hcju/isabelleps huggingface dataset:
swapEnv_sym: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "rho &[x \<and> y]_zs = rho &[y \<and> x]_zs"
type_definition_hmultiset: shows "type_definition Hereditary_Multiset.Rep_hmultiset Hereditary_Multiset.Abs_hmultiset {X. no_elem X}"
diff_fun_cong: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'e" assumes "diff_fun k charts f"   and "\<And>x. x \<in> manifold.carrier charts \<Longrightarrow> f x = g x" shows "diff_fun k charts g"
distinct_41: fixes x4 :: "nat"   and x71 :: "msg"   and x72 :: "msg" shows "Pan x4 \<noteq> \<lbrace>x71, x72\<rbrace>"
intros: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> A \<Gamma> \<Longrightarrow> d_OUT f x = weight \<Gamma> x" shows "linkage \<Gamma> f"
up_def: fixes R :: "('a, 'b) ring_scheme" shows "up R = {f \<in> UNIV \<rightarrow> carrier R. \<exists>n. bound \<zero>\<^bsub>R\<^esub> n f}"
dropWhile_drop: fixes f :: "'b \<Rightarrow> bool"   and l :: "'b list" shows "\<exists>x. dropWhile f l = drop x l"
almost_full_on_mono: fixes A :: "'a set"   and B :: "'a set"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x y. Q x y \<Longrightarrow> P x y"   and "almost_full_on Q B" shows "almost_full_on P A"
h7c: shows "derivative x p = sum f {0..MPoly_Type.degree p x - 1}"
refl: fixes a :: "'al" assumes "a \<in> U" shows "a \<le>\<^sub>o\<^sub>w a"
singular_boundary: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_relboundary p X {} c = (\<exists>d. singular_chain (Suc p) X d \<and> chain_boundary (Suc p) d = c)"
const_ms_aux: fixes basis :: "(real \<Rightarrow> real) list"   and c :: "real" assumes "basis_wf basis"   and "length basis = Suc (expansion_level TYPE('a))" shows "is_expansion_aux (const_ms_aux c) (\<lambda>_. c) basis"
of_nat_power_less_of_nat_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "nat"   and b :: "nat"   and w :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less (semiring_1.of_nat one plus zero x) (power.power one times (semiring_1.of_nat one plus zero b) w) = (x < b ^ w)"
last_message_hold_nth: fixes i :: "nat"   and xs :: "'b message_af list" assumes "i < length xs" shows "last_message_hold xs ! i = last_message (xs \<down> Suc i)"
IP: fixes \<sigma> :: "'\<sigma>"   and k :: "'u"   and it :: "'u set" assumes "c \<sigma>"   and "k \<in> it"   and "\<forall>j\<in>it. j \<le> k"   and "\<forall>j\<in>\<alpha> m - it. k \<le> j"   and "it \<subseteq> \<alpha> m"   and "I it \<sigma>" shows "I (it - {k}) (f k \<sigma>)"
Diff_iff: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
distinguishing_formula_distinguishes: fixes P :: "'state"   and Q :: "'state" assumes "\<not> P =\<cdot> Q" shows "distinguishing_formula P Q distinguishes P from Q"
restricted_subformulas_R_\<mu>: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" shows "subformulas\<^sub>\<mu> \<psi> \<subseteq> restricted_subformulas (\<phi> R\<^sub>n \<psi>)"
NotL: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n" shows "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
assms_2: shows "A \<in> obj\<^bsub>C\<^esub>"
S_transform_Tree\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('a, 'c, 'b) Tree\<^sub>\<alpha>" shows "p \<bullet> S_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha> = S_transform_Tree\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
