hcju/isabelleps huggingface dataset:
card_wellformed: fixes k :: "'b"   and A :: "'b set" assumes "k \<in> A" shows "enat (card {i \<in> A. i < k}) < esize A"
sallI: fixes S :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "\<And>s::'b set. s \<subseteq> (S::'b set) \<Longrightarrow> (P::'b set \<Rightarrow> bool) s" shows "\<forall>s\<subseteq>S::'b set. (P::'b set \<Rightarrow> bool) s"
mk_or_def: fixes x :: "'b ltln"   and y :: "'b ltln" shows "mk_or x y \<equiv> case x of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> y | _ \<Rightarrow> case y of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> x | _ \<Rightarrow> x or\<^sub>n y"
simps_3: fixes b1 :: "bexp"   and b2 :: "bexp"   and res :: "bool"   and S :: "'av Abs_State.st option" shows "bfilter (And b1 b2) res S = (if res then bfilter b1 True (bfilter b2 True S) else bfilter b1 False S \<squnion> bfilter b2 False S)"
mult_scaleC_right: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and a :: "complex"   and y :: "'d" assumes "class.complex_algebra scaleR scaleC plus zero minus uminus times" shows "times x (scaleC a y) = scaleC a (times x y)"
hby2: shows "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>g l - g m\<bar> \<le> y"
mute_preserves_dimensions_1: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.mute q k l (r, A) = (r', A')" shows "dim_row A' = dim_row A"
map_eq_map_tailrec: shows "list.map = map_tailrec"
in_homE: fixes f :: "nat"   and a :: "nat"   and b :: "nat" assumes "J'J.A.in_hom f a b" obtains "J'J.A.arr f" and "J'J.A.dom f = a" and "J'J.A.cod f = b"
ts_process_refusals: fixes T :: "'a list set"   and xs :: "'a list" assumes "trace_set T"   and "xs \<in> T" shows "refusals (ts_process T) xs = {X. \<forall>x\<in>X. xs @ [x] \<notin> T}"
four_x_squared: fixes x :: "real" shows "4 * x\<^sup>2 = (2 * x)\<^sup>2"
HBL1_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and P :: "'c" shows "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc P \<equiv> Encode var trm fmla Var FvarsT substT Fvars subst num enc \<and> Deduct2 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv \<and> HBL1_axioms var fmla Fvars subst prv bprv enc P"
simps_1: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" shows "thread_oks ts [] = True"
all_n_lists_iff: fixes cenum :: "'a list"   and P :: "'a list \<Rightarrow> bool"   and n :: "nat" shows "all_n_lists (Ball (set cenum)) P n = (\<forall>xs\<in>set (List.n_lists n cenum). P xs)"
option_pred_mono_strong: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option"   and P' :: "'c \<Rightarrow> bool" assumes "pred_option P x"   and "\<And>a. \<lbrakk>a \<in> set_option x; P a\<rbrakk> \<Longrightarrow> P' a" shows "pred_option P' x"
has_limits_of_shape_def: fixes J :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "J.has_limits_of_shape J \<equiv> \<forall>D. diagram J J D \<longrightarrow> (\<exists>a \<chi>. limit_cone J J D a \<chi>)"
det_iarrays_def: fixes A :: "'b iarray iarray" shows "det_iarrays A = (let A' = Gauss_Jordan_det_P_iarrays A in (\<Prod>i\<leftarrow>[0..<nrows_iarray A]. snd A' !! i !! i) / fst A')"
induct: fixes P :: "'a list list \<Rightarrow> bool"   and a0 :: "'a list list" assumes "\<And>xs ys zss. P zss \<Longrightarrow> P (xs # ys # zss)"   and "P []"   and "\<And>x. P [x]" shows "P a0"
tail_def: fixes t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.tail t = (case FingerTreeStruc.viewL t of Some (x, m) \<Rightarrow> m)"
exec_meth_append_xt: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and s :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and ins' :: "'a instr list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" assumes "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt t h s ta h' s'" shows "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P (ins @ ins') (xt @ xt') t h s ta h' s'"
