hcju/isabelleps huggingface dataset:
v11_injective: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = \<FF>\<lparr>ArrMap\<rparr>\<lparr>b\<rparr>" shows "a = b"
case_eq_if: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and f3 :: "'e \<Rightarrow> 'd \<Rightarrow> 'c"   and step :: "('e, 'd) step" shows "(case step of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = (if step = Done then f1 else if is_Skip step then f2 (un_Skip step) else f3 (un_Yield1 step) (un_Yield2 step))"
rel_map_1: fixes S1b :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S2b :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and i1 :: "'i \<Rightarrow> 'e"   and i2 :: "'j \<Rightarrow> 'g"   and x :: "'i + 'j"   and y :: "'f + 'h" shows "rel_sum S1b S2b (map_sum i1 i2 x) y = rel_sum (\<lambda>x. S1b (i1 x)) (\<lambda>x. S2b (i2 x)) x y"
simps: fixes state :: "State" shows "exhaustiveUnitPropagate state = (if getConflictFlag state \<or> getQ state = [] then state else exhaustiveUnitPropagate (applyUnitPropagate state))"
simps: fixes a1 :: "'tp"   and a2 :: "('c, 'd) lit list" shows "(a1 \<turnstile> a2) = ((\<exists>\<sigma> c. a1 = \<sigma> \<and> a2 = c \<and> infTp \<sigma>) \<or> (\<exists>c \<sigma>. a1 = \<sigma> \<and> a2 = c \<and> (\<forall>x\<in>nvC c. tpOfV x \<noteq> \<sigma>)))"
subst_vsubst_compose2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Z :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and y1 :: "'var" assumes "good X"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "X #[y1 // y]_ys #[Z / z]_zs = X #[Z / z]_zs #[Var ys y1 #[Z / z]_zs / y]_ys"
constraint_expand: fixes x :: "'a word"   and lower :: "'a word"   and upper :: "'a word" shows "(x \<in> {y. lower \<le> y \<and> y \<le> upper}) = (lower \<le> x \<and> x \<le> upper)"
assms: shows "h \<turnstile> get_disconnected_document child_node \<rightarrow>\<^sub>r disconnected_document"
list_comb_snoc: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "app (term.list_comb app f xs) y = term.list_comb app f (xs @ [y])"
less_le: fixes x :: "'b word"   and y :: "'b word" shows "(x <s y) = (x \<le>s y \<and> x \<noteq> y)"
1: shows "distributed M lborel (\<lambda>x. X i x + (\<Sum>i\<in>I. X i x)) (\<lambda>x. ennreal (normal_density (\<mu> i + sum \<mu> I) (sqrt ((\<sigma> i)\<^sup>2 + (sqrt (\<Sum>i\<in>I. (\<sigma> i)\<^sup>2))\<^sup>2)) x))"
word64_bounds_1: fixes x :: "64 word" shows "- (2 ^ (size x - 1)) = - 9223372036854775808"
adversary3_def: fixes \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf"   and g :: "'a" shows "adversary3 \<A> g = \<A> (g, g [^] x)"
homologous_rel_singular_subdivision: fixes p :: "nat"   and X :: "'c topology"   and T :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle p X T c" shows "homologous_rel p X T (singular_subdivision p c) c"
ordIso_imp_ordLeq: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "r =o r'" shows "r \<le>o r'"
empty: shows "GEN_OP emp3 {} (\<langle>B\<rangle>Rs3)"
a_Let: fixes x :: "Syntax.var"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and e\<^sub>1 :: "Syntax.term"   and eP\<^sub>1 :: "Syntax.term"   and eV\<^sub>1 :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and e\<^sub>2 :: "Syntax.term"   and eP\<^sub>2 :: "Syntax.term"   and eV\<^sub>2 :: "Syntax.term"   and \<tau>\<^sub>2 :: "ty" assumes "atom x \<sharp> (\<Gamma>, e\<^sub>1, eP\<^sub>1, eV\<^sub>1)"   and "\<Gamma> \<turnstile> e\<^sub>1, eP\<^sub>1, eV\<^sub>1 : \<tau>\<^sub>1"   and "\<Gamma>(x $$:= \<tau>\<^sub>1) \<turnstile> e\<^sub>2, eP\<^sub>2, eV\<^sub>2 : \<tau>\<^sub>2" shows "\<Gamma> \<turnstile> Syntax.Let e\<^sub>1 x e\<^sub>2, Syntax.Let eP\<^sub>1 x eP\<^sub>2, Syntax.Let eV\<^sub>1 x eV\<^sub>2 : \<tau>\<^sub>2"
