hcju/isabelleps huggingface dataset:
right_idem: fixes a :: "'a"   and b :: "'a" shows "lcm (lcm a b) b = lcm a b"
closed_proj_rel_euclidean: fixes b :: "'a"   and a :: "'a" assumes "\<And>c. c \<noteq> 0 \<Longrightarrow> b \<noteq> c *\<^sub>R a"   and "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "\<exists>A B. (0::'a) \<notin> A \<and> (0::'a) \<notin> B \<and> open A \<and> open B \<and> a \<in> A \<and> b \<in> B \<and> A \<times> B \<subseteq> - {(x, y). (x, y) \<noteq> 0 \<and> (\<exists>c. c \<noteq> 0 \<and> y = c *\<^sub>R x)}"
anonymous_profile_def: shows "anonymous_profile (permute_profile \<sigma> R) = image_mset (weak_ranking \<circ> permute_profile \<sigma> R) (mset_set agents)"
is_even_4: fixes n :: "nat" shows "is_even (4 * n)"
lunstream_simps_1: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" assumes "g s = Done" shows "lunstream g s = LNil"
B'_smaller: fixes a :: "'a" shows "B' a \<subseteq> B a"
le_add_diff_inverse2: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and b :: "'e"   and a :: "'e" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus (minus a b) b = a"
quasinorm_sum_def: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" shows "N1 +\<^sub>N N2 = quasinorm_of (max (defect N1) (defect N2), \<lambda>f. \<Sqinter> {eNorm N1 f1 + eNorm N2 f2 |f1 f2. f = f1 + f2})"
to_nat_from_nat_id: fixes i :: "nat" assumes "i < CARD('b)" shows "Bij_Nat.to_nat (Bij_Nat.from_nat i) = i"
S4: shows "(LEAST z. 0 < 2 ^ (z + 1)) \<le> 0"
UC: shows "uniformly_continuous_on (range to_metric_completion) (\<lambda>x. f (I x))"
assms_5: shows "\<not> ((\<exists>(A, X)\<in>Q_pos. Set.is_empty X) \<or> (\<exists>(A, X)\<in>Q_neg. Set.is_empty A \<and> \<not> Set.is_empty X))"
simps: fixes x :: "'c stone_phi_pair" shows "stone_phi_embed x = Abs_lifted_pair (\<lambda>f. if Rep_phi f = stone_phi then Rep_stone_phi_pair x else triple.pairs_uminus (Rep_phi f) (triple.pairs_uminus (Rep_phi f) (Rep_stone_phi_pair x)))"
a: shows "Ide a \<and> Arr a \<and> Dom a = a \<and> Cod a = a \<and> ide \<lbrace>a\<rbrace> \<and> ide \<lbrace>\<^bold>\<lfloor>a\<^bold>\<rfloor>\<rbrace> \<and> \<lbrace>a\<^bold>\<down>\<rbrace> \<in> hom \<lbrace>a\<rbrace> \<lbrace>\<^bold>\<lfloor>a\<^bold>\<rfloor>\<rbrace>"
i_2: shows "\<forall>n<n\<^sub>0. s (0\<^sup>\<infinity> \<triangleright> n) \<down>= 0"
strict_order_iff: fixes b :: "'a word"   and a :: "'a word" shows "(b <s a) = (a = signed.max a b \<and> a \<noteq> b)"
simpt: shows "subst_bv (Fv x \<tau>') (bind_fv (x, \<tau>') t) = t"
cases: fixes x :: "nat" obtains "x = 0"   | "x = Suc 0"   | h :: "nat" where "x = Suc (Suc h)"
closed_orbit_global_existence: fixes x :: "'a" assumes "closed_orbit x" shows "existence_ivl0 x = UNIV"
Nil_le_Cons: fixes x :: "'b list" shows "[] \<le> x"
has_domain_iff_has_codomain: fixes f :: "'a" shows "(domains f \<noteq> {}) = (codomains f \<noteq> {})"
s0_add_secret_def: fixes m :: "msg" shows "s0_add_secret m \<equiv> {(s, s'). m \<notin> synth (analz (ik s)) \<and> s' = s\<lparr>secret := insert m (secret s)\<rparr>}"
diff_le_diff_pow: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "2 \<le> k" shows "m - n \<le> k ^ m - k ^ n"
analz_synth_subset_Un1: fixes X :: "msg set"   and X' :: "msg set"   and Y :: "msg set" assumes "analz X \<subseteq> synth (analz X')" shows "analz (X \<union> Y) \<subseteq> synth (analz (X' \<union> Y))"
constr_dense_linorder_axioms: shows "constr_dense_linorder (\<sqsubseteq>) (\<sqsubset>) between"
