hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "'b prog"   and a2 :: "ty"   and a3 :: "bop"   and a4 :: "ty"   and a5 :: "ty" assumes "a1 \<turnstile> a2\<guillemotleft>a3\<guillemotright>a4 : a5" obtains P :: "'b prog" and T1 :: "ty" and T2 :: "ty" where "a1 = Pa__" and "a2 = T1a__" and "a3 = Eq" and "a4 = T2a__" and "a5 = Boolean"   | P :: "'b prog" and T1 :: "ty" and T2 :: "ty" where "a1 = Pb__" and "a2 = T1" and "a3 = NotEq" and "a4 = T2" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pc__" and "a2 = Integer" and "a3 = LessThan" and "a4 = Integer" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pd__" and "a2 = Integer" and "a3 = LessOrEqual" and "a4 = Integer" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pe__" and "a2 = Integer" and "a3 = GreaterThan" and "a4 = Integer" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pf__" and "a2 = Integer" and "a3 = GreaterOrEqual" and "a4 = Integer" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pg__" and "a2 = Integer" and "a3 = Add" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Ph__" and "a2 = Integer" and "a3 = Subtract" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pi__" and "a2 = Integer" and "a3 = Mult" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pj__" and "a2 = Integer" and "a3 = Div" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pk__" and "a2 = Integer" and "a3 = Mod" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pl__" and "a2 = Boolean" and "a3 = BinAnd" and "a4 = Boolean" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pm__" and "a2 = Integer" and "a3 = BinAnd" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pn__" and "a2 = Boolean" and "a3 = BinOr" and "a4 = Boolean" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Po__" and "a2 = Integer" and "a3 = BinOr" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pp__" and "a2 = Boolean" and "a3 = BinXor" and "a4 = Boolean" and "a5 = Boolean"   | P :: "'b prog" where "a1 = Pq__" and "a2 = Integer" and "a3 = BinXor" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Pr__" and "a2 = Integer" and "a3 = ShiftLeft" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = Ps__" and "a2 = Integer" and "a3 = ShiftRightZeros" and "a4 = Integer" and "a5 = Integer"   | P :: "'b prog" where "a1 = P" and "a2 = Integer" and "a3 = ShiftRightSigned" and "a4 = Integer" and "a5 = Integer"
states_PR: shows "dfa.states (nfa.Power_dfa (Reverse_nfa M)) = HF ` Pow (dfa.states M)"
snd_set_def: fixes s :: "('c \<times> 'b) set" shows "snd_set s = {a. \<exists>b. (b, a) \<in> s}"
lless_cong_r: fixes x :: "'a"   and y :: "'a"   and y' :: "'a" assumes "x \<sqsubset> y"   and "y .= y'"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "y' \<in> carrier L" shows "x \<sqsubset> y'"
subst_simps_2: fixes x :: "'a" shows "subst x (Var x) = Var"
lam_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>g : HF'.prod a b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>\<Lambda> a b c g : a \<rightarrow> exp b c\<guillemotright>"
\<F>: fixes X :: "'a set" assumes "X \<in> \<F>" shows "\<exists>a g. a \<notin> U \<and> continuous_on (X - {a}) g \<and> g ` (X - {a}) \<subseteq> T \<and> (\<forall>x\<in>X \<inter> K. g x = h x)"
UN_ball_bex_simps_4: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>x\<in>\<Union> (B ` A). P x) = (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
supp_3: fixes assn :: "assn"   and exp :: "exp" shows "supp (LetA assn exp) = supp exp \<union> supp assn - set (bn assn)"
inter_filter: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "finite A" shows "sum g {x \<in> A. P x} = (\<Sum>x\<in>A. if P x then g x else (0::'c))"