discrim_def: fixes X :: "exp" shows "discrim X = the_elem (\<Union>U\<in>Rep_Exp X. {freediscrim U})"
assms_3: shows "xs \<in>\<^sub>\<circ> Vset \<alpha>"
image_eqI: fixes b :: "'b"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and A :: "'c set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
chain_sup_subgraph: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph"   and j :: "nat" assumes "chain S" shows "subgraph (S j) (chain_sup S)"
conjugate_dist_add: fixes a :: "'a"   and b :: "'a" shows "conjugate (a + b) = conjugate a + conjugate b"
R13_aux_3: shows "pmf (sds R13) d = 1 - pmf (sds R13) a"
right_complemented_monoid_algebra: shows "class.right_complemented_monoid_algebra lesseq_r less_r (1::'a) (*) inf_rr (r\<rightarrow>)"
simps_2: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signals :: "signal \<Rightarrow> nat"   and chan :: "chan set"   and bad :: "agent set"   and more :: "'a" shows "l2_state.bad \<lparr>ik = ik, secret = secret, progress = progress, signals = signals, chan = chan, bad = bad, \<dots> = more\<rparr> = bad"
sporadic_on_e1: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<tau> :: "'b tag_const"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>"
simps_1: fixes x :: "octo"   and y :: "octo" shows "Ree (x - y) = Ree x - Ree y"
min_add_distrib_left: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "subset_mset.min x y + z = subset_mset.min (x + z) (y + z)"
assms_1: shows "strict_monoidal_category D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D"
simps: fixes R :: "('a, 'b) ring_scheme"   and a1 :: "nat"   and a2 :: "'a set"   and a3 :: "'a set" assumes "ring R" shows "ring.dimension R a1 a2 a3 = ((\<exists>K. a1 = 0 \<and> a2 = K \<and> a3 = {\<zero>\<^bsub>R\<^esub>}) \<or> (\<exists>v E n K. a1 = Suc n \<and> a2 = K \<and> a3 = ring.line_extension R K v E \<and> v \<in> carrier R \<and> v \<notin> E \<and> ring.dimension R n K E))"
loop: fixes p :: "assertion"   and c :: "command" assumes "prov_triple (p, c, p)" shows "prov_triple (p, Loop c, p)"
simps_9: fixes var :: "nat"   and a :: "atom fm" shows "freeIn var (Neg a) = freeIn var a"
inf_eq_neg_sup: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "inf a b = uminus (sup (uminus a) (uminus b))"
cindex_poly_0_2: fixes a :: "real"   and b :: "real"   and q :: "real poly" shows "cindex_poly a b q 0 = 0"
order_impl_l: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r"   and "less_eq a b" shows "imp_l a b = one"
pair_list_nonzero: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "rb_aux_inv (bs, ss, ps)"   and "p \<in> set ps" shows "poly_of_pair p \<noteq> 0"
strict_trans2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "ordering less_eq less"   and "less a b"   and "less_eq b c" shows "less a c"
Mul_CountInv_def: shows "Mul_CountInv \<equiv> \<lambda>s ind. \<lbrace>\<acute>(<) ind \<and> \<acute>((!) (Ma s)) = Black\<rbrace> \<subseteq> bc s"
vcg_of_SPEC_np_1: fixes f :: "'c nres"   and post :: "'c \<Rightarrow> bool"   and m :: "'c nres" assumes "f \<equiv> SPEC post"   and "SPEC post \<le> m" shows "f \<le> m"
val_zero: fixes f :: "nat \<Rightarrow> int"   and p :: "int" assumes "f \<in> padic_set p" shows "(padic_val p f = - 1) = (f = padic_zero p)"
multiset_diff_union_assoc: fixes C :: "'k multiset"   and B :: "'k multiset"   and A :: "'k multiset" assumes "C \<subseteq># B" shows "A + B - C = A + (B - C)"
get_const_sum: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<noteq> \<infinity>"   and "b \<noteq> \<infinity>"   and "get_const a \<in> \<int>"   and "get_const b \<in> \<int>" shows "get_const (a + b) \<in> \<int>"