image: shows "pi \<bullet> X = (\<bullet>) pi ` X"
fold_insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'a"   and z :: "'a" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold ((\<^bold>*) \<circ> g) z (insert x A) = ((\<^bold>*) \<circ> g) x (Finite_Set.fold ((\<^bold>*) \<circ> g) z A)"
abc_abd_acdadc: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c]]"   and "[[a b d]]"   and "c \<noteq> d" shows "[[a c d]] \<or> [[a d c]]"
snd_apfst: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e \<times> 'c" shows "snd (apfst f x) = snd x"
gba_to_idx_ext_correct: fixes ecnv :: "('Q, 'L, 'more) gba_rec_scheme \<Rightarrow> 'b" assumes "\<And>A. A \<in> F \<Longrightarrow> finite A" shows "gba_to_idx_ext ecnv G \<le> SPEC (\<lambda>G'. igba.accept G' = accept \<and> g_V G' = V \<and> g_E G' = E \<and> g_V0 G' = V0 \<and> igba_rec.more G' = ecnv G \<and> igba G')"
satisfiableAppend_1: fixes formula1 :: "Literal list list"   and formula2 :: "Literal list list" assumes "satisfiable (formula1 @ formula2)" shows "satisfiable formula1"
hyps_2: shows "shortestPath g m = Suc (shortestPath g (last ns))"
rho_char: fixes x :: "'a"   and y :: "'b" shows "up_filter (rho x y) = phi x \<sqinter> up_filter y"
continuous_on_Icc_at_rightD: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on {a..b} f"   and "a < b" shows "(f \<longlongrightarrow> f a) (at_right a)"
stake_sdrop: fixes n :: "nat"   and s :: "'d stream" shows "stake n s @- sdrop n s = s"
ex_mset: fixes X :: "'b multiset" shows "\<exists>xs. mset xs = X"
disj_one_right: fixes x :: "'a" shows "x \<^bold>\<squnion> \<one> = \<one>"
birkhoff_sum_diff: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real"   and n :: "nat"   and x :: "'a" shows "birkhoff_sum (\<lambda>x. f x - g x) n x = birkhoff_sum f n x - birkhoff_sum g n x"
Null_notin_Nodes: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and nb :: "nat"   and levellista :: "ref list list"   and var :: "ref \<Rightarrow> nat" assumes "Dag p low high t"   and "nb \<le> length levellista"   and "wf_ll t levellista var" shows "Null \<notin> Nodes nb levellista"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "Fun_fg.map f = C.null"
\<gamma>_ladder: shows "\<gamma> = ladder_\<gamma> (\<alpha> @ \<delta>) D L (length L - 1)"
consistent_eq_rank_ge: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "consistent A b = ((if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'b)) + 1 else 0) \<le> rank A)"
orthogonal_basis_exists': fixes V :: "(real, 'a) vec list" assumes "is_basis (set V)"   and "distinct V" shows "is_basis (set (Gram_Schmidt V)) \<and> distinct (Gram_Schmidt V) \<and> pairwise orthogonal (set (Gram_Schmidt V))"
convex_less_inS: fixes f :: "'b \<Rightarrow> ereal"   and S :: "'b set"   and a :: "ereal" assumes "Lower_Semicontinuous.convex_on UNIV f"   and "convex S"   and "S \<subseteq> domain f"   and "\<exists>x\<in>closure S. f x < a" shows "\<exists>x\<in>S. f x < a"
seqr_right_unit: fixes P :: "('b, 'c) urel" shows "P ;; II = P"
distinct_3: fixes x1 :: "ct list" shows "TopType x1 \<noteq> Bot"
N_def: shows "N = Tensor.mat_of_cols_list (dim_row B * dim_row D) (mult.Tensor (*) (mat_to_cols_list B) (mat_to_cols_list D))"
fact_num_eq_if: fixes m :: "nat" shows "fact m = (if m = 0 then 1::'c else of_nat m * fact (m - 1))"
low_equal_com: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "low_equal s2 s1"