less_log_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(y < log b x) = (b powr y < x)"
rep_eq: fixes x :: "'a dense"   and xa :: "'a dense" shows "(x < xa) = (Rep_dense x < Rep_dense xa)"
inner_zero_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x zero = 0"
cr_scalar_def: fixes R :: "V \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "V"   and y :: "'a" shows "cr_scalar R x y = (\<exists>a. x = [a]\<^sub>\<circ> \<and> R a y)"
aux1: fixes \<A> :: "object atom \<Rightarrow> bool" assumes "valuation M \<Turnstile> \<phi>"   and "\<forall>G\<in>M. \<A> \<Turnstile> G" shows "\<A> \<Turnstile> \<phi>"
definable_def: fixes d :: "ValD" shows "definable d \<equiv> \<exists>M. \<lbrakk>M\<rbrakk>env_empty = d"
assoc'_in_hom\<^sub>A\<^sub>W\<^sub>C_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<noteq> null"   and "g \<star> h \<noteq> null" shows "\<guillemotleft>local.inv \<a>[f, g, h] : f \<star> g \<star> h \<Rightarrow> (f \<star> g) \<star> h\<guillemotright>"
is_auto_reduced_def: fixes B :: "('a \<Rightarrow>\<^sub>0 'c) set" shows "punit.is_auto_reduced B \<equiv> \<forall>b\<in>B. \<not> punit.is_red (B - {b}) b"
simps: fixes a1 :: "'a \<rightharpoonup>\<^sub>f 'b fset"   and a2 :: "'a"   and a3 :: "'b" shows "has_literal' a1 a2 a3 = (\<exists>literals e lits lit. a1 = literals \<and> a2 = e \<and> a3 = lit \<and> fmlookup literals e = Some lits \<and> lit |\<in>| lits)"
case_prodI: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
Smith_1x2_eucl_JNF_works: fixes A :: "'b mat"   and S :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat 1 2"   and "(S, Q) = Smith_1x2_eucl_JNF A" shows "is_SNF A (1\<^sub>m 1, Smith_1x2_eucl_JNF A)"
IsProperInX_def: shows "IsProperInX \<equiv> map_fun (map_fun id eval\<o>) id (\<lambda>\<phi>. \<forall>x v. (\<exists>a. \<nu>\<upsilon> a = \<nu>\<upsilon> x \<and> \<phi> (a\<^sup>P) dj v) = \<phi> (x\<^sup>P) dj v)"
gf: fixes x :: "'a" assumes "x \<in> topspace X" shows "g (f x) = x"
R: fixes f :: "'a \<Rightarrow> 'a" assumes "continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1" shows "\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere (0::'a) 1) f (\<lambda>x. c)"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "P \<langle>\<rangle>"   and "\<And>uu_ r. P \<langle>\<langle>\<rangle>, uu_, r\<rangle>"   and "\<And>ll a lr b r. (ll \<noteq> \<langle>\<rangle> \<Longrightarrow> P ll) \<Longrightarrow> P \<langle>\<langle>ll, a, lr\<rangle>, b, r\<rangle>" shows "P a0"
S3_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S3 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> S3 rmhist p$ \<or> (S4 rmhist p \<or> S6 rmhist p)$"
stut_eq: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "STUTINV F"   and "STUTINV G" shows "STUTINV F = G"
abs_eq: fixes xa :: "'d"   and x :: "'d" shows "to_conjugate_space xa + to_conjugate_space x = to_conjugate_space (xa + x)"
os_is_empty_rule: fixes xs :: "'a list"   and b :: "'a node ref option" shows "<os_list xs b> os_is_empty b <\<lambda>r. os_list xs b * \<up> (r = (xs = []))>"
conjugate_dual_additive: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "bounded_lattice.conjugate (\<squnion>) top f g" shows "lattice.dual_additive (\<squnion>) (\<sqinter>) (uminus \<circ> f)"
P0: shows "p \<otimes>\<^bsub>P\<^esub> q = (p0 \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) \<otimes>\<^bsub>P\<^esub> q0 \<oplus>\<^bsub>P\<^esub> (p0 \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)"
one_sheaf_spec_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.one_sheaf_spec R addition multiplication zero unit U \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.one_rel (R\<setminus>\<pp>) R addition multiplication zero unit"
