hcju/isabelleps huggingface dataset:
DisP'': fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and q :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat"   and m :: "nat" assumes "(p \<^bold>\<or> q) at a in' set ((ps, a) # branch)"   and "A, n \<turnstile> (p # ps, a) # branch"   and "A, m \<turnstile> (q # ps, a) # branch" shows "A, max n m \<turnstile> (ps, a) # branch"
cblinfun_extension_existsI: fixes S :: "'a set"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and \<phi> :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> S \<Longrightarrow> B *\<^sub>V x = \<phi> x" shows "cblinfun_extension_exists S \<phi>"
pcompose_uminus: fixes p :: "'a poly"   and r :: "'a poly" shows "- p \<circ>\<^sub>p r = - (p \<circ>\<^sub>p r)"
WhileFalse: fixes b :: "state \<Rightarrow> bool"   and s :: "state"   and c :: "com" assumes "\<not> b s" shows "s -WHILE b DO c\<rightarrow> s"
add_mono_right: fixes c :: "'c"   and d :: "'c"   and a :: "'c" assumes "c \<le> d" shows "a + c \<le> a + d"
residual_blocks_nempty: fixes bld :: "'a set" assumes "bld \<in># \<B>\<^sup>R"   and "multiplicity bl = 1" shows "bld \<noteq> {}"
of_nat_ceiling: fixes r :: "'a" shows "r \<le> of_nat (nat \<lceil>r\<rceil>)"
ci_stk_offer_def: fixes ci :: "'c instr \<Rightarrow> (nat \<times> nat \<times> 'c instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog \<Rightarrow> 'd \<Rightarrow> 'c val list \<Rightarrow> 'c val list \<Rightarrow> String.literal \<Rightarrow> String.literal \<Rightarrow> nat \<Rightarrow> 'c frame list \<Rightarrow> bool" shows "ci_stk_offer ci = (\<forall>ins P h stk stk' loc C M pc frs. ci ins P h stk loc C M pc frs \<longrightarrow> ci ins P h (stk @ stk') loc C M pc frs)"
smult_vec_def: fixes a :: "'a"   and v :: "'a vec" shows "a \<cdot>\<^sub>v v \<equiv> vec (dim_vec v) (\<lambda>i. a * v $ i)"
share_mono: fixes A :: "nat \<Rightarrow> bool option"   and B :: "nat \<Rightarrow> bool option"   and xs :: "'b memref list" assumes "dom A \<subseteq> dom B" shows "dom (share xs A) \<subseteq> dom (share xs B)"
append1_eq_conv: fixes xs :: "'g list"   and x :: "'g"   and ys :: "'g list"   and y :: "'g" shows "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
TR4_mem: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme"   and m :: "'f"   and n :: "'h"   and a :: "'a" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "m \<in> carrier M"   and "n \<in> carrier N"   and "a \<in> carrier R" shows "(m, a \<cdot>\<^sub>s\<^bsub>N\<^esub> n) \<plusminus>\<^bsub>P\<^esub> -\<^sub>a\<^bsub>P\<^esub> a \<cdot>\<^sub>s\<^bsub>P\<^esub> (m, n) \<in> TR\<^bsub>R\<^esub> M N P"
iMOD_iFROM_subset_conv: fixes r' :: "nat"   and m' :: "nat"   and n :: "nat" shows "([ r', mod m' ] \<subseteq> [n\<dots>]) = (n \<le> r')"
EM_remainder_cnj: fixes a :: "int"   and f :: "real \<Rightarrow> complex"   and n :: "nat" assumes "continuous_on {real_of_int a..} f" shows "EM_remainder n (\<lambda>x. cnj (f x)) a = cnj (EM_remainder n f a)"
zero_neq_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "zero \<noteq> numeral.numeral one plus n"
simps_2: fixes p :: "'a poly"   and q :: "'a poly" shows "polynate (Add p q) = polynate p +\<^sub>p polynate q"
has_bochner_integral_restrict_space: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b" assumes "\<Omega> \<inter> space M \<in> sets M" shows "has_bochner_integral (restrict_space M \<Omega>) f i = has_bochner_integral M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) i"
translation_invertible: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c" assumes "left_translations_of_group G composition unit"   and "a \<in> G" shows "monoid.invertible (left_translations_of_monoid.translation G composition ` G) (compose G) (identity G) (left_translations_of_monoid.translation G composition a)"