app: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm" assumes "A >>> A'"   and "B >>> B'"   and "\<And>x T M. A \<noteq> Fn x T M" shows "App A B >>> App A' B'"
elims: fixes x :: "int \<Rightarrow> int \<Rightarrow> bool"   and xa :: "value option"   and xb :: "value option"   and y :: "trilean" assumes "MaybeBoolInt x xa xb = y" obtains f :: "int \<Rightarrow> int \<Rightarrow> bool" and a :: "int" and b :: "int" where "x = f" and "xa = Some (Num a)" and "xb = Some (Num b)" and "y = (if f a b then true else false)"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> bool" and uw_ :: "value option" where "x = uu___" and "xa = None" and "xb = uw___" and "y = invalid"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> bool" and va :: "String.literal" and uw_ :: "value option" where "x = uua___" and "xa = Some (value.Str va__)" and "xb = uw_" and "y = invalid"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> bool" and uv_ :: "value option" where "x = uub___" and "xa = uv___" and "xb = None" and "y = invalid"   | uu_ :: "int \<Rightarrow> int \<Rightarrow> bool" and uv_ :: "value option" and va :: "String.literal" where "x = uu_" and "xa = uv_" and "xb = Some (value.Str va)" and "y = invalid"
oiifs: shows "list_all (\<lambda>m. oiface (fst (snd m)) = ifaceAny) ard"
app\<^sub>1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" shows "(\<Gamma>, App e x, S) \<Rightarrow> (\<Gamma>, e, Arg x # S)"
intro: fixes U\<^sub>1 :: "'a set"   and inf\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top\<^sub>1 :: "'a"   and U\<^sub>2 :: "'b set"   and inf\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top\<^sub>2 :: "'b" assumes "bounded_lattice_top_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 top\<^sub>1"   and "bounded_lattice_top_ow U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2 top\<^sub>2" shows "bounded_lattice_top_pair_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 top\<^sub>1 U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2 top\<^sub>2"
adjoin_subset: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "a \<in> carrier G" shows "adjoin G H a \<subseteq> carrier G"
language_def: shows "Collapse.language \<equiv> {xs. Collapse.nextl (init Collapse_dfa) xs \<in> final Collapse_dfa}"
\<L>_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.\<L> \<NN> \<TT> \<RR> \<SS> = {v |v. CFG.is_word \<TT> v \<and> CFG.is_derivation \<NN> \<TT> \<RR> \<SS> v}"
card_le_one_alt: fixes X :: "'b set" assumes "finite X" shows "(card X \<le> 1) = (X = {} \<or> (\<exists>x. X = {x}))"
partition_on_singletons: fixes A :: "'c set" shows "partition_on A ((\<lambda>x. {x}) ` A)"
in_homE: fixes f :: "'a1"   and a :: "'a1"   and b :: "'a1" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>1 b\<guillemotright>" obtains "A1.arr f" and "A1.dom f = a" and "A1.cod f = b"
subfield_of_algebraics: fixes K :: "'a set" assumes "subfield K R" shows "subfield {x \<in> carrier R. (algebraic over K) x} R"
write_tt_def: fixes new_tt_val :: "word8"   and tbr_val :: "word32" shows "write_tt new_tt_val tbr_val \<equiv> let tmp = tbr_val AND 8589930511 in tmp OR (ucast new_tt_val << 4)"
rel_envir_conv_rel_fun: shows "rel_envir = rel_fun (=)"
len: shows "(length xs + 4) div 5 = nat \<lceil>real (length xs) / 5\<rceil>"
tautology_def: fixes C :: "'b Literal set" shows "tautology C \<equiv> \<exists>A. Pos A \<in> C \<and> Neg A \<in> C"