exec_c1: shows "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> \<Rightarrow> t"
ranking_stuck_odd: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and f :: "nat \<times> 'd \<Rightarrow> nat"   and v :: "nat \<times> 'd"   and r :: "(nat \<times> 'd) stream" assumes "ranking A w f"   and "v \<in> gunodes A w"   and "gurun A w r v" obtains n :: "nat" where "\<forall>a\<in>sset (smap f (gtrace (sdrop n r) (gtarget (stake n r) v))). odd a"
init_worklist_is_sorted: fixes r :: "nat list \<Rightarrow> nat list \<Rightarrow> bool"   and step :: "nat \<Rightarrow> nat list \<Rightarrow> (nat \<times> nat list) list"   and \<tau>s :: "nat list list" shows "Sorted_Less.sorted (unstables r step \<tau>s)"
eq: shows "defNode g \<phi>\<^sub>s = defNode g s"
f_approx_pos: fixes x :: "real" assumes "x\<^sub>0 \<le> x" shows "0 < f_approx x"
induct: fixes P :: "bool list \<times> bool list \<Rightarrow> (nat, nat) atomic \<Rightarrow> bool"   and a0 :: "bool list \<times> bool list"   and a1 :: "(nat, nat) atomic" assumes "\<And>bs1 bs2 m. P (bs1, bs2) (Fo m)"   and "\<And>bs1 bs2 m n. P (bs1, bs2) (Eq_Const None m n)"   and "\<And>bs1 bs2 m1 m2. P (bs1, bs2) (Less None m1 m2)"   and "\<And>bs1 bs2 m1 m2. P (bs1, bs2) (Eq_FO False m1 m2)"   and "\<And>bs1 bs2 m1 m2 n. P (bs1, bs2) (Plus_FO None m1 m2 n)"   and "\<And>bs1 bs2 M1 M2. P (bs1, bs2) (Eq_SO M1 M2)"   and "\<And>bs1 bs2 bl M1 M2. P (bs1, bs2) (Suc_SO False bl M1 M2)"   and "\<And>bs1 bs2 M. P (bs1, bs2) (Empty M)"   and "\<And>bs1 bs2 M. P (bs1, bs2) (Singleton M)"   and "\<And>bs1 bs2 M1 M2. P (bs1, bs2) (Subset M1 M2)"   and "\<And>bs1 bs2 m M. P (bs1, bs2) (In False m M)"   and "\<And>bs1 bs2 m M. P (bs1, bs2) (Eq_Max False m M)"   and "\<And>bs1 bs2 m M. P (bs1, bs2) (Eq_Min False m M)"   and "\<And>bs1 bs2 M1 M2 M3. P (bs1, bs2) (Eq_Union M1 M2 M3)"   and "\<And>bs1 bs2 M1 M2 M3. P (bs1, bs2) (Eq_Inter M1 M2 M3)"   and "\<And>bs1 bs2 M1 M2 M3. P (bs1, bs2) (Eq_Diff M1 M2 M3)"   and "\<And>bs1 bs2 M1 M2. P (bs1, bs2) (Disjoint M1 M2)"   and "\<And>bs1 bs2 M n. P (bs1, bs2) (Eq_Presb None M n)"   and "\<And>uu_ vc va vb. P uu_ (Eq_Const (Some vc) va vb)"   and "\<And>uu_ vc va vb. P uu_ (Less (Some vc) va vb)"   and "\<And>uu_ vd va vb vc. P uu_ (Plus_FO (Some vd) va vb vc)"   and "\<And>uu_ va vb. P uu_ (Eq_FO True va vb)"   and "\<And>uu_ va vb vc. P uu_ (Suc_SO True va vb vc)"   and "\<And>uu_ va vb. P uu_ (In True va vb)"   and "\<And>uu_ va vb. P uu_ (Eq_Max True va vb)"   and "\<And>uu_ va vb. P uu_ (Eq_Min True va vb)"   and "\<And>uu_ vc va vb. P uu_ (Eq_Presb (Some vc) va vb)" shows "P a0 a1"
aux2: fixes acc :: "'Q \<Rightarrow> nat set"   and num_acc :: "nat"   and i :: "nat" assumes "F = {{q. i \<in> acc q} |i. i < num_acc}"   and "\<Union> (range acc) \<subseteq> {0..<num_acc}" shows "finite {q. i \<in> acc q}"
frechet_derivative_transform_within_open: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and X :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "f differentiable at x"   and "open X"   and "x \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x" shows "frechet_derivative f (at x) = frechet_derivative g (at x)"
pp_surjective: fixes x :: "'a" shows "surjective (- - x) = dense (top * x)"
mono_rqs_cost_exp: shows "incseq rqs_cost_exp"
induct: fixes P :: "'b list \<Rightarrow> 'b trie \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b trie" assumes "\<And>b m. P [] (Nd b m)"   and "\<And>x xs b m. (\<And>x2. m x = Some x2 \<Longrightarrow> P xs x2) \<Longrightarrow> P (x # xs) (Nd b m)" shows "P a0 a1"
