hcju/isabelleps huggingface dataset:
intros_1: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and k' :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and \<sigma> :: "'var \<Rightarrow> 'val"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "cs\<^bsup>\<pi>'\<^esup> k' \<prec> cs\<^bsup>\<pi>\<^esup> k"   and "\<pi> = path \<sigma>"   and "\<pi>' = path \<sigma>'"   and "\<pi> (Suc (\<pi> \<exclamdown> cs\<^bsup>\<pi>'\<^esup> k')) \<noteq> \<pi>' (Suc k')" shows "(\<sigma>', k') \<cc> (\<sigma>, k)"
simps_1: fixes c :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and c' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" shows "abs.meval_sched_abv c [] c' = (c = c')"
vsv_axioms: shows "vsv (xs #\<^sub>\<circ> x)"
power_poly_sum_mod_ring: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c mod_ring poly" assumes "finite A" shows "sum f A ^ CARD('c) = (\<Sum>i\<in>A. f i ^ CARD('c))"
mult_zero_right: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "z *\<^sub>h 0\<^sub>h = 0\<^sub>h"
assms_2: shows "cart2_append1 p = cart2_append1 q"
unfold_optimize_ruleset_CHAIN_def: fixes optimize :: "'a match_expr \<Rightarrow> 'a match_expr"   and chain_name :: "char list"   and default_action :: "action"   and rs :: "char list \<Rightarrow> 'a rule list option" shows "unfold_optimize_ruleset_CHAIN optimize chain_name default_action rs = (let rs = repeat_stabilize 1000 (optimize_matches opt_MatchAny_match_expr) (optimize_matches optimize (rw_Reject (rm_LogEmpty (repeat_stabilize 10000 (process_call rs) [Rule MatchAny (Call chain_name), Rule MatchAny default_action])))) in if simple_ruleset rs then Some rs else None)"
enum_set_get_one_empty: fixes ss :: "'b list" shows "enum_set_get_one ss {} = None"
\<beta>\<^sub>g: shows "\<guillemotleft>\<beta>\<^sub>g : tab\<^sub>1 g \<star> w\<^sub>g \<rightarrow> tab\<^sub>1 g \<star> w\<^sub>g'\<guillemotright>"
field_differentiable_inverse: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and S :: "'a set" assumes "f field_differentiable at a within S"   and "f a \<noteq> (0::'a)" shows "(\<lambda>z. inverse (f z)) field_differentiable at a within S"
unique_session_keys: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and nb :: "msg"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and B' :: "agent"   and na' :: "msg"   and nb' :: "msg"   and X' :: "msg" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace> \<in> set evs"   and "Says Server A' \<lbrace>Crypt (shrK A') \<lbrace>Agent B', Key K, na', nb'\<rbrace>, X'\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "A = A' \<and> B = B' \<and> na = na' \<and> nb = nb'"
fold_insert_idem: fixes A :: "'a set"   and z :: "'a"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold inf z (insert x A) = inf x (Finite_Set.fold inf z A)"
g: shows "tendsto_ine\<^sub>N (\<LL> 2 (lebesgue_on S)) f g"
tllist_all2_is_TNilD: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist" assumes "tllist_all2 P Q xs ys" shows "is_TNil xs = is_TNil ys"
traverse_cast_Dyn_to_all_Dyn: fixes xs :: "('a, 'b, 'c) unboxed list"   and ys :: "'a list" assumes "traverse cast_Dyn xs = Some ys" shows "list_all (\<lambda>x. typeof x = None) xs"
induce_subgraph_conn_2: shows "induced_subgraph (G \<restriction> verts (Digraph_Component.union c d)) G"