gen2_nth0: fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "Algorithm.gen2 A B a b ! 0 = (zeroes (length a), zeroes (length b))"
homeomorphic_locally: fixes S :: "'a set"   and T :: "'b set"   and P :: "'a set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool" assumes "S homeomorphic T"   and "\<And>X Y. X homeomorphic Y \<Longrightarrow> P X = Q Y" shows "locally P S = locally Q T"
shows_list_gen_append: fixes xs :: "'a list"   and showsx :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and e :: "char list"   and l :: "char list"   and sep :: "char list"   and r :: "char list"   and s :: "char list"   and t :: "char list" assumes "\<And>r s. \<forall>x\<in>set xs. showsx x (r @ s) = showsx x r @ s" shows "shows_list_gen showsx e l sep r xs (s @ t) = shows_list_gen showsx e l sep r xs s @ t"
deduct_if_deduct_num: fixes M :: "('fun, 'var) Term.term set"   and k :: "nat"   and t :: "('fun, 'var) Term.term" assumes "\<langle>M; k\<rangle> \<turnstile>\<^sub>n t" shows "M \<turnstile> t"
phi_next: shows "uninst_code.ssa.phi phis g (snd next) = Some next_args"
compat_def: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "compat r r' f \<equiv> \<forall>a b. (a, b) \<in> r \<longrightarrow> (f a, f b) \<in> r'"
simps_14: fixes f1 :: "'b"   and f2 :: "BitList \<Rightarrow> 'b \<Rightarrow> 'b"   and f3 :: "BitList \<Rightarrow> 'b \<Rightarrow> 'b"   and x3 :: "BitList" shows "rec_BitList f1 f2 f3 (Bit1 x3) = f3 x3 (rec_BitList f1 f2 f3 x3)"
sum_Un: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B" shows "sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
sel_2: fixes x1 :: "'c"   and x2 :: "'b" shows "snd (x1, x2) = x2"
deg: fixes w :: "'a" assumes "w \<in> verts G"   and "u \<noteq> w"   and "v \<noteq> w" shows "in_degree G w = out_degree G w"
simps_2: fixes ao :: "'a option"   and y :: "'a"   and ys :: "'a list" shows "set_less_aux_Compl ao [] (y # ys) = (proper_interval ao (Some y) \<or> set_less_aux_Compl (Some y) [] ys)"
INF_lower2: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and i :: "'c::type"   and A :: "'c::type set"   and f :: "'c::type \<Rightarrow> 'b::type"   and u :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(i::'c::type) \<in> (A::'c::type set)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((f::'c::type \<Rightarrow> 'b::type) (i::'c::type)) (u::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Inf::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set))) (u::'b::type)"
simps_2: fixes t :: "('c, 'd) bintree" shows "add (Some t) [] = [Some t]"
span_sum: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'b"   and S :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S" shows "sum f A \<in> span S"
strict_trans1: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<sqsubseteq> a"   and "c \<sqsubset> b" shows "c \<sqsubset> a"
len_snoc: fixes \<Pi> :: "('a, 'b) chain"   and x :: "'b"   and P :: "'a" shows "chainlen (cSnoc \<Pi> x P) = 1 + chainlen \<Pi>"
singular_relcycle_restrict: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "singular_relcycle p X (topspace X \<inter> S) = singular_relcycle p X S"
A: fixes f :: "'a \<Rightarrow> 'b dres"   and x :: "'a" assumes "\<And>x. f x \<noteq> dSUCCEED" shows "B f x \<noteq> dSUCCEED"
power_one_over: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and 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 a :: "'c"   and n :: "nat" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "power.power one times (divide one a) n = divide one (power.power one times a n)"
leq_trans: fixes x :: "'a llist"   and y :: "'a llist"   and z :: "'a llist" assumes "lprefix x y"   and "lprefix y z" shows "lprefix x z"
cos_aux_1: fixes prec :: "nat"   and n :: "nat"   and x :: "float" shows "real_of_float (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum>i = 0..<n. (- 1) ^ i * (1 / fact (2 * i)) * real_of_float x ^ (2 * i))"