maxR_code: fixes r :: "('a \<times> 'a) set"   and linear_cf :: "'a set \<Rightarrow> 'a set"   and x :: "'a"   and y :: "'a" assumes "linear_cf r linear_cf" shows "MaxR.maxR r x y = (if (x, y) \<in> r then y else x)"
M_wf: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M"
satisfies_nFNot: fixes \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i" assumes "lformula \<phi>" shows "(\<AA> \<Turnstile> nFNot \<phi>) = (\<AA> \<Turnstile> FNot \<phi>)"
vrat_of_rat_def: fixes x :: "rat" shows "x\<^sub>\<rat> = (real_of_rat x)\<^sub>\<real>"
hfprod_def: fixes RR :: "('e \<Rightarrow> 'g \<Rightarrow> assn) \<times> ('e \<Rightarrow> 'g \<Rightarrow> assn)"   and SS :: "('f \<Rightarrow> 'h \<Rightarrow> assn) \<times> ('f \<Rightarrow> 'h \<Rightarrow> assn)" shows "RR *\<^sub>a SS \<equiv> (fst RR \<times>\<^sub>a fst SS, snd RR \<times>\<^sub>a snd SS)"
eventually_not_iff: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>\<^sub>F x in \<U>. \<not> P x) = (\<not> eventually P \<U>)"
of_hypreal_minus: fixes x :: "real star" shows "of_hypreal (- x) = - of_hypreal x"
ngba_igbg_def: fixes A :: "('d, 'c) ngba" shows "ngba_igbg A \<equiv> graph_rec.extend (ngba_g A) \<lparr>igbg_num_acc = length (accepting A), igbg_acc = ngba_acc (accepting A)\<rparr>"
tendsto_ennrealD: fixes f :: "'a \<Rightarrow> real"   and x :: "real"   and F :: "'a filter" assumes "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "0 \<le> x" shows "(f \<longlongrightarrow> x) F"
plus_diff_simp: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int"   and X :: "nat \<Rightarrow> int"   and Y :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c \<in> carrier Zp"   and "X = a \<ominus> b"   and "Y = c \<ominus> a" shows "X \<oplus> Y = c \<ominus> b"
irrefl_less_nmultiset: fixes X :: "'b nmultiset" assumes "X < X" shows "False"
length_insort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "length (linorder.insort_key less_eq f x xs) = Suc (length xs)"
pres_type_lift: fixes A :: "'a set"   and n :: "nat"   and app :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list" assumes "\<forall>s\<in>A. \<forall>p<n. app p s \<longrightarrow> (\<forall>(q, s')\<in>set (step p s). s' \<in> A)" shows "pres_type (err_step n app step) n (err A)"
T4: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and v :: "'d \<Rightarrow> 'e"   and w :: "'d \<Rightarrow> 'f" shows "\<turnstile> \<box>[P]_v \<longrightarrow> \<box>[_actrans P v]_w"
surj_prop: fixes g :: "'a" assumes "g \<in> carrier G" shows "\<phi> g ` E = E"
reds_throw: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>t :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "throw_of e = \<lfloor>e\<^sub>t\<rfloor>" shows "\<exists>e\<^sub>t'. throw_of e' = \<lfloor>e\<^sub>t'\<rfloor>"
Max_in: fixes A :: "'a word set" assumes "finite A"   and "A \<noteq> {}" shows "signed.Max A \<in> A"
card_of_ordLeq_infinite: fixes A :: "'a set"   and B :: "'b set" assumes "|A| \<le>o |B|"   and "infinite A" shows "infinite B"
runit_in_hom_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.in_hom V (prebicategory_with_homs_and_units.runit V H \<a> \<i> src f) (H f (src f)) f"
simps_4: fixes a :: "Ferrack.num" shows "bound0 (Le a) = numbound0 a"
aux4: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y + x \<cdot> - y = x"
oriented_simplex_def: fixes p :: "nat"   and l :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "oriented_simplex p l \<equiv> \<lambda>x\<in>standard_simplex p. \<lambda>i. \<Sum>j\<le>p. l j i * x j"