theta_F: fixes f :: "ValD \<rightarrow> ValD"   and f' :: "(('o ValK \<rightarrow> 'o) \<rightarrow> 'o) \<rightarrow> ('o ValK \<rightarrow> 'o) \<rightarrow> 'o" assumes "\<And>x x'. \<eta>: x \<mapsto> x' \<Longrightarrow> \<eta>: f\<cdot>x \<mapsto> f'\<cdot>x'" shows "\<theta>: f \<mapsto> f'"
\<Phi>_hps_ge0: fixes hs :: "'b hp list" shows "0 \<le> \<Phi>_hps hs"
thmD11_helper: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and X :: "symbol \<times> character list"   and T :: "(symbol \<times> character list) set"   and q :: "(symbol \<times> character list) list"   and N :: "symbol"   and \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and r :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and \<gamma> :: "symbol list"   and k' :: "nat"   and x :: "item"   and I :: "item set" assumes "p \<in> \<PP>"   and "charslength p = k"   and "X \<in> T"   and "T \<subseteq> \<X> k"   and "q = p @ [X]"   and "(N, \<alpha> @ \<beta>) \<in> \<RR>"   and "r \<le> length q"   and "LeftDerivation [\<SS>] D (terminals (take r q) @ [N] @ \<gamma>)"   and "leftderives \<alpha> (terminals (drop r q))"   and "k' = k + length (chars_of_token X)"   and "x = Item (N, \<alpha> @ \<beta>) (length \<alpha>) (charslength (take r q)) k'"   and "I = items_le k' (\<pi> k' {} (Scan T k (Gen (Prefixes p))))" shows "x \<in> I"
subst0_I: fixes p :: "fm"   and b :: "real"   and bs :: "real list"   and a :: "MIR.num" assumes "qfree p" shows "Ifm (b # bs) (subst0 a p) = Ifm (Inum (b # bs) a # bs) p"
fract_collapse_1: fixes k :: "'c" shows "Fraction_Field.Fract (0::'c) k = 0"
x_prop: shows "(\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow>* (a6,a7)" obtains ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" where "a1 = Sa__" and "a2 = fa__" and "a3 = msa__" and "a4 = sa__" and "a5 = []" and "a6 = msa__" and "a7 = sa__" and "length msa__ = length sa__"   | S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as' :: "'edge list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = last as # as'" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')" and "S,f \<turnstile> (ms',s') =as'\<Rightarrow>* (ms'',s'')"
d\<mu>_def: fixes i :: "nat"   and j :: "nat" shows "fs'.d\<mu> i j = int_of_rat (rat_of_int (fs'.d fs' (Suc j)) * fs'.gs.\<mu> i j)"
deleteMin_invar: fixes bq :: "('c, 'd) BinomialTree list" assumes "invar bq"   and "bq \<noteq> []" shows "invar (deleteMin bq)"
has_bochner_integral_implies_finite_norm: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "has_bochner_integral M f x" shows "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>"
assms_1: shows "externals (ioa.asig B) = externals (ioa.asig A)"
assms_3: fixes dip :: "nat" shows "sqn (rt \<xi>) dip \<le> sqn (rt \<xi>') dip"
openin_final_topology: fixes X :: "'c set"   and Y :: "'d \<Rightarrow> 'e topology"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c" shows "openin (final_topology X Y f) = (\<lambda>U. U \<subseteq> X \<and> (\<forall>i. openin (Y i) (f i -` U \<inter> topspace (Y i))))"
fs_scalar_Ints: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j < length fs" shows "fs ! i \<bullet> fs ! j \<in> \<int>"
geodesic_segment_betweenI: fixes g :: "real \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and G :: "'b set" assumes "g 0 = x"   and "g (dist x y) = y"   and "isometry_on {0..dist x y} g"   and "G = g ` {0..dist x y}" shows "geodesic_segment_between G x y"
ldfix: shows "LeftDerivationIntro (\<alpha> @ \<delta>) i r ix D j (\<beta> @ \<delta>')"
step: shows "(ts, m, \<S>) \<Rightarrow> (ts', m', \<S>')"
atLeast_Suc: fixes k :: "nat" shows "{Suc k..} = {k..} - {k}"