godel_code_append_cons: fixes xs :: "nat list"   and i :: "nat"   and y :: "nat"   and ys :: "nat list" assumes "length xs = i" shows "godel_code' (xs @ y # ys) (Suc 0) = godel_code' xs (Suc 0) * Pi (Suc i) ^ y * godel_code' ys (i + 2)"
simplicial_simplex_oriented_simplex: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and l :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "simplicial_simplex p S (oriented_simplex p l) = ((\<lambda>x i. \<Sum>j\<le>p. l j i * x j) ` standard_simplex p \<subseteq> S)"
symE: fixes r :: "('c \<times> 'c) set"   and b :: "'c"   and a :: "'c" assumes "sym r"   and "(b, a) \<in> r" obtains "(a, b) \<in> r"
MsgCN: shows "msgs cN = (\<lambda>m. if \<exists>p. m = <p, inM True> then 1 else 0)"
skip_r_alpha_eq: shows "II = ($\<^bold>v\<acute> =\<^sub>u $\<^bold>v)"
vrat_inverse_closed: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "x\<inverse>\<^sub>\<rat> \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
quote_all_PfP_ssubst: fixes \<beta> :: "fm"   and V :: "name set" assumes "{} \<turnstile> \<beta>"   and "V \<subseteq> V'"   and "supp \<beta> \<subseteq> atom ` V'" shows "quote_all p' V \<turnstile> PfP (ssubst \<lfloor>\<beta>\<rfloor>V V F')"
a0: fixes m :: "nat" assumes "m < 2 ^ n" shows "length (bin_rep_aux n m) = n + 1"
fun_upd_comp: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'d"   and y :: "'f" shows "f \<circ> g(x := y) = (f \<circ> g)(x := f y)"
eq_imp_eq_mod_nat: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a = b" shows "[a = b] (mod m)"
exi: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "exi x \<phi> \<in> fmla"
q_avail: shows "i1.orig.quorum I\<^sub>2"
transpose_code: fixes A :: "(('c, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "($h) (Finite_Cartesian_Product.transpose A) = transpose_row A"
K_stable: fixes B :: "'a set" assumes "B \<in> K" shows "induced_map A -` B \<in> K"
d_gt_0: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real" assumes "diamond i j d" shows "0 < d"
simps_6: fixes vc :: "'b match_expr"   and vd :: "'b match_expr"   and va :: "action"   and rs :: "'b rule list" shows "rmMatchFalse (Rule (MatchNot (MatchAnd vc vd)) va # rs) = Rule (MatchNot (MatchAnd vc vd)) va # rmMatchFalse rs"
wf_Base_foo: shows "wf_mdecl tprg Base Base_foo"
simps_5: fixes m :: "'a"   and x :: "nat" shows "parent_node (Actual_out (m, x)) = m"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'f list" assumes "i < length xs" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
not_coprimeI: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c a"   and "dvd.dvd times c b"   and "\<not> dvd.dvd times c one" shows "\<not> algebraic_semidom.coprime times one a b"
sel_rews_5: fixes a :: "'b"   and list :: "['b]" shows "head\<cdot>(a : list) = a"
PDQ_diag: shows "(P'', D, Q'') = local.diagonalize A bezout"
abs_leI: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b"   and "- a \<le> b" shows "\<bar>a\<bar> \<le> b"
strict_mono_on_ordertype: fixes M :: "V set" assumes "M \<subseteq> ON"   and "small M" obtains f :: "V \<Rightarrow> V" where "f \<in> elts (tp M) \<rightarrow> M" and "strict_mono_on f (elts (tp M))"
is_OpUbx1_def: fixes unboxed :: "('a, 'b, 'c) unboxed" shows "is_OpUbx1 unboxed = (\<exists>x2. unboxed = OpUbx1 x2)"
Int_empty_right: fixes A :: "'a set" shows "A \<inter> {} = {}"
list_of_dlist_Dlist: fixes xs :: "'d list" shows "list_of_dlist (Dlist xs) = remdups xs"