end_point_at_most_one: fixes x :: "'a" assumes "path x" shows "is_inj (end_points x)"
nn: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in># X"   and "y \<in># Y"   and "z \<in># Z"   and "(x, y) \<in> ns"   and "(y, z) \<in> ns" shows "(x, z) \<in> ns"
isomorphic_symmetric: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "category.isomorphic C f g" shows "category.isomorphic C g f"
count_repeat_mset: fixes i :: "nat"   and A :: "'a multiset"   and a :: "'a" shows "count (repeat_mset i A) a = i * count A a"
"4_1": shows "card {p. p partitions m \<and> sum p {..m} = k} = Partition m k"
degree_m_eq: fixes f :: "int poly" assumes "lead_coeff f mod p ^ n \<noteq> 0"   and "1 < p ^ n" shows "pn.degree_m f = degree f"
kleene_qfp_is_least: fixes q :: "'a" assumes "\<forall>q\<in>A. \<forall>x\<in>A. f q \<sim> q \<longrightarrow> x \<sqsubseteq> f q \<longrightarrow> x \<sqsubseteq> q"   and "extreme_bound A (\<sqsubseteq>) {f^n \<bottom> |. n} q" shows "extreme {s \<in> A. f s \<sim> s} (\<lambda>x y. y \<sqsubseteq> x) q"
dense_complement_convex: fixes T :: "'a set"   and S :: "'a set" assumes "aff_dim T < aff_dim S"   and "convex S" shows "closure (S - T) = closure S"
nfa_startnode_def: fixes A :: "bool list bdd list \<times> bool list" shows "nfa_startnode A = (replicate (length (fst A)) False)[0 := True]"
butlast_map: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "xs \<noteq> []" shows "butlast (map f xs) = map f (butlast xs)"
const_def: fixes x :: "'d" shows "pure x = (\<lambda>_. x)"
\<gamma>3m: shows "set (labels_conv \<gamma>3) \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step}"
assoc: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" shows "(a \<cdot> b) \<cdot> c = a \<cdot> b \<cdot> c"
deriv_strip: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P strip_guards (- F) c UNIV,UNIV"
tm_comp_range: fixes x :: "real"   and e' :: "nat \<Rightarrow> real"   and tf :: "taylor_model"   and e :: "nat \<Rightarrow> real"   and tg :: "taylor_model"   and ga :: "float"   and I :: "float interval list"   and a :: "float interval list"   and prec :: "nat"   and ord :: "nat" assumes "x \<in>\<^sub>i range_tm e' tf"   and "e' 0 \<in>\<^sub>i range_tm e (tm_sub tg (tm_const ga))"   and "num_params (tm_poly tf) \<le> 1"   and "num_params (tm_poly tg) \<le> length I"   and "develops_at_within e a I" shows "x \<in>\<^sub>i range_tm e (tm_comp prec ord I a ga tf tg)"
fin_nrename: fixes M :: "trm"   and z :: "name"   and x :: "name"   and y :: "name" assumes "fin M z"   and "z \<sharp> (x, y)" shows "fin (M[x\<turnstile>n>y]) z"
ideal_prod_one: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set" assumes "ring R"   and "ideal I R" shows "I \<cdot>\<^bsub>R\<^esub> carrier R = I"
snd_pair: fixes a :: "'b"   and b :: "'a" shows "snd (a, b) = b"
intros_2: fixes xs :: "'a list"   and x :: "'a" assumes "\<forall>y\<in>set xs. x \<prec> y"   and "local.sorted xs" shows "local.sorted (x # xs)"
zero_eq_1_divide_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(zero = divide one a) = (a = zero)"
index_not_zero_impl_diff_group: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set"   and G :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> point_set"   and "card ps = 2"   and "int (block_collection index ps) = index"   and "G \<in> groups" shows "\<not> ps \<subseteq> G"
smult_l_null: fixes x :: "'a Matrix.vec" assumes "x \<in> carrier_vec n" shows "(0::'a) \<cdot>\<^sub>v x = 0\<^sub>v n"
snap_p_i: shows "ps (S t i) p \<noteq> None"