LPDi_safe_fv_regex: fixes mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex"   and n :: "nat" assumes "safe_regex Futu Strict (from_mregex mr \<phi>s)"   and "ms \<in> LPDi n mr" shows "fv_regex (from_mregex ms \<phi>s) = fv_regex (from_mregex mr \<phi>s)"
simps: fixes u :: "'b trm"   and v :: "'b trm"   and s :: "('b \<times> 'b trm) list" shows "subst_equation (Eq u v) s = Eq (u \<lhd> s) (v \<lhd> s)"
open_existence_ivl: fixes x0 :: "'a" shows "open (existence_ivl t0 x0)"
f_shrink_length: fixes xs :: "'d message_af list"   and k :: "nat" shows "length (xs \<div> k) = length xs div k"
\<tau>diverge_bisim_inv: fixes s1 :: "'c"   and s2 :: "'e" assumes "bisim23 s1 s2" shows "s1 -\<tau>2\<rightarrow> \<infinity> = trsys3.\<tau>diverge s2"
add_poly_p_add_mset_comb3: fixes A :: "(char list multiset \<times> int) multiset"   and Aa :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and ys :: "char list multiset"   and m :: "int" assumes "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#}) ({#}, {#}, r)" shows "add_poly_p\<^sup>*\<^sup>* (A, add_mset (ys, m) Aa, {#}) ({#}, {#}, add_mset (ys, m) r)"
minus_fun_def: fixes A :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd" shows "A - B = (\<lambda>x. A x - B x)"
tmod4_4: shows "[mn_] \<in> defined_mods s"
summable_LIMSEQ': fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "(\<lambda>n. sum f {..n}) \<longlonglongrightarrow> suminf f"
inters_is_singleton: fixes l :: "Lines"   and m :: "Lines"   and P :: "Points"   and Q :: "Points" assumes "l \<noteq> m"   and "P \<in> inters l m"   and "Q \<in> inters l m" shows "P = Q"
index_row_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "Matrix.row A i $v j = A $$ (i, j)"
nn_integral_has_integral_lborel: fixes f :: "'a \<Rightarrow> real"   and I :: "real" assumes "f \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "(f has_integral I) UNIV" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal I"
binary_functor_axioms: shows "binary_functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) S.map"
iTrigger_not_iSince_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "finite I" shows "(P t'. t' \<T> t I. Q t) = (\<not> (\<not> P t'. t' \<S> t I. \<not> Q t))"
opaque_dist_def: shows "opaque_dist = fold (\<lambda>i y. y \<diamondop> Opaque i)"
AE_T_enabled: fixes s :: "'b" shows "almost_everywhere (K2.T s) (K2.enabled s)"
mod_partition_iT_Div_Int2: fixes k :: "nat"   and d :: "nat"   and n :: "nat"   and A :: "nat set" assumes "0 < k"   and "0 < d"   and "n mod k = 0"   and "d mod k = 0" shows "A \<inter> [n\<dots>,d - Suc 0] \<oslash> k = (A \<oslash> k) \<inter> ([n\<dots>,d - Suc 0] \<oslash> k)"
fx: fixes x :: "'a"   and t :: "'b" assumes "x \<in> U"   and "t \<in> cbox a b" shows "((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
bts_dfs_rev_distinct: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and ts :: "('d, 'e) bintree list" assumes "distinct (bts_dfs f ts)" shows "distinct (bts_dfs f (rev ts))"
g: fixes j :: "nat"   and r :: "nat" shows "eval g [j, r, a, b] \<down>= e_snoc r (e_nth b j)"
induct: fixes P :: "trm \<Rightarrow> bool"   and trm :: "trm" assumes "P zer"   and "\<And>x. P (Var x)"   and "\<And>x. P x \<Longrightarrow> P (suc x)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (pls x1 x2a)"   and "\<And>x1 x2a. \<lbrakk>P x1; P x2a\<rbrakk> \<Longrightarrow> P (tms x1 x2a)" shows "P trm"
neutral: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "is_pref_profile R" shows "sds (permute_profile \<sigma> R) = map_pmf \<sigma> (sds R)"
Max_image_cong: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "S = T"   and "\<And>x. x \<in> T \<Longrightarrow> f x = g x" shows "Max (f ` S) = Max (g ` T)"