lessThan_non_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.no_bot less_eq less" shows "ord.lessThan less x \<noteq> {}"
augment_cf_impl_refine: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list"   and x :: "'a" assumes "RGraph c s t cf"   and "Graph.isSimplePath cf s p t" shows "Network.augment_cf_impl c cf p x \<le> return (Graph.augment_cf cf (set p) x)"
is_elsewhere_def: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" shows "is_elsewhere p i branch xs \<equiv> \<exists>w w' ps. (w, w') \<notin> xs \<and> branch !. w = Some (ps, i) \<and> (p = Nom i \<or> ps !. w' = Some p)"
cross_once_notin: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and R :: "'b set"   and a :: "'b"   and b :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "Q \<in> \<P>"   and "R \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "b \<in> R"   and "a \<noteq> b"   and "Q \<noteq> R" shows "a \<notin> R"
causal_Read: shows "causal_program_history (Read volatile a t # is\<^sub>s\<^sub>b) sb"
hyps: shows "mat (rat_poly.row_length (blockmat ls)) (length (blockmat ls)) (blockmat ls)"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "seq (src g) (src f)"
productive_imp_produces_Max_literal: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset" assumes "production N C \<noteq> {}" shows "production N C = {atm_of (Max_mset C)}"
entail_set_all_formulas: fixes N1 :: "'b literal multiset set"   and N2 :: "'b literal multiset set" shows "(N1 \<TTurnstile>e N2) = (\<forall>C\<in>N2. N1 \<TTurnstile>e {C})"
inv_unit_factor_eq_0_iff: fixes a :: "'a" shows "((1::'a) div unit_factor a = (0::'a)) = (a = (0::'a))"
summable_k: fixes x :: "'v"   and y :: "'v" shows "summable (\<lambda>i. (k_i' i (x, y) - k_i' i (y, x)) * indicat_real \<^bold>E (x, y))"
dg_Par_Arr_iff: fixes x :: "V"   and \<alpha> :: "V" shows "(x \<in>\<^sub>\<circ> dg_Par \<alpha>\<lparr>Arr\<rparr>) = arr_Par \<alpha> x"
subset_Image1_Image1_iff: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "Preorder r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(r `` {a} \<subseteq> r `` {b}) = ((b, a) \<in> r)"
graph_0: fixes A :: "('d, 'c) nba"   and w :: "'d stream" shows "graph A w 0 = gunodes A w"
alternating_list_split: fixes m :: "nat"   and n :: "nat"   and s :: "'a"   and t :: "'a" shows "alternating_list (m + n) s t = alternating_list m s t @ (if gcd_nat.greater_eq m 2 then alternating_list n s t else alternating_list n t s)"
components_are_iso: fixes a :: "'d" assumes "D.ide a" shows "D.iso (Adj.\<epsilon> a)"
d_pos: fixes x :: "'a"   and e :: "real" assumes "x \<in> S"   and "0 < e" shows "0 < d x e"
aiN: fixes xs :: "floatarith list"   and p :: "nat"   and zs :: "((real \<times> real pdevs) \<times> real) list"   and ys :: "floatarith list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> approx_floatarith p x zs \<noteq> None"   and "\<And>x. x \<in> set ys \<Longrightarrow> approx_floatarith p x zs \<noteq> None" shows "approx_floatarith p (inner_floatariths xs ys) zs \<noteq> None"
5: shows "(\<box>a)\<box> \<cong> \<box>a"
cases: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c" assumes "Domainp r a" obtains aa :: "'c" and b :: "'d" where "a = aa" and "r aa b"
inject_3: fixes x41 :: "('a, 'b, 'c) com"   and x42 :: "('a, 'b, 'c) com"   and y41 :: "('a, 'b, 'c) com"   and y42 :: "('a, 'b, 'c) com" shows "(Seq x41 x42 = Seq y41 y42) = (x41 = y41 \<and> x42 = y42)"