assms_1: shows "Group H \<and> order.greater_eq G H"
FSbImorph_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'g, 'h, 'i) model_scheme" shows "FSbImorph h hA termMOD MOD = termFSbImorph h hA MOD"
vec_to_iarray_scaleR: fixes a :: "real"   and x :: "('c, 'd) vec" shows "vec_to_iarray (a *\<^sub>R x) = a *\<^sub>R vec_to_iarray x"
simps_1: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "Applicative.ap (pure f) (pure x) = pure (f x)"
nat_to_sch_of_pair_by_index: fixes n1 :: "nat"   and n2 :: "nat" shows "nat_to_sch (pair_by_index n1 n2) = Pair_op (nat_to_sch n1) (nat_to_sch n2)"
simps_2: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'a"   and x :: "'b"   and xs :: "'b list" shows "foldl f a (x # xs) = foldl f (f a x) xs"
r_null: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "semiring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
bounded_iff: fixes b :: "'t"   and c :: "'t"   and a :: "'t" shows "(ord_term_lin.max b c \<preceq>\<^sub>t a) = (b \<preceq>\<^sub>t a \<and> c \<preceq>\<^sub>t a)"
diff_diff_right: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b" shows "c - (b - a) = c + a - b"
assms_4: shows "e = Binop_f t fop \<or> e = Relop_f t frop"
rel_mset_via_perm: fixes rel :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "rel_mset rel (mset xs) (mset ys) = (\<exists>zs. xs <~~> zs \<and> list_all2 rel zs ys)"
int_and_numerals_1: fixes x :: "num"   and y :: "num" shows "numeral (num.Bit0 x) AND numeral (num.Bit0 y) = 2 * (numeral x AND numeral y)"
rb_sp_one: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv1 :: "'d \<times> nat"   and re :: "('d \<times> nat) edge"   and rv2 :: "'d \<times> nat" assumes "RedBlack prb" shows "SubRel.subpath (red prb) rv1 [re] rv2 (subs prb) = (sub_rel_of (red prb) (subs prb) \<and> (rv1 = src re \<or> (rv1, src re) \<in> subs prb) \<and> re \<in> edges (red prb) \<and> (tgt re = rv2 \<or> (tgt re, rv2) \<in> subs prb))"
tt: shows "(take n Strat @ [Strat ! n]) ! n = Strat ! n"
inject_2: fixes x3 :: "'d"   and x2 :: "name list"   and x1 :: "'d"   and y3 :: "'d"   and y2 :: "name list"   and y1 :: "'d" shows "(x3\<lparr>\<nu>*x2\<rparr>\<langle>x1\<rangle> = y3\<lparr>\<nu>*y2\<rparr>\<langle>y1\<rangle>) = (x3 = y3 \<and> x2 = y2 \<and> x1 = y1)"
poly2_monom: fixes a :: "'a poly"   and n :: "nat"   and x :: "'a"   and y :: "'a" shows "poly2 (monom a n) x y = poly a x * y ^ n"
splitAt_2: fixes v :: "'c"   and vs :: "'c list"   and a :: "'c list"   and b :: "'c list"   and ram :: "'c" assumes "v \<in> set vs"   and "(a, b) = splitAt ram vs" shows "v \<in> set a \<or> v \<in> set b \<or> v = ram"
lt_lt_1_ccontr: fixes a :: "int"   and b :: "int" assumes "a < b"   and "b < a + 1" shows "False"
s_mul_ext_trans: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "trans s"   and "trans ns"   and "compatible_l ns s"   and "compatible_r ns s"   and "refl ns"   and "(A, B) \<in> s_mul_ext ns s"   and "(B, C) \<in> s_mul_ext ns s" shows "(A, C) \<in> s_mul_ext ns s"
fresh_16: fixes a :: "atom"   and "term" :: "Syntax.term" shows "a \<sharp> Unauth term = a \<sharp> term"
mheads_type_accessible: fixes emh :: "ref_ty \<times> mhead"   and G :: "prog"   and S :: "qtname"   and T :: "ref_ty"   and sig :: "sig" assumes "emh \<in> mheads G S T sig"   and "wf_prog G" shows "G \<turnstile> RefT T accessible_in pid S"
alw_invar: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "\<phi> xs"   and "alw (\<lambda>xs. \<phi> xs \<longrightarrow> nxt \<phi> xs) xs" shows "alw \<phi> xs"