simps: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and xs :: "'b list" shows "foldl1 f (x # xs) = foldl f x xs"
order_le_less_subst2: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "a \<le> b"   and "f b < c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a < c"
triangle_left: shows "(\<epsilon> \<star> f) \<cdot> (f \<star> \<eta>) = f"
strict_iff_order: fixes b :: "'d"   and a :: "'d" shows "(b < a) = (b \<le> a \<and> a \<noteq> b)"
cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a pmf"   and q :: "'b pmf" assumes "rel_pmf R p q" obtains pq :: "('a \<times> 'b) pmf" where "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y" and "map_pmf fst pq = p" and "map_pmf snd pq = q"
dis: fixes C :: "'a set" assumes "C \<in> components (T - S)"   and "bounded C" shows "C \<inter> L \<noteq> {}"
listset_def: shows "listset \<equiv> rec_list {[]} (\<lambda>A As. set_Cons A)"
rep_eq: fixes x :: "'b"   and xa :: "'b multiset" shows "count (add_mset x xa) = (\<lambda>b. if b = x then Suc (count xa b) else count xa b)"
symrun_interp_set_lifting: fixes \<Gamma> :: "'a constr list"   and \<Gamma>' :: "'a constr list" assumes "set \<Gamma> = set \<Gamma>'" shows "\<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>' \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
simps_2: fixes l :: "'a \<Rightarrow> 'a"   and r :: "'a \<Rightarrow> 'a" shows "traverse_dir l r R = r"
unset_break_status_def: shows "unset_break_status \<equiv> \<lambda>\<sigma>. Some ((), \<sigma>\<lparr>break_status := False\<rparr>)"
abs_eq: fixes x :: "nat \<Rightarrow> rat" assumes "eq_onp Abstract_Linear_Poly.inv x x" shows "coeff (Abs_linear_poly x) = fun_coeff x"
le_less_linear: fixes x :: "'d DBMEntry"   and y :: "'d DBMEntry" shows "x \<preceq> y \<or> y \<prec> x"
ctor2_dtor2: fixes x :: "'d IF2" shows "ctor2 (dtor2 x) = x"
lstrict_prefix_def: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lstrict_prefix xs ys \<equiv> xs \<sqsubseteq> ys \<and> xs \<noteq> ys"
rec_inseparable_def: fixes A :: "nat set"   and B :: "nat set" shows "rec_inseparable A B \<equiv> \<forall>X. A \<subseteq> X \<and> B \<subseteq> - X \<longrightarrow> \<not> computable X"
par_neq1: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "A B Par C D" shows "A \<noteq> B"
lexordp_append_rightI: fixes ys :: "'g list"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "ys \<noteq> []" shows "ord.lexordp less xs (xs @ ys)"
squarefree_prod_coprime: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A; a \<noteq> b\<rbrakk> \<Longrightarrow> coprime (f a) (f b)"   and "\<And>a. a \<in> A \<Longrightarrow> squarefree (f a)" shows "squarefree (prod f A)"
prems_2: fixes p :: "'a" assumes "p \<in># add_mset p__ A" shows "prime p"
hyps: fixes ws :: "'a list list" assumes "ws \<in> lists {u}"   and "\<^bold>|ws\<^bold>| = k" shows "concat ws = u \<^sup>@ k"
openin_subtopology_empty: fixes U :: "'a topology"   and S :: "'a set" shows "openin (subtopology U {}) S = (S = {})"
exhaust_sel: fixes pred :: "'b Predicate.pred" obtains "pred = pred.Pred (pred.eval pred)"
iwlsFSw_iterAll_unique_ipresCons: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and us :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "iwlsFSw MOD"   and "ipresCons h hA MOD" shows "(wls s X \<longrightarrow> h X = iter MOD X) \<and> (wlsAbs (us, s') A \<longrightarrow> hA A = iterAbs MOD A)"
Q_of_Liminf_state_inf: shows "Q_of_state (Liminf_state Sts') \<subseteq> Q_of_state (Liminf_state Sts)"
ex_y: shows "\<forall>x. x \<in># X \<longrightarrow> (\<exists>y. y \<in># Y \<and> gt y x)"
FAssRedsThrow2: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and e :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2,b\<^sub>2\<rangle>"
acomplete_subtreeR: fixes l :: "'c tree"   and x :: "'c"   and r :: "'c tree" assumes "acomplete \<langle>l, x, r\<rangle>" shows "acomplete r"