ltln_expand_Release: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = \<xi> \<Turnstile>\<^sub>n \<psi> and\<^sub>n \<phi> or\<^sub>n X\<^sub>n (\<phi> R\<^sub>n \<psi>)"
smap_sconst: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "smap f (sconst x) = sconst (f x)"
append_eq_appendI: fixes xs :: "'b list"   and xs1 :: "'b list"   and zs :: "'b list"   and ys :: "'b list"   and us :: "'b list" assumes "xs \<cdot> xs1 = zs"   and "ys = xs1 \<cdot> us" shows "xs \<cdot> ys = zs \<cdot> us"
countable_or_card_of: fixes A :: "'f set" assumes "countable A" shows "finite A \<and> |A| <o |UNIV| \<or> infinite A \<and> |A| =o |UNIV|"
nv2T_vars: fixes x :: "Preliminaries.var"   and T :: "'b trm" assumes "x \<in> nv2T T" shows "x \<in> vars T"
l9_9: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p" assumes "P Q TS A B" shows "\<not> P Q OS A B"
fimage_inter_dom_1: fixes m :: "('d, 'c) fmap"   and A :: "'d fset" shows "fmimage m (fmdom m |\<inter>| A) = fmimage m A"
assms_2: fixes i :: "nat" assumes "i < length xs" shows "xs ! i = ys ! f i"
arctan_series': fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda>k. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) sums arctan x"
cscc_invar_unchanged: fixes v0 :: "'v"   and D0 :: "'v set"   and l :: "'v set list"   and p :: "'v set list"   and D :: "'v set"   and pE :: "('v \<times> 'v) set"   and p' :: "'a"   and pE' :: "'b" assumes "cscc_invar v0 D0 (l, p, D, pE)" shows "cscc_invar_part (l, p', D, pE')"
rule_translate_def: shows "rule_translate \<equiv> map (\<lambda>(u, a, b). SimpleRule u (if a = Accept \<and> b = Accept then Accept else Drop))"
1: shows "(ll \<noteq> {}) = (\<exists>\<alpha>. finished_item \<alpha> \<in> \<II>)"
ag_single_gen_def: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e" shows "ag_single_gen A a = (aGroup A \<and> carrier A = \<Inter> {H. A +> H \<and> a \<in> H})"
IH_1: shows "(\<Sum>m\<in>monomials p1. MPoly_Type.coeff p1 m * (\<Prod>k\<in>keys m. f k ^ lookup m k)) = (\<Sum>m\<in>monomials (lowerPoly i j (liftPoly i j p1)). MPoly_Type.coeff (lowerPoly i j (liftPoly i j p1)) m * (\<Prod>k\<in>keys m. f k ^ lookup m k))"
lnth_ltl: fixes xs :: "'b llist"   and n :: "nat" assumes "\<not> lnull xs" shows "lnth (ltl xs) n = lnth xs (Suc n)"
Next_ge_max: fixes S :: "'a set" assumes "finite S"   and "S \<noteq> {}" shows "Max S < Next S"
natural_transformation_def: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" shows "natural_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_transformation_axioms A B F G \<tau>"
hoare_triple_D: shows "\<turnstile>\<^sub>p {Q} D {adjoint exM0 * P' * exM0 + adjoint exM1 * Q * exM1}"
isnull: shows "ltakeWhile (Not \<circ> P) (ldrop (enat (lfind P lst)) lst) = LNil"
U'_def: fixes x :: "'s option"   and y :: "'s option" shows "K2.U' x y = \<integral>\<^sup>+ x. ennreal_of_enat (eSuc (sfirst (HLD {y}) x)) \<partial>K2.T x"
\<psi>_in_terms_of_\<epsilon>: fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<psi> x g = \<epsilon> x \<cdot>\<^sub>C F g"
isUb_def: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" shows "isUb R S x = (S *<= x \<and> x \<in> R)"
ring_iso_memE_5: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_iso R S" shows "bij_betw h (carrier R) (carrier S)"
ins_barrier_locs_def: shows "mut_m.ins_barrier_locs = prefixed ''store_ins''"