entails_equiv_backward: fixes P :: "assn"   and Q :: "assn" assumes "P = Q" shows "Q \<Longrightarrow>\<^sub>A P"
le_contour_def: fixes b :: "contour"   and b' :: "contour" shows "(b \<le> b') = (length (Rep_contour b) \<le> length (Rep_contour b'))"
pairs_uminus_closed: fixes x :: "'a"   and y :: "'b Filters.filter" shows "pairs_uminus (x, y) \<in> pairs"
at_ds8: fixes pi1 :: "('a \<times> 'a) list"   and pi2 :: "('a \<times> 'a) list" assumes "at TYPE('a)" shows " (pi1 @ pi2) \<triangleq> (pi1 \<bullet> pi2 @ pi1) "
DisjointPaths_axioms: shows "DisjointPaths G v0 v1 paths"
zip_with_altdef: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'d list" shows "zip_with f xs ys = map2 f xs ys"
local: fixes i :: "'a" assumes "i \<in> I" shows "\<sigma>' i = \<sigma> i"
smult_1_left: fixes p :: "'b poly" shows "Polynomial.smult (1::'b) p = p"
F_aprog_uhalt: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "'a"   and F_ap :: "abc_inst list"   and rs_pos :: "nat"   and a_md :: "nat"   and suflm :: "nat list" assumes "tm_wf0 tp"   and "\<forall>stp. \<not> TSTD (steps0 (Suc 0, Bk \<up> l, <lm>) tp stp)"   and "rec_ci rec_F = (F_ap, rs_pos, a_md)" shows "{\<lambda>nl. nl = [code tp, bl2wc (<lm>)] @ 0 \<up> (a_md - rs_pos) @ suflm} F_ap \<up>"
1: shows "(\<exists>x. linear_orderable_1 x) = (\<exists>x. linear_orderable_4 x)"
lfp_unfold: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "lfp f = f (lfp f)"
splus_def: fixes t :: "'a"   and v :: "'a" shows "punit.splus t v = fst (t + punit.pp_of_term v, punit.component_of_term v)"
array_set'_def: fixes a :: "'a array" shows "array_set' a \<equiv> array_set a \<circ> nat_of_integer"
CapSS: fixes \<psi> :: "'fmla"   and f :: "'trm \<Rightarrow> 'trm" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {inp}"   and "f \<in> F" shows "ssap \<psi> \<langle>encF f\<rangle> = \<langle>instInpP \<psi> 0 (instInp (encF f) \<langle>encF f\<rangle>)\<rangle>"
eSwapAbs_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'e, 'd) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" shows "eSwapAbs MOD zs z1 z2 ERR = ERR"
eval: shows "P\<^sub>1 \<turnstile>\<^sub>1 \<langle>body,(h, ls)\<rangle> \<Rightarrow> \<langle>e',(h', ls')\<rangle>"
sop_distr1_def: fixes R :: "('e, 'f) Ring_scheme"   and s :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and i :: "'g \<Rightarrow> 'g"   and A :: "'g set"   and z :: "'g" shows "sop_distr1 R s f i A z = (\<forall>a\<in>carrier R. \<forall>b\<in>carrier R. \<forall>x\<in>aug_pm_set z i A. (a \<plusminus>\<^bsub>R\<^esub> b) \<^sub>s\<cdot> x = a \<^sub>s\<cdot> x \<^sub>f+ b \<^sub>s\<cdot> x)"
equivalentFormulaeReflexivity: fixes formula :: "Literal list list" shows "equivalentFormulae formula formula"
simps_4: fixes x31 :: "comb"   and x32 :: "comb" shows "K \<noteq> x31 \<bullet> x32"
gauss_lemma: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "Legendre a (int p) = (- 1) ^ card (GAUSS.E p a)"
random_serial_dictatorship_foldl: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite agents" shows "random_serial_dictatorship agents alts R = pmf_of_set (permutations_of_set agents) \<bind> (\<lambda>agents'. pmf_of_set (foldl (\<lambda>alts i. Max_wrt_among (R i) alts) alts agents'))"
runiq_alt: fixes R :: "('a \<times> 'b) set" shows "runiq R = (\<forall>x. trivial (R `` {x}))"