subset_segment_4: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a<--<b} \<subseteq> {c<--<d}) = (a = b \<or> a \<in> {c--d} \<and> b \<in> {c--d})"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and msg :: "msg" shows "P (case msg of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = (\<not> ((\<exists>x11 x12 x13 x14 x15 x16 x17 x18. msg = Rreq x11 x12 x13 x14 x15 x16 x17 x18 \<and> \<not> P (f1 x11 x12 x13 x14 x15 x16 x17 x18)) \<or> (\<exists>x21 x22 x23 x24 x25. msg = Rrep x21 x22 x23 x24 x25 \<and> \<not> P (f2 x21 x22 x23 x24 x25)) \<or> (\<exists>x31 x32. msg = Rerr x31 x32 \<and> \<not> P (f3 x31 x32)) \<or> (\<exists>x41 x42. msg = Newpkt x41 x42 \<and> \<not> P (f4 x41 x42)) \<or> (\<exists>x51 x52 x53. msg = Pkt x51 x52 x53 \<and> \<not> P (f5 x51 x52 x53))))"
simps_7: fixes \<nu> :: "'a ltln"   and \<nu>' :: "'a ltln" shows "pure_universal (\<nu> W\<^sub>n \<nu>') = (pure_universal \<nu> \<and> pure_universal \<nu>' \<or> \<nu>' = false\<^sub>n)"
SF1: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "|~ P \<and> _actrans N v \<longrightarrow> \<circle>P \<or> \<circle>Q"   and "|~ P \<and> \<langle>N \<and> A\<rangle>_v \<longrightarrow> \<circle>Q"   and "\<turnstile> \<box>P \<and> \<box>[N]_v \<and> \<box>F \<longrightarrow> \<diamond>Enabled \<langle>A\<rangle>_v"   and "|~ P \<and> Unchanged v \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>[N]_v \<and> SF(A)_v \<and> \<box>F \<longrightarrow> P \<leadsto> Q"
rel_witness_converter_def: fixes A :: "'a \<Rightarrow> 'i \<Rightarrow> bool"   and A' :: "'i \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'j \<Rightarrow> bool"   and R' :: "'j \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_witness_converter A A' B C R R' = corec_converter (\<lambda>(conv1, conv2). map_gpv (map_prod id Inr \<circ> rel_witness_prod) id \<circ> rel_witness_gpv (rel_prod B (rel_converter (A OO A') B C (R OO R'))) C R R' \<circ> rel_witness_fun A A' (run_converter conv1, run_converter conv2))"
PRelQ: shows "(\<Psi>, P, Q) \<in> Rel"
assms_2: shows "(\<xi>, p) \<in> reachable A I"
vars\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('e, 'd) stateful_strand_step"   and A :: "('e, 'd) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t (a # A) = vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> vars\<^sub>s\<^sub>s\<^sub>t A"
eq: shows "space (subprob_algebra N) = space (subprob_algebra L)"
prems_2: shows "w \<le>p s \<cdot> w"
subspace_ideal: fixes F :: "('d \<Rightarrow>\<^sub>0 'c) set" shows "phull.subspace (ideal F)"
cfcomp2: fixes f :: "'g \<rightarrow> 'e"   and g :: "'f \<rightarrow> 'g"   and x :: "'f" shows "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
unsigned_of_nat: fixes n :: "nat" shows "unsigned (word_of_nat n) = semiring_1_class.of_nat (take_bit LENGTH('l) n)"
diagonalize_axioms: shows "Diagonalize.diagonalize diagonalize_JNF"
gcd_poly_code_def: fixes p :: "'a poly"   and q :: "'a poly" shows "gcd_poly_code p q = (if p = 0 then normalize q else if q = 0 then normalize p else Polynomial.smult (gcd (content p) (content q)) (gcd_poly_code_aux (primitive_part p) (primitive_part q)))"
indicator_UN_disjoint: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c set"   and x :: "'c" assumes "finite A"   and "disjoint_family_on f A" shows "indicator (\<Union> (f ` A)) x = (\<Sum>y\<in>A. indicator (f y) x)"
emeasure_notin_sets: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<notin> sets M" shows "emeasure M A = 0"