arcsin_le_iff: fixes x :: "real"   and y :: "real" assumes "- 1 \<le> x"   and "x \<le> 1"   and "- pi / 2 \<le> y"   and "y \<le> pi / 2" shows "(arcsin x \<le> y) = (x \<le> sin y)"
backflows_finite: fixes E :: "('b \<times> 'b) set" assumes "finite E" shows "finite (backflows E)"
perm_append1: fixes xs :: "'a list"   and ys :: "'a list"   and l :: "'a list" assumes "xs <~~> ys" shows "l @ xs <~~> l @ ys"
Leaf_def: shows "Leaf \<equiv> Old_Datatype.Atom \<circ> Inl"
rel_option_reflI: fixes y :: "'a option"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. x \<in> set_option y \<Longrightarrow> P x x" shows "rel_option P y y"
intro: fixes set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr h h' ptr'. \<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')" shows "l_set_mode_get_disconnected_nodes_axioms set_mode_locs get_disconnected_nodes_locs"
Abs_seqauto_inverse: fixes y :: "'a set \<times> 'a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) set \<times> ('a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) \<times> 'a) set" assumes "y \<in> seqauto" shows "Rep_seqauto (Abs_seqauto y) = y"
sub: fixes a :: "'a set" assumes "a \<in> induced.Ker" shows "a \<in> contained.Partition"
abelian_groupE_1: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
simps_4: fixes v :: "'d"   and va :: "'a \<Rightarrow> nat"   and vb :: "'a \<Rightarrow> 'b"   and vc :: "('a, 'b, 'c, 'd) seqp"   and vd :: "('a, 'b, 'c, 'd) seqp" shows "stermsl ({v}unicast(va, vb) . vc \<triangleright> vd) = {{v}unicast(va, vb) . vc \<triangleright> vd}"
l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors_di :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_di_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs get_ancestors_di get_ancestors_di_locs \<equiv> get_ancestors_di = l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_di get_parent get_host get_disconnected_document \<and> get_ancestors_di_locs = l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_di_locs get_parent_locs get_host_locs get_disconnected_document_locs"
preorder_family_def: fixes dom :: "'c set"   and carrier :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" shows "preorder_family dom carrier R \<equiv> dom \<noteq> {} \<and> (\<forall>i. i \<in> dom \<longrightarrow> Order_Predicates.preorder_on carrier (R i)) \<and> (\<forall>i x y. i \<notin> dom \<longrightarrow> \<not> R i x y)"
set_sort1: fixes sort1 :: "('b list \<Rightarrow> 'b) \<Rightarrow> 'b list list \<Rightarrow> 'b list list"   and f :: "'b list \<Rightarrow> 'b"   and xss :: "'b list list" assumes "Radix_Sort sort1" shows "set (sort1 f xss) = set xss"
Asub: fixes j :: "nat"   and i :: "nat" assumes "j \<le> i" shows "A j \<subseteq> A i"
trivial_cartesian: fixes X :: "'c set"   and Y :: "'d set" assumes "trivial X"   and "trivial Y" shows "trivial (X \<times> Y)"
red_dfs_correct: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and reds :: "'a set"   and onstack :: "'a set"   and u0 :: "'a" assumes "red_dfs_inv E U reds onstack"   and "u0 \<in> U"   and "u0 \<notin> reds" shows "red_dfs E onstack reds u0 \<le> SPEC (\<lambda>(reds', cyc). case cyc of None \<Rightarrow> red_dfs_inv E U reds' onstack \<and> u0 \<in> reds' \<and> reds' \<subseteq> reds \<union> E\<^sup>* `` {u0} | Some (p, v) \<Rightarrow> v \<in> onstack \<and> p \<noteq> [] \<and> path E u0 p v)"
dist_at2: fixes vs :: "'a list"   and a :: "'a list"   and r :: "'a"   and b :: "'a list"   and c :: "'a list"   and d :: "'a list" assumes "distinct vs"   and "vs = a @ r # b"   and "vs = c @ r # d" shows "b = d"