augmented_Sset_minimum: fixes D :: "'c Order"   and X :: "'c set"   and a :: "'c"   and Y :: "'c set"   and y :: "'c set"   and x :: "'c" assumes "Order D"   and "X \<subseteq> carrier D"   and "a \<in> carrier D"   and "Y - {a} \<subseteq> X"   and "y - {a} \<noteq> {}"   and "minimum_elem (SIod D X) (Y - {a}) x"   and "\<forall>x\<in>X. x \<preceq>\<^bsub>D\<^esub> a" shows "minimum_elem (SIod D (insert a X)) Y x"
rep_eq: fixes x :: "'b ratfps" shows "quot_of_ratfps x = quot_of_fract (Rep_ratfps x)"
assms_4: shows "D \<in> carrier_mat nr2 nc2"
Min_in: fixes A :: "'c Word.word set" assumes "finite A"   and "A \<noteq> {}" shows "signed.Min A \<in> A"
normalize_lcm_right: fixes a :: "'a"   and b :: "'a" shows "lcm a (normalize b) = lcm a b"
function_ring_defs_2: shows "(\<otimes>\<^bsub>F\<^esub>) = function_mult S R"
evbA: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "A $$ (i, i) = A $$ (j, j)"
summable'': fixes p :: "nat"   and A :: "nat set" assumes "prime p" shows "(\<lambda>x. f (p ^ x)) abs_summable_on A"
card1: shows "card {p. p partitions Suc m \<and> sum p {..Suc m} = Suc k \<and> p 1 \<noteq> 0} = Partition m k"
if_eq_is_pr3: fixes p1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and p2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and q1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and q2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "p1 \<in> PrimRec3"   and "p2 \<in> PrimRec3"   and "q1 \<in> PrimRec3"   and "q2 \<in> PrimRec3" shows "(\<lambda>x y z. if p1 x y z = p2 x y z then q1 x y z else q2 x y z) \<in> PrimRec3"
outside_connected_component_le: fixes S :: "'a set" shows "outside S = {x. \<forall>B. \<exists>y. B \<le> norm y \<and> connected_component (- S) x y}"
ID_ccompare_interval: fixes ccompare :: "('b \<Rightarrow> 'b \<Rightarrow> order) option"   and cproper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "class.cproper_interval ccompare cproper_interval"   and "ID ccompare = Some c"   and "finite UNIV" shows "class.linorder_proper_interval (le_of_comp c) (lt_of_comp c) cproper_interval"
2: shows "Dom (\<mu>' \<bullet> \<mu>) = Dom \<mu> \<and> Dom (\<nu>' \<bullet> \<nu>) = Dom \<nu> \<and> Cod (\<mu>' \<bullet> \<mu>) = Cod \<mu>' \<and> Cod (\<nu>' \<bullet> \<nu>) = Cod \<nu>'"
cons_pre_rule: fixes P :: "assn"   and P' :: "assn"   and c :: "'c Heap"   and Q :: "'c \<Rightarrow> assn" assumes "P \<Longrightarrow>\<^sub>A P'"   and "<P'> c <Q>" shows "<P> c <Q>"
open_UNION_box: fixes M :: "'c set" assumes "open M" shows "M = (\<Union>f\<in>{f \<in> Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) \<subseteq> M}. box (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i))"
sets_measure_of_conv: fixes \<Omega> :: "'c set"   and A :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "sets (measure_of \<Omega> A \<mu>) = (if A \<subseteq> Pow \<Omega> then sigma_sets \<Omega> A else {{}, \<Omega>})"
ldropn_Suc_conv_ldropn: fixes n :: "nat"   and xs :: "'a llist" assumes "enat n < llength xs" shows "LCons (lnth xs n) (ldropn (Suc n) xs) = ldropn n xs"
simps_11: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and \<theta>\<^sub>1 :: "('sf, 'sz) trm"   and \<theta>\<^sub>2 :: "('sf, 'sz) trm"   and \<sigma> :: "('sf, 'sc, 'sz) subst" shows "step_result (SG, C) (i, CQ \<theta>\<^sub>1 \<theta>\<^sub>2 \<sigma>) = (closeI SG i, C)"