Red_preserves_defass: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "\<D> e \<lfloor>dom (lcl s)\<rfloor>" shows "\<D> e' \<lfloor>dom (lcl s')\<rfloor>"
powr_less_mono2: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "0 < a"   and "0 \<le> x"   and "x < y" shows "x powr a < y powr a"
ltl_prop_unfold_equiv_gt_and_lt_3: shows "(\<sim>\<^sub>Q) \<le> (\<sim>\<^sub>L)"
closed_Collect_conj: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "class.topological_space open"   and "topological_space.closed open {x. P x}"   and "topological_space.closed open {x. Q x}" shows "topological_space.closed open {x. P x \<and> Q x}"
distinct_1: fixes x2 :: "'d" shows "Marker \<noteq> Msg x2"
that: fixes \<sigma>' :: "nat \<Rightarrow> 'g"   and \<zeta>' :: "'l net_state" assumes "((\<sigma>, \<zeta>), connect(i, i'), \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n)"   and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>' j = \<sigma> j"   and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')" shows "thesis"
abc_steps_l_0: fixes ac :: "nat \<times> nat list"   and ap :: "abc_inst list" shows "abc_steps_l ac ap 0 = ac"
Int_space_eq2: fixes x :: "'a set"   and M :: "'a measure" assumes "x \<in> sets M" shows "x \<inter> space M = x"
semi_complete_onI: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "WN_on r A"   and "CR_on r A" shows "semi_complete_on r A"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "unit_preserving_hom hom"   and "irreducibility_hom hom" shows "factor_preserving_hom hom"
valid_history_nth_update: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'b"   and sb :: "'b memref list"   and "is" :: "instr list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "valid_history program_step ts"   and "i < length ts"   and "program.history_consistent program_step \<theta> (hd_prog p sb) sb" shows "valid_history program_step (ts[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
that: fixes n :: "nat" assumes "\<And>m f. \<lbrakk>n \<le> m; f \<in> Poly_Mapping.keys ((singular_subdivision p ^^ m) c)\<rbrakk> \<Longrightarrow> \<exists>V\<in>\<C>. f ` standard_simplex p \<subseteq> V" shows "thesis"
p_circ_pt_def: shows "p_circ_pt \<equiv> linepath (p + complex_of_real (h p)) (p + complex_of_real (h p))"
assms_2: fixes i :: "'b"   and j :: "'c"   and k :: "'d"   and l :: "'e" assumes "(i, j, k, l) \<in> A" shows "P i j k l is H"
wffp_append_iff: fixes sl :: "('St, 'U, 'C) state list"   and sl1 :: "('St, 'U, 'C) state list" shows "wffp (sl @ sl1) = (wffp sl \<and> sl1 = [] \<or> sl = [] \<and> wffp sl1 \<or> wffp sl \<and> wffp sl1 \<and> hd sl1 \<in> \<delta> (last sl))"
lr_of_tran_s3_def: fixes ifs :: "char list list"   and ard :: "('b \<times> 32 simple_match \<times> char list \<times> simple_action) list" shows "lr_of_tran_s3 ifs ard = concat (map (\<lambda>(p, r, c, a). map (\<lambda>b. (p, b, case a of Accept \<Rightarrow> [Forward c] | Drop \<Rightarrow> [])) (simple_match_to_of_match r ifs)) ard)"
simps_5: fixes b :: "bexp"   and c :: "com" shows "D (WHILE b DO c) = lfp (W (bval b) (D c))"
les: fixes X :: "(real \<times> real \<times> real) list" shows "(\<exists>(a, b, c)\<in>set X. les_fun a b c eq les leq neq) = (\<exists>x\<in>F\<epsilon> ` \<Union> (roots ` set X). x)"
arr_inv: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C"   and "category.iso C f" shows "partial_magma.arr C (category.inv C f)"
