hcju/isabelleps huggingface dataset:
s_Prj1: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Prj1 e\<ggreater> m\<rightarrow> \<lless>\<pi>', Prj1 e'\<ggreater>"
simps_18: fixes vb :: "('a, 'b, 'c, 'd) Tree"   and v :: "'d"   and va :: "'b" shows "(tNot vb =\<^sub>\<alpha> tPred v va) = False"
monoD: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<le> f y"
convex_halfspace_ge: fixes b :: "real"   and a :: "'a" shows "convex {x. b \<le> a \<bullet> x}"
unreach_on_path: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and Q :: "'b set"   and b :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "a \<in> MinkowskiPrimitive.unreachable_subset \<E> \<P> Q b" shows "a \<in> Q"
simps_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" shows "efficient_funpow f y x 0 = y"
eq_lin: fixes p :: "tm" assumes "tmbound0 p" shows "islin (eq p)"
set_cases: fixes e :: "'a"   and a :: "'a ns_constraint" assumes "e \<in> set_ns_constraint a" obtains z1 :: "linear_poly" where "a = LEQ_ns z1__ e"   | z1 :: "linear_poly" where "a = GEQ_ns z1 e"
binop_f: fixes t :: "t"   and \<C> :: "t_context"   and uw :: "binop_f" assumes "is_float_t t" shows "\<C> \<turnstile> [Binop_f t uw] : [t, t] _> [t]"
count_eq_zero_iff: fixes M :: "'b multiset"   and x :: "'b" shows "(count M x = 0) = (x \<notin># M)"
vec_component: fixes x :: "'a"   and i :: "'b" shows "vec x $ i = x"
mem_simps_3: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
thetaParZOT_ZOretrT: shows "thetaParZOT \<subseteq> ZOretrT (thetaParZOT \<union> ZObisT)"
ff: fixes n :: "nat" shows "(flow0 x \<circ> r) n = ((\<lambda>x. flow0 x t) \<circ> (flow0 x \<circ> s)) n"
signed_take_bit_int_eq_self_iff: fixes n :: "nat"   and k :: "int" shows "(signed_take_bit n k = k) = (- (2 ^ n) \<le> k \<and> k < 2 ^ n)"
simps_3: fixes uv :: "'c match_expr"   and rs :: "'c rule list" shows "rm_LogEmpty (Rule uv Log # rs) = rm_LogEmpty rs"
compose_first_None: fixes xs :: "('e \<times> 'd) list"   and k :: "'e"   and ys :: "('d \<times> 'f) list" assumes "map_of xs k = None" shows "map_of (compose xs ys) k = None"
arr_to_hfun_hfun_to_arr: fixes B :: "hf"   and C :: "hf"   and F :: "hf" assumes "hfun B C F" shows "arr_to_hfun (hfun_to_arr B C F) = F"
distrib_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
inv_left_eq_iff: fixes f :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "bijection f" shows "(inv f a = b) = (f b = a)"
P: shows "P = (\<lambda>l s. \<exists>k. 1 + s ''x'' = 2 ^ k)"
decr_mult_lemma: fixes d :: "int"   and P :: "int \<Rightarrow> bool"   and k :: "int" assumes "0 < d"   and "\<forall>x. P x \<longrightarrow> P (x - d)"   and "0 \<le> k" shows "\<forall>x. P x \<longrightarrow> P (x - k * d)"
mono_PE_base_lf_rep: shows "mono PE_base_lf_rep"
mult_not_zero: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" assumes "a \<odot> b \<noteq> mbot" shows "a \<noteq> mbot \<and> b \<noteq> mbot"
hyp: fixes k :: "nat"   and cs :: "('b, 'a, 'c) com list" assumes "f k = (cs, css, Stuck)" shows "False"
mono_neutral_right: fixes T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'a)" shows "prod g T = prod g S"
wellformed_eq: fixes p :: "('b, 'c) Combinators list" shows "DenyAll \<in> set p \<longrightarrow> wellformed_policy1 p = wellformed_policy1_strong p"
simps_9: fixes n :: "nat"   and p :: "fm"   and q :: "fm" shows "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