H_fcs_eq_for_a: fixes b :: "'b" assumes "b \<in> arcs rev.H \<inter> G.face_cycle_set a'" shows "rev.H.face_cycle_set b = rev.S"
a: shows "x:M \<in> ORLEFT (B OR C) (\<parallel>(B)\<parallel>) (\<parallel>(C)\<parallel>)"
field_poly_irreducible_dvd_mult: fixes p :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" assumes "irreducible p" shows "(p dvd q * r) = (p dvd q \<or> p dvd r)"
Dom_in_Obj: fixes f :: "(unit, unit) concrete_category.arr" assumes "arr f" shows "Dom f \<in> {}"
dim_null_space: shows "vec.dim (null_space A) = 0"
inf_sup_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (x + y) = x"
sum_path_weights_append_singleton: fixes xs :: "('c \<times> 'sum \<times> 'd) list"   and l :: "'c"   and x :: "'sum"   and l' :: "'d" shows "sum_weights (map (\<lambda>(s, l, t). l) (xs @ [(l, x, l')])) = followed_by (sum_weights (map (\<lambda>(s, l, t). l) xs)) x"
funof_def: fixes r :: "('b \<times> 'c) set" shows "funof r \<equiv> \<lambda>x. THE y. (x, y) \<in> r"
\<phi>_simps_3: shows "trg f\<^sub>0g\<^sub>1.\<phi> = trg (tab\<^sub>1 g)"
generator_inject: fixes x :: "('f, 'e) generator"   and y :: "('f, 'e) generator" shows "(generator x = generator y) = (x = y)"
int_sdiv_simps_2: fixes a :: "int" shows "a sdiv 0 = 0"
Zfun_norm_iff: fixes f :: "'d \<Rightarrow> 'e"   and F :: "'d filter" shows "Zfun (\<lambda>x. norm (f x)) F = Zfun f F"
fnc: fixes X :: "complex set" assumes "open X"   and "X \<subseteq> S"   and "X \<noteq> {}" shows "\<not> f constant_on X"
antisym_conv1: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.order less_eq less"   and "\<not> less x y" shows "less_eq x y = (x = y)"
permute_fun_app_eq: fixes p :: "perm"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "p \<bullet> f x = (p \<bullet> f) (p \<bullet> x)"
plus_monotone_left: fixes s :: "'x \<Rightarrow>\<^sub>0 nat"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and u :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm s t" shows "lex_pm (u + s) (u + t)"
finite_atLeastLessThan: fixes l :: "nat"   and u :: "nat" shows "finite {l..<u}"
partition_on_mset_eq_imp_eq_carrier: fixes A :: "'a multiset"   and P :: "'a multiset multiset"   and B :: "'a multiset" assumes "partition_on_mset A P"   and "partition_on_mset B P" shows "A = B"
qtable_unit_table: fixes n :: "nat"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool" assumes "\<And>x. \<lbrakk>wf_tuple n {} x; P x\<rbrakk> \<Longrightarrow> Q x" shows "qtable n {} P Q (unit_table n)"
foldl_ins_treap: fixes t :: "('a \<times> 'b) tree"   and xs :: "('a \<times> 'b) list" assumes "treap t" shows "treap (foldl (\<lambda>t' (x, p). ins x p t') t xs)"
subset_empty: fixes A :: "'b set" shows "(A \<subseteq> {}) = (A = {})"
dim_subset_UNIV_cart_gen: fixes S :: "('a, 'b) vec set" shows "vec.dim S \<le> CARD('b)"
elims: fixes x :: "nat list"   and y :: "nat" assumes "Nat_Bijection.list_encode x = y" obtains "x = []" and "y = 0"   | xa :: "nat" and xs :: "nat list" where "x = xa # xs" and "y = Suc (prod_encode (xa, Nat_Bijection.list_encode xs))"
bounded_linear_right: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a" assumes "bounded_bilinear prod" shows "bounded_linear (prod a)"
span_eq: fixes S :: "'b set"   and T :: "'b set" shows "(span S = span T) = (S \<subseteq> span T \<and> T \<subseteq> span S)"
vector_mult_component: fixes x :: "('d, 'e) Finite_Cartesian_Product.vec"   and y :: "('d, 'e) Finite_Cartesian_Product.vec"   and i :: "'e" shows "(x * y) $h i = x $h i * y $h i"
hom_components_nonzero: fixes q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "q \<in> hom_components p" shows "q \<noteq> 0"
Var_eq_iff: fixes x :: "nat"   and s :: "ty"   and y :: "nat"   and t :: "ty" shows "(Var x s = Var y t) = (x = y \<and> s = t)"
assms_1: shows "\<lbrace>Q\<rbrace> d \<lbrace>R\<rbrace>"