array_copy_length: fixes n :: "nat"   and xs :: "'b list"   and xs' :: "'b list" assumes "n \<le> length xs"   and "n \<le> length xs'" shows "length (array_copy xs xs' n) = length xs'"
simps_83: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x31 :: "char list"   and x32 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> \<lparr>x31\<rparr>x32"
positively_responsive_prefer_witness: fixes P :: "'i \<Rightarrow> ('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "profile A Is P"   and "hasw [x, y] A"   and "card {i \<in> Is. y \<^bsub>(P i)\<^esub>\<prec> x} < card {i \<in> Is. x \<^bsub>(P i)\<^esub>\<prec> y}" obtains P' :: "'i \<Rightarrow> ('a \<times> 'a) set" and k :: "'i" where "profile A Is P'" and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P' i)\<^esub>\<prec> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P i)\<^esub>\<prec> y" and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P' i)\<^esub>\<approx> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P i)\<^esub>\<preceq> y" and "k \<in> Is \<and> x \<^bsub>(P' k)\<^esub>\<approx> y \<and> x \<^bsub>(P k)\<^esub>\<prec> y" and "card {i \<in> Is. x \<^bsub>(P' i)\<^esub>\<prec> y} = card {i \<in> Is. y \<^bsub>(P' i)\<^esub>\<prec> x}"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case R of L \<Rightarrow> f1 | R \<Rightarrow> f2) = f2"
valid_mbind_mt: fixes \<sigma> :: "'d"   and f :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option"   and P :: "'e list \<Rightarrow> bool" shows "(\<sigma> \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e [] f) (unit\<^sub>S\<^sub>E (P s))) = P []"
map_poly_homo_polyadd: fixes f :: "'a \<Rightarrow> 'b"   and p1 :: "'a poly"   and p2 :: "'a poly" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>x. (f x = (0::'b)) = (x = (0::'a))" shows "map_poly f (p1 +\<^sub>p p2) = map_poly f p1 +\<^sub>p map_poly f p2"
red_rtrancl_ord: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "(red F)\<^sup>*\<^sup>* p q" shows "q \<preceq>\<^sub>p p"
conn1: fixes bid :: "'bid"   and k :: "cnf" assumes "\<parallel>bid\<parallel>\<^bsub>k\<^esub>" shows "bbns (\<sigma>\<^bsub>bid\<^esub>k) = (\<Union>kid\<in>{kid. \<parallel>kid\<parallel>\<^bsub>k\<^esub>}. {ksns (\<sigma>\<^bsub>kid\<^esub>k)})"
induced_arrowI_1: fixes \<chi>' :: "'j \<Rightarrow> 'b"   and a' :: "'b" assumes "\<chi>' \<in> \<chi>a.D.cones a'" shows "\<guillemotleft>\<chi>a.induced_arrow a' \<chi>' : a' \<rightarrow>\<^sub>B l a\<guillemotright>"
infix_sum: shows "(\<Sum>i = i..<j + 1. if postrecording_event t i then card {j \<in> {i + 1..<length t}. prerecording_event t j} else 0) = (\<Sum>ia = i..<j + 1. if postrecording_event (swap_events i j t) ia then card {ja \<in> {ia + 1..<length (swap_events i j t)}. prerecording_event (swap_events i j t) ja} else 0) + 1"
boundedI: fixes b :: "'d"   and a :: "'d"   and c :: "'d" assumes "b \<le> a"   and "c \<le> a" shows "sup b c \<le> a"
compatible_modes_def: fixes mdss :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" shows "conc.compatible_modes mdss = (\<forall>i x. i < length mdss \<longrightarrow> (x \<in> (mdss ! i) AsmNoReadOrWrite \<longrightarrow> (\<forall>j<length mdss. j \<noteq> i \<longrightarrow> x \<in> (mdss ! j) GuarNoReadOrWrite)) \<and> (x \<in> (mdss ! i) AsmNoWrite \<longrightarrow> (\<forall>j<length mdss. j \<noteq> i \<longrightarrow> x \<in> (mdss ! j) GuarNoWrite)))"
linorder_cases: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less" obtains "less x y"   | "x = y"   | "less y x"
is_ring: shows "ring (RDirProd R (RDirProd_list Rs))"
set_sublists_eq: fixes xs :: "'a list" shows "set (sublists xs) = {ys. sublist ys xs}"