prod_pOp_func: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_pOp I A \<in> carr_prodag I A \<rightarrow> carr_prodag I A \<rightarrow> carr_prodag I A"
double_add_le_zero_iff_single_add_le_zero: fixes a :: "'a" shows "(a + a \<le> (0::'a)) = (a \<le> (0::'a))"
padic_one_def: fixes p :: "int" shows "padic_one p \<equiv> \<lambda>n. if n = 0 then 0 else 1"
isInAlternativeLista: fixes aa :: "'b"   and a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "aa \<in> set (net_list_aux [a])" shows "aa \<in> set (net_list_aux (a # p))"
the_eq_trivial: fixes a :: "'d" shows "(THE x. x = a) = a"
cong_add_rcancel: fixes x :: "'a"   and a :: "'a"   and y :: "'a"   and n :: "'a" shows "[x + a = y + a] (mod n) = [x = y] (mod n)"
restrict_map_to_empty: fixes m :: "'c \<Rightarrow> 'd option" shows "m |` {} = Map.empty"
quot_set_image_times: fixes x :: "'a"   and A :: "'a set" assumes "inj ((*) x)" shows "(*) x ` A \<div> x = A"
simE_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
approx: fixes xs :: "real list"   and vs :: "float interval option list"   and prec :: "nat"   and arith :: "floatarith"   and ivl :: "float interval" assumes "bounded_by xs vs"   and "approx prec arith vs = Some ivl" shows "interpret_floatarith arith xs \<in>\<^sub>r ivl"
hom_induced_compose': fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and Z :: "'d topology"   and g :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and p :: "int"   and x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "continuous_map Y Z g"   and "g ` T \<subseteq> U" shows "hom_induced p Y T Z U g (hom_induced p X S Y T f x) = hom_induced p X S Z U (g \<circ> f) x"
vsv_ex1: fixes r :: "V"   and a :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "\<exists>!b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
\<tau>rtrancl3p_step: fixes s :: "'s2"   and tl :: "'tl2"   and s' :: "'s2"   and tls :: "'tl2 list"   and s'' :: "'s2" assumes "s -2-tl\<rightarrow> s'"   and "\<not> \<tau>move2 s tl s'"   and "s' -\<tau>2-tls\<rightarrow>* s''" shows "s -\<tau>2-tl # tls\<rightarrow>* s''"
assms: fixes m :: "'a" assumes "invar m" shows "map_iterator_rev_linord (ritoi m) (\<alpha> m)"
assms_1: shows "{} \<turnstile>\<^sub>c t"
card_of_ordIso_czero_iff_empty: fixes A :: "'c set" shows "(|A| =o czero) = (A = {})"
natural_filtration: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure" assumes "infinite_cts_filtration p M F" shows "F = infinite_coin_toss_space.nat_filtration M"
add_le_impl: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a + b < a + c" shows "b < c"
connected_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and v :: "nat" shows "Graph.connected c u v \<equiv> \<exists>p. Graph.isPath c u p v"
power_down_even_nonneg: fixes n :: "nat"   and p :: "nat"   and x :: "real" assumes "even n" shows "0 \<le> power_down p x n"
In0_def: fixes M :: "('b, 'c) Old_Datatype.node set" shows "In0 M \<equiv> Scons (Numb 0) M"
test_iteration_annir: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "!p \<cdot> (p \<cdot> x)\<^sup>\<infinity> = !p"
nsp: fixes cd :: "jvm_method cdecl" assumes "cd \<in> set P" shows "(fst cd, Start) \<notin> (subcls1 P)\<^sup>*"
eq: shows "(\<lambda>x. finmap_of (J x) (f x)) -` Pi' K S \<inter> space N = (if \<exists>x\<in>space N. K = J x \<and> finite K then if K = {} then {x \<in> space N. J x = K} else \<Inter>i\<in>K. (\<lambda>x. f x i) -` S i \<inter> {x \<in> space N. J x = K} else {})"
Sup_image_eadd1: fixes Y :: "enat set"   and x :: "enat" assumes "Y \<noteq> {}" shows "(\<Squnion>y\<in>Y. y + x) = \<Squnion> Y + x"