Complexs_minus_iff: fixes a :: "'b" shows "(- a \<in> \<complex>) = (a \<in> \<complex>)"
dom_simp: fixes f :: "'a1 \<times> 'a2" assumes "arr f" shows "local.dom f = (C1.dom (fst f), C2.dom (snd f))"
sgn_squared_ereal: fixes l :: "ereal" assumes "l \<noteq> 0" shows "sgn l * sgn l = 1"
assms_1: shows "C \<ge>\<^sub>m 0\<^sub>m n n"
xs': shows "(x, s') \<in> set_spmf (exec_gpv callee gpv s)"
ipresSubstAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'c, 'd, 'i) Iteration.model_scheme" shows "ipresSubstAll h hA MOD \<equiv> ipresSubst h MOD \<and> ipresSubstAbs h hA MOD"
Lim_within_union: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and x :: "'b"   and s :: "'b set"   and t :: "'b set" shows "(f \<longlongrightarrow> l) (at x within s \<union> t) = ((f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t))"
simps_22: fixes G :: "prog"   and t1 :: "ty"   and t2 :: "ty" shows "wt_binop G CondOr t1 t2 = (t1 = PrimT Boolean \<and> t2 = PrimT Boolean)"
seq_plus_closed: fixes s :: "nat \<Rightarrow> 'a"   and s' :: "nat \<Rightarrow> 'a" assumes "s \<in> carrier R\<^bsup>\<omega>\<^esup>"   and "s' \<in> carrier R\<^bsup>\<omega>\<^esup>" shows "s \<oplus>\<^bsub>R\<^bsup>\<omega>\<^esup>\<^esub> s' \<in> carrier R\<^bsup>\<omega>\<^esup>"
reduction_oracle_def: fixes seed :: "'seed"   and x :: "unit"   and b :: "'\<beta>" shows "reduction_oracle seed x b = Generative_Probabilistic_Value.Pause (h seed b) (\<lambda>x. Generative_Probabilistic_Value.Done (x, ()))"
bisim1Throw2': fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and n :: "nat"   and a :: "'addr"   and xs :: "'addr val list" assumes "\<And>xs. P,e,n,h \<turnstile>' (e, xs) \<leftrightarrow> ([], xs, 0, None)" shows "P,throw e,n,h \<turnstile>' (Throw a, xs) \<leftrightarrow> ([Addr a], xs, length (compE2 e), \<lfloor>a\<rfloor>)"
blocksRedsFinal: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and vs' :: "val list"   and e :: "expr"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>0 :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option"   and l\<^sub>2 :: "char list \<Rightarrow> val option" assumes "wf_prog wf_md P"   and "length Vs = length Ts"   and "length vs = length Ts"   and "distinct Vs"   and "P \<turnstile> Ts Casts vs to vs' "   and "P,E(Vs [\<mapsto>] Ts) \<turnstile> \<langle>e,(h\<^sub>0, l\<^sub>0(Vs [\<mapsto>] vs'))\<rangle> \<rightarrow>* \<langle>e',(h\<^sub>1, l\<^sub>1)\<rangle>"   and "final e'"   and "l\<^sub>2 = override_on l\<^sub>1 l\<^sub>0 (set Vs)" shows "P,E \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h\<^sub>0, l\<^sub>0)\<rangle> \<rightarrow>* \<langle>e',(h\<^sub>1, l\<^sub>2)\<rangle>"
restrictedDomain: fixes R :: "('c \<times> 'd) set"   and X :: "'c set" shows "Domain (R || X) = Domain R \<inter> X"
minimal_ttree: fixes S :: "'a ttree" shows "TTree.empty \<sqsubseteq> S"
thm2_ind2b: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and k :: "nat"   and l :: "nat" assumes "long_ch_by_ord2 f X"   and "finite X"   and "0 < k \<and> k < l \<and> l < card X" shows "[[f (k - 1) f k f l]]"
NewFail\<^sub>1: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and h :: "heap"   and P :: "nat exp prog"   and l :: "val list" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "new_Addr h = None" shows "P \<turnstile>\<^sub>1 \<langle>new C,(h, l, sh)\<rangle> \<Rightarrow> \<langle>THROW OutOfMemory,(h, l, sh)\<rangle>"
ev: shows "\<forall>\<^sub>F z in nhds s. z \<noteq> 1"