goedel_numberingE: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "goedel_numbering \<psi>"   and "\<chi> \<in> \<P>\<^sup>2" obtains c :: "nat \<Rightarrow> nat option" where "c \<in> \<R>" and "\<forall>i. \<chi> i = \<psi> (the (c i))"
complete_natural': fixes j :: "natural"   and k :: "natural"   and f :: "natural \<Rightarrow> (bool \<times> term list) option" shows "(\<exists>n\<ge>j. n \<le> k \<and> is_some (f n)) = is_some (Quickcheck_Exhaustive.exhaustive_natural' f k j)"
Iface_less_eq_empty: fixes x :: "char list" assumes "Iface x \<le> Iface []" shows "x = []"
flip: shows "bounded_bilinear (\<lambda>x y. y *\<^sub>v x)"
assms_1: fixes \<A> :: "nat \<Rightarrow> ('a + 'b, 'c + 'd) distinguisher_obsf" assumes "\<And>\<eta>. interaction_any_bounded_by (\<A> \<eta>) (bound \<eta>)"   and "\<And>\<eta>. lossless \<Longrightarrow> plossless_gpv (exception_\<I> (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>)) (\<A> \<eta>)" shows "constructive_security_obsf' real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common \<A>"
subset_mset_def: fixes A :: "'c multiset"   and B :: "'c multiset" shows "(A \<subset># B) = (A \<subseteq># B \<and> A \<noteq> B)"
False: shows "S \<inter> {x. a \<bullet> x = b} \<noteq> {}"
assms_3: shows "lose t = [False] \<longrightarrow> ack t = [y] \<and> st_out t = y"
le_matrix_def: fixes A :: "'a matrix"   and B :: "'a matrix" shows "(A \<le> B) = (\<forall>j i. Rep_matrix A j i \<le> Rep_matrix B j i)"
dim_vec_lincomb: fixes F :: "complex vec set"   and f :: "complex vec \<Rightarrow> complex"   and n :: "nat" assumes "finite F"   and "f \<in> F \<rightarrow> UNIV"   and "F \<subseteq> carrier_vec n" shows "dim_vec (module.lincomb (module_cpx_vec n) f F) = n"
contlub_cfun_fun: fixes F :: "nat \<Rightarrow> 'a \<rightarrow> 'b"   and x :: "'a" assumes "chain F" shows "(\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
walk: shows "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"
Vhom: shows "top_of_set V homeomorphic_space Euclidean_space (dim V)"
P2_OT_14_FF: fixes m0 :: "bool"   and m1 :: "bool"   and m2 :: "bool"   and m3 :: "bool" shows "R2_14 (m0, m1, m2, m3) (False, False) = funct_OT_14 (m0, m1, m2, m3) (False, False) \<bind> (\<lambda>(out1, out2). S2_14 (False, False) out2)"
elims: fixes x :: "'a tree23"   and y :: "nat" assumes "height x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "'a tree23" and uu_ :: "'a" and r :: "'a tree23" where "x = \<langle>l__, uu_, r__\<rangle>" and "y = Suc (max (height l__) (height r__))"   | l :: "'a tree23" and uv_ :: "'a" and m :: "'a tree23" and uw_ :: "'a" and r :: "'a tree23" where "x = \<langle>l, uv_, m, uw_, r\<rangle>" and "y = Suc (max (height l) (max (height m) (height r)))"
geom_rand_walk_borel_measurable: fixes n :: "nat" shows "random_variable borel (geom_proc n)"
mmulti_join'_opt_True: fixes n :: "nat"   and A_zs :: "nat set list"   and A_x :: "nat set"   and A_xs :: "nat set list"   and A_y :: "nat set"   and A_ys :: "nat set list"   and A_neg :: "nat set list"   and zs :: "'b option list set list"   and x :: "'b option list set"   and xs :: "'b option list set list"   and y :: "'b option list set"   and ys :: "'b option list set list"   and L_neg :: "'b option list set list" assumes "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ A_x # A_xs @ A_y # A_ys) @ A_neg) ((zs @ x # xs @ y # ys) @ L_neg)"   and "length A_xs = length xs"   and "length A_ys = length ys"   and "length A_zs = length zs" shows "mmulti_join' (A_zs @ A_x # A_xs @ A_y # A_ys) A_neg ((zs @ x # xs @ y # ys) @ L_neg) = mmulti_join' (A_zs @ (A_x \<union> A_y) # A_xs @ A_ys) A_neg ((zs @ join x True y # xs @ ys) @ L_neg)"