span_alt: fixes B :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "phull B = {\<Sum>x | f x \<noteq> (0::'b). f x \<cdot> x |f. {x. f x \<noteq> (0::'b)} \<subseteq> B \<and> finite {x. f x \<noteq> (0::'b)}}"
elims: fixes x :: "('f, 'b) Term.term"   and xa :: "('f, 'b) Term.term"   and y :: "bool \<times> bool" assumes "kbo x xa = y" obtains s :: "('f, 'b) Term.term" and t :: "('f, 'b) Term.term" where "x = s" and "xa = t" and "y = (if vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> weight t \<le> weight s then if weight t < weight s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict (f, length ss) (g, length ts) then (True, True) else if pr_weak (f, length ss) (g, length ts) then lex_ext_unbounded kbo ss ts else (False, False) else (False, False))"
negligible_le: fixes g :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real" assumes "negligible g"   and "\<And>\<eta>. \<bar>f \<eta>\<bar> \<le> g \<eta>" shows "negligible f"
eq_min_height_0: fixes kdt :: "'b kdt" shows "(min_height kdt = 0) = (\<exists>p. kdt = Leaf p)"
set_sel_1: fixes a :: "'b list" assumes "a \<noteq> []" shows "list.hd a \<in> set a"
trcless: fixes n :: "nat" shows "trace (complex_of_real c \<cdot>\<^sub>m matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>) n) \<le> 1"
add_0_left: fixes a :: "'ag" assumes "a \<in> U" shows "0\<^sub>o\<^sub>w +\<^sub>o\<^sub>w a = a"
circ_decompose_9: shows "top = top \<sqinter> top \<sqinter> top"
landau_product_preprocess_10: fixes m :: "num" shows "num.Bit1 m + num.One = num.Bit0 (m + num.One)"
hyps: fixes A :: "'a set" assumes "finite A"   and "number_partition (card A) N" shows "\<exists>P. partition_on A P \<and> image_mset card (mset_set P) = N"
mk_release_atoms: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "atoms_ltln (mk_release \<phi> \<psi>) \<subseteq> atoms_ltln \<phi> \<union> atoms_ltln \<psi>"
nb: shows "bound0 (list_disj [minusinf (simpfm p), plusinf (simpfm p), simpfm (subst0 (CP 0\<^sub>p) (simpfm p)), evaldjf (\<lambda>(c, t). msubst2 (simpfm p) (c *\<^sub>p (- 2)\<^sub>p) t) (remdups (uset (simpfm p))), evaldjf (\<lambda>((b, a), d, c). msubst2 (simpfm p) ((- 2)\<^sub>p *\<^sub>p b *\<^sub>p d) (tm.Add (tm.Mul d a) (tm.Mul b c))) (alluopairs (remdups (uset (simpfm p))))])"
liminf_shift: fixes u :: "nat \<Rightarrow> 'b" shows "liminf (\<lambda>n. u (n + 1)) = liminf u"
transitive_def: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" shows "transitive x = (\<forall>y z u. x y z \<longrightarrow> x z u \<longrightarrow> x y u)"
fsup: fixes x :: "char list" shows "finite (support (\<lambda>l. wpw b c (l x) Q l))"
l_remove_child_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_remove_child_axioms type_wf known_ptrs remove_child remove_child_locs get_owner_document get_child_nodes get_disconnected_nodes \<equiv> ((\<forall>object_ptr h child h'. writes (remove_child_locs object_ptr |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r) (remove_child object_ptr child) h h') \<and> (\<forall>w ptr owner_document h h'. w \<in> remove_child_locs ptr owner_document \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> object_ptr_kinds h = object_ptr_kinds h') \<and> (\<forall>w ptr owner_document h h'. w \<in> remove_child_locs ptr owner_document \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h = type_wf h')) \<and> ((\<forall>h ptr child h' owner_document