to_nat_on_infinite: fixes S :: "'a set" assumes "countable S"   and "infinite S" shows "bij_betw (to_nat_on S) S UNIV"
contf: shows "continuous_on (convex hull {a, b, c}) f"
kE_wtE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "I.wtE (kE \<xi>)"
mset_set_size_card_count: fixes A :: "nat multiset" shows "(\<Sum>x\<in>#A. x) = (\<Sum>x\<in>set_mset A. x * count A x)"
errMOD_igSubstAbsIPresIGWlsAbsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igSubstAbsIPresIGWlsAbs MOD"   and "igWlsAbsDisj MOD"   and "igWlsAbsIsInBar MOD" shows "igSubstAbsIPresIGWlsAbsSTR (errMOD MOD)"
assms_3: shows "D r \<tau> \<sigma> \<sigma>' \<tau>'"
floor_less_iff: fixes x :: "'a"   and z :: "int" shows "(\<lfloor>x\<rfloor> < z) = (x < of_int z)"
simps_2: fixes \<sigma> :: "'e \<Rightarrow> ('c monom \<times> 'd) list"   and x :: "'e"   and p :: "nat"   and m :: "('e \<times> nat) list" shows "monom_list_subst \<sigma> ((x, p) # m) = poly_mult (poly_power (\<sigma> x) p) (monom_list_subst \<sigma> m)"
\<tau>eq: shows "loopfp\<tau> the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>))** I X = game_sem I the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>))** X"
rel_star_def: fixes X :: "('b \<times> 'b) set" shows "X\<^sup>* = \<Union> (range (rel_dioid.power X))"
assms_2: shows "is_mgu \<sigma> (subst_set (subst x t) E)"
right_neutral: fixes a :: "'g" shows "a * (1::'g) = a"
simps_2: fixes T :: "typ"   and body :: "Core.term"   and n :: "nat"   and u :: "Core.term" shows "subst_bv2 (Abs T body) n u = Abs T (subst_bv2 body (n + 1) (lift u 0))"
simps_5: fixes f1 :: "ename \<Rightarrow> 'a"   and f2 :: "'a" shows "(case This of EName x \<Rightarrow> f1 x | This \<Rightarrow> f2) = f2"
difference_eq: fixes x :: "i"   and y :: "i" shows "x \<ominus> y = (THE z. \<forall>w. P w z = (P w x \<and> \<not> O w y))"
induct: fixes x1 :: "'a list"   and x2 :: "'a list"   and r :: "('a \<times> 'a) set"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "(x1, x2) \<in> genPrefix r"   and "P [] []"   and "\<And>xs ys x y. \<lbrakk>(xs, ys) \<in> genPrefix r; P xs ys; (x, y) \<in> r\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>xs ys zs. \<lbrakk>(xs, ys) \<in> genPrefix r; P xs ys\<rbrakk> \<Longrightarrow> P xs (ys @ zs)" shows "P x1 x2"
RedSeq: fixes v :: "val"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>Val v;; e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e\<^sub>2,s\<rangle>"
q4: fixes f :: "'a" assumes "\<forall>c>0. 2 * Re \<langle>h - k, f\<rangle> \<le> c * \<parallel>f\<parallel>\<^sup>2" shows "\<forall>c>0. 2 * Re \<langle>h - k, f\<rangle> \<le> c"
M_index: fixes i :: "nat"   and j :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)"   and "j < length (map of_int_hom.vec_hom fs)" shows "gs1.M (length (map of_int_hom.vec_hom fs)) $$ (i, j) = gs1.\<mu> i j"
hyps: fixes a :: "int"   and b :: "int" assumes "\<forall>a\<in>#ps. prime a"   and "odd (int (\<Prod>i\<in>#ps. i))"   and "coprime a b"   and "a\<^sup>2 + 3 * b\<^sup>2 = int (\<Prod>i\<in>#ps. i) ^ 3" shows "is_cube_form a b"
elims_1: fixes x :: "'b val"   and y :: "bool" assumes "is_Addr x = y" obtains a :: "'b" where "x = Addr a" and "y"   | "x = Unit" and "\<not> y"   | "x = Null" and "\<not> y"   | v :: "bool" where "x = Bool v__" and "\<not> y"   | v :: "32 word" where "x = Intg v" and "\<not> y"
cComm1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "distinct xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K" shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'"
assms_3: shows "Coplanar C1 C2 A2 B1"
zero_fun_def: shows "0 = (\<lambda>x. 0::'b)"