update_def: fixes x :: "'c"   and y :: "'d"   and t :: "('c \<times> 'd) bro" shows "update x y t = tree (upd x y t)"
continuous_on_Lambert_W: shows "continuous_on {- exp (- 1)..} Lambert_W"
sint_uint: fixes w :: "'b word" shows "sint w = signed_take_bit (LENGTH('b) - Suc 0) (uint w)"
converged_cd_diverge_return: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l :: "nat"   and m' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> 0 = \<pi>' 0"   and "\<nexists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"   and "\<pi>' m' = local.return" obtains k :: "nat" and k' :: "nat" where "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'" and "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" and "\<pi> (Suc k) \<noteq> \<pi>' (Suc k')"
psiFreshVec_1: fixes xvec :: "name list"   and M :: "'d"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* M\<langle>N\<rangle>.P = (xvec \<sharp>* M \<and> xvec \<sharp>* N \<and> xvec \<sharp>* P)"
lb_pi_def: fixes prec :: "nat" shows "lb_pi prec = (let A = lapprox_rat prec 1 5; B = rapprox_rat prec 1 239 in Float 1 2 * float_plus_down prec (Float 1 2 * float_round_down prec (A * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (A * A)))) (- float_round_up prec (B * ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (float_round_down (Suc prec) (B * B)))))"
Min_le_iff: fixes A :: "'c Word.word set"   and x :: "'c Word.word" assumes "finite A"   and "A \<noteq> {}" shows "(signed.Min A \<le>s x) = (\<exists>a\<in>A. a \<le>s x)"
unitary_bound: fixes P :: "'b \<Rightarrow> real" assumes "unitary P" shows "bounded_by 1 P"
Append_def: fixes Z :: "(symbol \<times> character list) set"   and k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "Append Z k P = P \<union> {p @ [t] |p t. p \<in> by_length k P \<and> t \<in> Z \<and> admissible (p @ [t])}"
PO_m1_obs_secrecy: shows "oreach m1 \<subseteq> m1_secrecy"
equiv_F_grounding: fixes q :: "'a literal multiset set"   and C1 :: "'a literal multiset"   and C2 :: "'a literal multiset" assumes "q \<in> UNIV"   and "C1 \<doteq> C2" shows "\<G>_F C1 \<subseteq> \<G>_F C2"
test_huntington2: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "!(p + q) + !(p + !q) = !p"
dismantle_cases': fixes t :: "trm"   and k :: "stack"   and r :: "trm" assumes "t \<star> k \<mapsto> r"   and "NEUT t" obtains t' :: "trm" where "t \<mapsto> t'" and "r = t' \<star> k"   | k' :: "stack" where " k \<mapsto> k' " and "r = t \<star> k'"
sel_4: fixes x21 :: "char list"   and x22 :: "'a list" shows "get_terms (Neg x21 x22) = x22"
bisim_must_sync_preserve2: fixes t :: "'t"   and x1 :: "'x1"   and m1 :: "'m1"   and x2 :: "'x2"   and m2 :: "'m2" assumes "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "t \<turnstile> \<langle>x2, m2\<rangle> \<wrong>2" shows "t \<turnstile> \<langle>x1, m1\<rangle> \<wrong>1"
bind_def: fixes x :: "('a, 'c) state"   and f :: "'c \<Rightarrow> ('a, 'b) state" shows "x \<bind> f = State (\<lambda>s. case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
msg_nonempty1: fixes a :: "nat \<Rightarrow> 'a list"   and t :: "nat"   and aa :: "'a"   and l :: "'a list" assumes "msg (Suc 0) a"   and "a t = aa # l" shows "l = []"
VUnion_def: fixes A :: "V" shows "\<Union>\<^sub>\<circ> A = set {a. \<exists>x\<in>elts A. a \<in>\<^sub>\<circ> x}"
cos_double_less_one: fixes x :: "real" assumes "0 < x"   and "x < 2" shows "cos (2 * x) < 1"