disc_nodes. known_ptrs h \<longrightarrow> h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h' \<longrightarrow> h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<rightarrow>\<^sub>r owner_document \<longrightarrow> h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes \<longrightarrow> child \<in> set disc_nodes) \<and> (\<forall>h ptr child. h \<turnstile> ok remove_child ptr child \<longrightarrow> ptr |\<in>| object_ptr_kinds h)) \<and> (\<forall>h ptr' child h'. h \<turnstile> remove_child ptr' child \<rightarrow>\<^sub>h h' \<longrightarrow> child |\<in>| node_ptr_kinds h) \<and> (\<forall>h parent child h' ptr children children'. known_ptrs h \<longrightarrow> type_wf h \<longrightarrow> h \<turnstile> remove_child parent child \<rightarrow>\<^sub>h h' \<longrightarrow> h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<longrightarrow> h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children' \<longrightarrow> set children' \<subseteq> set children)"
outside_union_restrict: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "P = P outside X \<union> P || X"
distinct_adj_mapD: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "distinct_adj (map f xs)" shows "distinct_adj xs"
rep_eq: fixes x :: "'a interval" shows "upper x = snd (bounds_of_interval x)"
rec_nat_0_imp: fixes f :: "nat \<Rightarrow> 'a"   and f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "f = rec_nat f1 f2" shows "f 0 = f1"
Un_closed: fixes T :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "topology T"   and "x closed\<^bsub>T\<^esub>"   and "y closed\<^bsub>T\<^esub>" shows "x \<union> y closed\<^bsub>T\<^esub>"
topological_space_quotient: fixes f :: "'d \<Rightarrow> 'c" assumes "surj f" shows "class.topological_space (openin (map_topology f euclidean))"
filter_for_biflows_def: fixes E :: "('b \<times> 'b) list" shows "filter_for_biflows E \<equiv> filter (\<lambda>e. (snd e, fst e) \<in> set E) E"
prems: shows "rsquarefree ([:a, - (1::'a):] * p)"
abc_comp_frist_steps_halt_eq': fixes lm :: "nat list"   and A :: "abc_inst list"   and n :: "nat"   and lm' :: "nat list"   and B :: "abc_inst list" assumes "abc_steps_l (0, lm) A n = (length A, lm')"   and "A \<noteq> []" shows "\<exists>n'. abc_steps_l (0, lm) (A [+] B) n' = (length A, lm')"
sub_designII: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and \<U> :: "'b set"   and \<A> :: "'b set multiset" assumes "design point_set block_collection"   and "design \<U> \<A>"   and "sub_incidence_system point_set block_collection \<U> \<A>" shows "sub_design point_set block_collection \<U> \<A>"
vsubset_vinsert: fixes A :: "V"   and x :: "V"   and B :: "V" shows "(A \<subseteq>\<^sub>\<circ> vinsert x B) = (if x \<in>\<^sub>\<circ> A \<Rightarrow> A -\<^sub>\<circ> set {x} \<subseteq>\<^sub>\<circ> B | otherwise \<Rightarrow> A \<subseteq>\<^sub>\<circ> B)"
assms_5: fixes zs :: "('a \<times> 'b) list \<times> 'o" assumes "oalist_inv zs" shows "g zs = map_val_raw (\<lambda>k v. f k v (0::'c)) zs"
zeta_3_linear_combination_bounds: obtains A :: "int" and B :: "int" where "0 < A" and "real_of_int A * Re (zeta 3) + real_of_int B \<in> {0<..2 * Re (zeta 3) * real (Lcm {1..n} ^ 3) / 27 ^ n}"
zero_lowner_le_positiveD: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "0\<^sub>m n n \<le>\<^sub>L A" shows "positive A"
simps_3: fixes x :: "nat"   and e :: "cexpr"   and e1 :: "cexpr"   and e2 :: "cexpr" shows "cexpr_comp_aux x e <e1, e2>\<^sub>c = <cexpr_comp_aux x e e1, cexpr_comp_aux x e e2>\<^sub>c"
