix k w \<Turnstile> \<phi>)"
some_gcd_ff_list_divides: fixes x :: "'b fract"   and xs :: "'b fract list" assumes "x \<in> set xs" shows "divides_ff (some_gcd_ff_list xs) x"
get_return_edges_unique: fixes a :: "'edge"   and a' :: "'edge"   and a'' :: "'edge" assumes "valid_edge a"   and "a' \<in> get_return_edges a"   and "a'' \<in> get_return_edges a" shows "a' = a''"
r_univ_almost_kleene_nf: shows "r_univ \<simeq> (let u = Cn 3 r_dec [r_result]; t = Cn 3 r_not [r_result] in Cn 2 u [Mn 2 t, recf.Id 2 0, recf.Id 2 1])"
transitive_star: fixes x :: "'a" assumes "x \<sqinter> x \<le> x" shows "top = top \<squnion> x"
assms: shows "II \<subseteq> {brnL cl n..<brnL cl n + brn (cl ! n)}"
domIff: fixes a :: "'d"   and m :: "'d \<Rightarrow> 'e option" shows "(a \<in> dom m) = (m a \<noteq> \<bottom>)"
span_neg: fixes x :: "('a, 'b) vec"   and S :: "('a, 'b) vec set" assumes "x \<in> vec.span S" shows "- x \<in> vec.span S"
arrE: fixes f :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f" shows "arr f \<and> (ide (local.dom f) \<and> Diag (DOM (local.dom f))) \<and> ide (cod f) \<and> Diag (DOM (cod f))"
cl10_var: fixes R :: "('a \<times> 'a set) set" shows "Multirelations.d (R - 1\<^sub>\<pi>) = 1\<^sub>\<sigma> \<inter> ((R - 1\<^sub>\<pi>) \<cdot> NC)"
lunit_naturality: fixes f :: "'a" assumes "arr f" shows "\<l>\<^sub>1[cod f] \<cdot> (\<I>\<^sub>1 \<otimes>\<^sub>1 f) = f \<cdot> \<l>\<^sub>1[local.dom f]"
oracle'_def: shows "oracle' = (\<lambda>(s, \<sigma>) x. (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len_plain) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs))) | Some bs \<Rightarrow> return_spmf (bs, \<sigma>)) \<bind> (\<lambda>(h, \<sigma>'). return_spmf (h, {x} \<union> s, \<sigma>')))"