specAnno_def: fixes P :: "'g \<Rightarrow> 'd set"   and c :: "'g \<Rightarrow> ('d, 'e, 'f) com"   and Q :: "'g \<Rightarrow> 'd set"   and A :: "'g \<Rightarrow> 'd set" shows "specAnno P c Q A = c undefined"
decisive4: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} y z" shows "decisive swf A Is {j} y x"
scast_down_scast_id: fixes d :: "'b word \<Rightarrow> 'c word"   and w :: "'c word" assumes "is_down d" shows "scast (scast w) = w"
RN_deriv_nn_integral: fixes N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "absolutely_continuous M N"   and "sets N = sets M"   and "f \<in> borel_measurable M" shows "integral\<^sup>N N f = \<integral>\<^sup>+ x. RN_deriv M N x * f x \<partial>M"
r_def: shows "r = (\<lambda>x. \<MM> x - ln x - c)"
finite_real_of_nat_less_real: fixes u :: "real" shows "finite {n. real n < u}"
mk_alt_consistency_closed: fixes C :: "('a, 'b) form set set" assumes "subset_closed C" shows "subset_closed (mk_alt_consistency C)"
uint32_msb_test_bit: fixes x :: "uint32" shows "msb x = bit x 31"
DOM_dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "monoidal_language.Dom C (free_monoidal_category.rep C (partial_magma.dom (free_monoidal_category.comp C) f)) = monoidal_language.Dom C (free_monoidal_category.rep C f)"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and list :: "'b list" shows "(case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (if list = [] then f1 else f2 (hd list) (tl list))"
quality_increasesI: fixes \<xi> :: "state"   and \<xi>' :: "state" assumes "\<And>dip. dip \<in> kD (rt \<xi>) \<Longrightarrow> dip \<in> kD (rt \<xi>')"   and "\<And>dip. \<lbrakk>dip \<in> kD (rt \<xi>); dip \<in> kD (rt \<xi>')\<rbrakk> \<Longrightarrow> rt \<xi> \<sqsubseteq>\<^bsub>dip\<^esub> rt \<xi>'"   and "\<And>dip. sqn (rt \<xi>) dip \<le> sqn (rt \<xi>') dip" shows "quality_increases \<xi> \<xi>'"
R_id: shows "R \<inter> {u. u \<turnstile> g} = R"
strict_order_strict: fixes q :: "('a \<times> 'a) set" assumes "strict_order q" shows "(\<lambda>x y. (x, y) \<in> q\<^sup>= \<and> (y, x) \<notin> q\<^sup>=) = (\<lambda>a b. (a, b) \<in> q)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "\<And>xs. \<lbrakk>\<not> length xs \<le> 1 \<Longrightarrow> P (take (length xs div 2) xs); \<not> length xs \<le> 1 \<Longrightarrow> P (drop (length xs div 2) xs)\<rbrakk> \<Longrightarrow> P xs" shows "P a0"
exists_in_alphabet: fixes t :: "'b tree" shows "\<exists>a. a \<in> alphabet t"
density_states_def: shows "density_states = {\<rho> \<in> carrier_mat d d. partial_density_operator \<rho>}"
leadsTo_UN_distrib: fixes F :: "'c program"   and A :: "'d \<Rightarrow> 'c set"   and I :: "'d set"   and B :: "'c set" shows "(F \<in> \<Union> (A ` I) \<longmapsto> B) = (\<forall>i\<in>I. F \<in> A i \<longmapsto> B)"
less_supI1: fixes x :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "x \<subset># a" shows "x \<subset># a \<union># b"
embed_bool_bounded: fixes P :: "'c \<Rightarrow> bool" shows "Expectations.bounded \<guillemotleft> P \<guillemotright>"
inf2D2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "inf A B x y" shows "B x y"
norm_equivalence: fixes A :: "((real, 'b) vec, 'c) vec"   and x :: "(real, 'b) vec" shows "(Finite_Cartesian_Product.transpose A *v (A *v x) = 0) = (A *v x = 0)"
CardinalityPrognosisShape_axioms: shows "CardinalityPrognosisShape prognosis"
generate_ring_is_subring: fixes H :: "'a set" assumes "H \<subseteq> carrier R" shows "subring (generate_ring R H) R"
Mem_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> t IN u IFF t' IN u'"
Suc_3: shows "(\<sigma>', coll') \<in> csmall_nstep P \<sigma> (Suc n1)"
Nil_notin_lex: fixes ys :: "'g list"   and r :: "('g \<times> 'g) set" shows "([], ys) \<notin> lex r"