wellformed_items_Gen: fixes P :: "(symbol \<times> character list) list set" shows "wellformed_items (Gen P)"
Inf_aform': fixes p :: "nat"   and X :: "'a \<times> 'a pdevs" shows "Inf_aform' p X \<le> Inf_aform X"
cut4: fixes A :: "fm"   and B :: "fm"   and C :: "fm"   and D :: "fm"   and E :: "fm"   and H :: "fm set" assumes "{A, B, C, D} \<turnstile> E"   and "H \<turnstile> A"   and "H \<turnstile> B"   and "H \<turnstile> C"   and "H \<turnstile> D" shows "H \<turnstile> E"
Select0_def: fixes d :: "d data" shows "Select0 d = Sel0 (d !P! 0)"
intros: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and x :: "nat" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "length as = x"   and "\<forall>as'. n -as'\<rightarrow>\<^sub>\<iota>* n' \<longrightarrow> x \<le> length as'" shows "distance n n' x"
f: fixes i :: "'a" assumes "i \<in> I" shows "0 \<le> f i"
rep_eq: fixes x :: "uint16" shows "Rep_uint32 (uint32_of_uint16 x) = ucast (Rep_uint16 x)"
that: fixes u :: "int"   and v :: "int" assumes "coprime u v"   and "\<not> (odd u \<and> odd v)"   and "x = 2 * u * v \<and> y = u\<^sup>2 - v\<^sup>2 \<or> x = u\<^sup>2 - v\<^sup>2 \<and> y = 2 * u * v" shows "thesis"
euler_char_def: shows "euler_char \<equiv> int (card (verts (with_proj (list_digraph G)))) - int (card (arcs (with_proj (list_digraph G))) div 2) + int (card face_cycle_sets)"
weakStatImpParPres: fixes R :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and xvec :: "name list"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>A\<^sub>R \<Psi>\<^sub>R. \<lbrakk>extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>; A\<^sub>R \<sharp>* \<Psi>; A\<^sub>R \<sharp>* P; A\<^sub>R \<sharp>* Q\<rbrakk> \<Longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<lessapprox><Rel> Q"   and "xvec \<sharp>* \<Psi>"   and "eqvt Rel"   and "\<And>\<Psi>' S T A\<^sub>U \<Psi>\<^sub>U U. \<lbrakk>(\<Psi>' \<otimes> \<Psi>\<^sub>U, S, T) \<in> Rel; extractFrame U = \<langle>A\<^sub>U, \<Psi>\<^sub>U\<rangle>; A\<^sub>U \<sharp>* \<Psi>'; A\<^sub>U \<sharp>* S; A\<^sub>U \<sharp>* T\<rbrakk> \<Longrightarrow> (\<Psi>', S \<parallel> U, T \<parallel> U) \<in> Rel'"   and "\<And>\<Psi>' S T yvec. \<lbrakk>(\<Psi>', S, T) \<in> Rel'; yvec \<sharp>* \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>*yvec\<rparr>S, \<lparr>\<nu>*yvec\<rparr>T) \<in> Rel'"   and "\<And>\<Psi>' S T \<Psi>''. \<lbrakk>(\<Psi>', S, T) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', S, T) \<in> Rel" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<parallel> R \<lessapprox><Rel'> \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R"
normalize_poly_p_poly_of_mset: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset" assumes "normalize_poly_p p q" shows "polynomial_of_mset p = polynomial_of_mset q"
finfun_apply_def: shows "finfun_apply = (\<lambda>f a. finfun_rec (\<lambda>b. b) (\<lambda>a'. If (a = a')) f)"
asset_iv_state: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set"   and C :: "nat"   and B :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset n, Token n (Auth_PriK n) B C SK) \<in> s" shows "\<exists>A D. fst (snd SK) = {A, B} \<and> snd (snd SK) = {C, D} \<and> (Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s \<and> (Asset n, \<lbrace>Num 4, PubKey D\<rbrace>) \<in> s \<and> Crypt (SesK SK) (PubKey D) \<in> used s \<and> (Asset n, PubKey B) \<in> s"