topology_eq_iff: fixes t :: "'c topology"   and s :: "'c topology" shows "(t = s) = (topspace t = topspace s \<and> (\<forall>x\<subseteq>topspace t. openin t x = openin s x))"
le_fininf_range: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2" shows "set w\<^sub>1 \<subseteq> sset w\<^sub>2"
order_irr: shows "order (mult_of R) = CARD('a) ^ degree f - 1"
g_set_xx_loc_def: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('a, 'd, 'e) set_ops_scheme" shows "g_set_xx_loc ops1 ops2 \<equiv> StdSet ops1 \<and> StdSet ops2"
inv_split_min: fixes t :: "('b \<times> color) tree"   and x :: "'b"   and t' :: "('b \<times> color) tree" assumes "split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>"   and "invh t"   and "invc t" shows "invh t' \<and> (color t = Red \<longrightarrow> bheight t' = bheight t \<and> invc t') \<and> (color t = Black \<longrightarrow> bheight t' = bheight t - 1 \<and> invc2 t')"
N_wf_trms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s N"
prems: shows "(WHILE b DO c, s) \<rightarrow>(n1) (c2, t)"
get_scdom_component_contains_get_dom_component: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_scdom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and thesis :: "bool"   and ptr :: "(_) object_ptr" assumes "l_get_dom_component_get_scdom_component get_owner_document heap_is_wellformed type_wf known_ptr known_ptrs get_scdom_component get_dom_component"   and "\<And>c. \<lbrakk>h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c; set c \<subseteq> set sc\<rbrakk> \<Longrightarrow> thesis"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "thesis"
suminf_upper: fixes f :: "nat \<Rightarrow> ereal"   and N :: "nat" assumes "\<And>n. 0 \<le> f n" shows "sum f {..<N} \<le> (\<Sum>n. f n)"
h1: shows "\<Gamma> \<turnstile> s \<equiv> t : T"
trace_and_start_determines_end: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.trace channel trans send recv c t c'"   and "distributed_system.trace channel trans send recv c t d'" shows "c' = d'"
card_subset_eq: fixes B :: "'c set"   and A :: "'c set" assumes "finite B"   and "A \<subseteq> B"   and "card A = card B" shows "A = B"
sink_nondet_stimcomm: fixes A :: "'a"   and B :: "'a"   and C :: "'a" assumes "A \<rightarrow>\<^sub>\<S> B"   and "\<forall>s t. s \<in> \<S>\<^sub>a \<and> t \<in> \<S>\<^sub>a \<and> t \<le>\<^sub>\<S> \<lambda> (s, A) \<longrightarrow> \<not> (t \<circ> B \<le>\<^sub>\<K> B + C \<and> t \<circ> C \<le>\<^sub>\<K> B + C)" shows "A \<rightarrow>\<^sub>\<S> B + C"
order_less_imp_le: fixes x :: "'d"   and y :: "'d" assumes "x < y" shows "x \<le> y"
Inverse_relI: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g"   and "f ;;\<^bsub>C\<^esub> g = id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f)"   and "g ;;\<^bsub>C\<^esub> f = id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f)" shows "cinv\<^bsub>C\<^esub> f g"
p_def: shows "p \<equiv> \<^bold>\<not> (\<^bold>@ i Nom i)"
sets_SUP: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and N :: "'c measure" assumes "\<And>x. x \<in> I \<Longrightarrow> sets (M x) = sets N"   and "I \<noteq> {}" shows "sets (Sup (M ` I)) = sets N"
gval_fold_not_invalid_all_valid_contra: fixes G :: "'b gexp list"   and s :: "'b \<Rightarrow> value option" assumes "\<exists>g\<in>set G. gval g s = invalid" shows "gval (fold gAnd G (Bc True)) s = invalid"
eq_span_insert_eq: fixes x :: "'b"   and y :: "'b"   and S :: "'b set" assumes "x - y \<in> span S" shows "span (insert x S) = span (insert y S)"
space_T: fixes s :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "space (T s) = space (stream_space (count_space UNIV))"
sem\<^sub>e\<^sub>s\<^sub>t_d_split_left: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<A>' :: "('fun, 'var) extstrand_step list" assumes "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> (\<A> @ \<A>')" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> \<A>"