rep_eq: fixes x :: "complex"   and xa :: "'a ccsubspace" shows "space_as_set (x *\<^sub>C xa) = (*\<^sub>C) x ` space_as_set xa"
Cod_f': shows "\<BB>\<lparr>Cod\<rparr>\<lparr>f'\<rparr> = b'"
sdom_def: fixes \<rho> :: "(nat \<times> exp) list" shows "sdom \<rho> \<equiv> {x. \<exists>v. lookup \<rho> x = Some v \<and> v \<noteq> EVar x}"
pow_mono: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'d"   and n :: "nat" assumes "SN_Orders.class.ordered_semiring_1 one times plus zero less_eq"   and "less_eq b a"   and "less_eq zero b" shows "less_eq (power.power one times b n) (power.power one times a n) \<and> less_eq zero (power.power one times b n)"
subst_dnf'_iff: fixes x :: "'b ltln list"   and A :: "'b ltln list set"   and m :: "'b ltln \<Rightarrow> 'b ltln option" shows "(x \<in> subst_dnf' A m) = (\<exists>\<Phi>\<in>A. x \<in> subst_clause' \<Phi> m)"
emeasure_def: fixes M :: "'c measure" shows "emeasure M = snd (snd (Rep_measure M))"
mat_of_vec_mult_one_dim_vec: fixes a :: "'c vec"   and d :: "nat"   and c :: "'c" assumes "a \<in> carrier_vec d" shows "mat d 1 (\<lambda>(i, j). a $ i) *\<^sub>v vec 1 (\<lambda>k. c) = c \<cdot>\<^sub>v a"
assms: shows "h \<turnstile> get_ancestors_di child \<rightarrow>\<^sub>r ancestors"
extremum_unique: fixes a :: "'a" shows "(a \<le> bot) = (a = bot)"
measurement_def: fixes d :: "nat"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat" shows "measurement d n M = ((\<forall>j<n. M j \<in> carrier_mat d d) \<and> matrix_sum d (\<lambda>j. adjoint (M j) * M j) n = 1\<^sub>m d)"
snd_sup: fixes x :: "'b \<times> 'a"   and y :: "'b \<times> 'a" shows "snd (x \<squnion> y) = snd x \<squnion> snd y"
mono_Chains: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "r \<subseteq> s" shows "Chains r \<subseteq> Chains s"
resr_comp2: fixes y :: "'a"   and x :: "'a" shows "y \<rightarrow> y \<cdot> (y \<rightarrow> x) = y \<rightarrow> x"
deriv_Polygamma: fixes z :: "'a"   and m :: "nat" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "deriv (Polygamma m) z = Polygamma (Suc m) z"
min_absorb1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "min x y = x"
assoc_naturality: fixes \<mu> :: "'d"   and \<nu> :: "'d"   and \<tau> :: "'d" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "D.arr \<tau>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<tau>" shows "\<a>\<^sub>D[D.cod \<mu>, D.cod \<nu>, D.cod \<tau>] \<cdot>\<^sub>D ((\<mu> \<star>\<^sub>D \<nu>) \<star>\<^sub>D \<tau>) = (\<mu> \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>) \<cdot>\<^sub>D \<a>\<^sub>D[D.dom \<mu>, D.dom \<nu>, D.dom \<tau>]"
simps_1: fixes edge :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and capacity :: "'a \<times> 'a \<Rightarrow> ennreal"   and source :: "'a"   and sink :: "'a"   and more :: "'b" shows "capacity \<lparr>edge = edge, capacity = capacity, source = source, sink = sink, \<dots> = more\<rparr> = capacity"
curry_uncurry_id: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "curry (uncurry f) = f"
scaleR_simps_2: fixes a :: "real"   and x :: "'a" shows "- a *\<^sub>R x = - (a *\<^sub>R x)"
continuous_imp_integrable: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on (cbox a b) f" shows "integrable (lebesgue_on (cbox a b)) f"
map_def: shows "Mapping.map \<equiv> map_fun id (map_fun id (map_fun mapping.rep mapping.Mapping)) (\<lambda>f g m. map_option g \<circ> m \<circ> f)"
scaleR_right_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "real"   and b :: "real"   and x :: "'e" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "a \<le> b"   and "less_eq zero x" shows "less_eq (scaleR a x) (scaleR b x)"
extremum_uniqueI: fixes a :: "'g" assumes "a \<le> bot" shows "a = bot"