simps_1: fixes x0 :: "'b"   and y0 :: "'b" assumes "complete_space_with_middle TYPE('b)" shows "complete_space_with_middle.geod x0 y0 0 0 = x0"
monad_writer_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and tell :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_writer return bind tell" shows "monad_writer (return_env return) (bind_env bind) (tell_env tell)"
cases: fixes a1 :: "('a, 'b, 'c, 'd) Tree"   and a2 :: "('a, 'b, 'c, 'd) Tree" assumes "(a1, a2) \<in> Tree_wf" obtains t :: "('a, 'b, 'c, 'd) Tree" and tset :: "('a, 'b, 'c, 'd) Tree set['a]" where "a1 = t__" and "a2 = tConj tset" and "t__ \<in> set_bset tset"   | t :: "('a, 'b, 'c, 'd) Tree" where "a1 = ta__" and "a2 = tNot ta__"   | t :: "('a, 'b, 'c, 'd) Tree" and f :: "'d" and \<alpha> :: "'c" where "a1 = t" and "a2 = tAct f \<alpha> t"
trss_env_increasing: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" shows "c \<subseteq># c'"
sat_eq_reduc_sat: fixes N :: "'a set" shows "saturated N = reduc_saturated N"
that: fixes x :: "'c" assumes "x \<in> X" shows "set_of (f x) \<subseteq> set_of (g x)"
sum_ideals_cont: fixes I :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "ideal R I"   and "A \<subseteq> I"   and "B \<subseteq> I" shows "A \<minusplus> B \<subseteq> I"
min_single_2: fixes x :: "('c, 'd) bintree option" assumes "Binomial_Queue.min [x] = None" shows "x = None"
infinite_solution_set_imp_consistent: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "infinite (solution_set A b)" shows "consistent A b"
r_maxletr_def: shows "r_maxletr \<equiv> Pr 1 Z (Cn 3 r_ifle [r_dummy 2 (Cn 1 r_triangle [S]), recf.Id 3 2, Cn 3 S [recf.Id 3 0], recf.Id 3 1])"
canI_unitor_0_2: fixes f :: "'a" assumes "ide f" shows "\<r>[f] = can \<^bold>\<langle>f\<^bold>\<rangle> (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>src f\<^bold>\<rangle>\<^sub>0)"
max_bot: fixes bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.order_bot bot less_eq less" shows "ord.max less_eq bot x = x"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_5: fixes l :: "'e strand_label"   and t :: "('f, 'g) Term.term"   and s :: "('f, 'g) Term.term"   and A :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, delete\<langle>t,s\<rangle>) # A) = (l, delete\<langle>t,s\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
p_eq_SUP_v: fixes s :: "'s" assumes "s \<in> S" shows "p s = \<Squnion> (v ` cfg_on s)"
bind_prob_def: shows "Monad_Overloading.bind = (\<bind>)"
split_part: fixes P :: "bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<lambda>(a, b). P \<and> Q a b) = (\<lambda>ab. P \<and> (case ab of (x, xa) \<Rightarrow> Q x xa))"
Un_insert_left: fixes a :: "'b"   and B :: "'b set"   and C :: "'b set" shows "Set.insert a B \<union> C = Set.insert a (B \<union> C)"
IH: fixes w :: "nat \<Rightarrow> real" assumes "y < output_size (shallow_model' Z M N)" shows "cprank_max1 (tensors_from_net (insert_weights s (shallow_model' Z M N) w) $ y)"
r_inv_ex: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (homology_group p X)" shows "\<exists>y\<in>carrier (homology_group p X). x \<otimes>\<^bsub>homology_group p X\<^esub> y = \<one>\<^bsub>homology_group p X\<^esub>"
find_shortest_augmenting_spec_cf_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity" shows "find_shortest_augmenting_spec_cf cf \<equiv> assert (RGraph c s t cf) \<bind> (\<lambda>_. spec x. case x of None \<Rightarrow> \<not> Graph.connected cf s t | Some p \<Rightarrow> Graph.isShortestPath cf s p t)"