l1_iagreement_def: shows "l2_iagreement \<equiv> {s. \<forall>A B N. l1_state.signals s (Commit A B N) \<le> l1_state.signals s (Running A B N)}"
weak_composition_with_homs_axioms: shows "weak_composition_with_homs (\<cdot>) (\<star>) src trg"
has_derivativeI: fixes f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "bounded_linear f'"   and "((\<lambda>y. (f y - f x - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> (0::'c)) (at x within s)" shows "(f has_derivative f') (at x within s)"
equivalence_functor_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e" shows "equivalence_functor C D G \<equiv> (category C \<and> category D) \<and> functor C D G \<and> equivalence_functor_axioms C D G"
Ik_intT_int: fixes T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wt T"   and "I.wtE \<xi>"   and "\<And>\<sigma>. infTp \<sigma> \<or> (\<forall>x\<in>nvT T. tpOfV x \<noteq> \<sigma>)" shows "Ik_intT (tpOf T) (I.int \<xi> T)"
left_closed: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Congruence" shows "a \<in> G"
elims: fixes x :: "'b tree"   and y :: "nat" assumes "height x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "'b tree" and a :: "'b" and r :: "'b tree" where "x = \<langle>l, a, r\<rangle>" and "y = max (height l) (height r) + 1"
elims: fixes x :: "'e \<Rightarrow> 'c \<times> 'd"   and xa :: "'e net_state"   and y :: "(nat \<Rightarrow> 'c option) \<times> 'd net_state" assumes "netgmap x xa = y" obtains sr :: "'e \<Rightarrow> 'c \<times> 'd" and i :: "nat" and s :: "'e" and R :: "nat set" where "x = sra__" and "xa = NodeS i s R" and "y = ([i \<mapsto> fst (sra__ s)], NodeS i (snd (sra__ s)) R)"   | sr :: "'e \<Rightarrow> 'c \<times> 'd" and s\<^sub>1 :: "'e net_state" and s\<^sub>2 :: "'e net_state" where "x = sr" and "xa = SubnetS s\<^sub>1 s\<^sub>2" and "y = (let (\<sigma>\<^sub>1, ss) = netgmap sr s\<^sub>1; (\<sigma>\<^sub>2, tt) = netgmap sr s\<^sub>2 in (\<sigma>\<^sub>1 ++ \<sigma>\<^sub>2, SubnetS ss tt))"
is_serial_sol_then_valid_plan_1: fixes ops :: "(nat, nat) sas_plus_operator list"   and \<pi> :: "char list" assumes "is_serial_solution_for_problem abs_prob ops"   and "\<pi> \<in> set ((map (fst \<circ> the \<circ> lookup_action) \<circ> rem_condless_ops I) ops)" shows "lookup_operator \<pi> \<noteq> None"
add_mset_remove_trivial: fixes x :: "'a"   and M :: "'a multiset" shows "(x, M) - (x, {#}) = M"
rtranclp_imp_relpowp: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "P\<^sup>*\<^sup>* x y" shows "\<exists>n. (P ^^ n) x y"
three_neq_6: assumes "three TYPE('d)" shows "three.three\<^sub>3 \<noteq> three.three\<^sub>2"
f1: fixes x :: "'a" assumes "x \<in> sphere a r" shows "f1 x = x"
Quotient3_rel: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and r :: "'e"   and s :: "'e" assumes "Quotient3 R Abs Rep" shows "(R r r \<and> R s s \<and> Abs r = Abs s) = R r s"
arctan_tan: fixes x :: "real" assumes "- (pi / 2) < x"   and "x < pi / 2" shows "arctan (tan x) = x"
completeAux: fixes P :: "pi"   and Q :: "pi"   and n :: "nat" assumes "depth P + depth Q \<le> n"   and "valid P"   and "valid Q"   and "uhnf P"   and "uhnf Q"   and "P \<sim> Q" shows "P \<equiv>\<^sub>e Q"
mem_Sigma_iff: fixes a :: "'b"   and b :: "'d"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'd set" shows "((a, b) \<in> Sigma A B) = (a \<in> A \<and> b \<in> B a)"
snd_quotient_plus_leq: fixes x :: "rat"   and y :: "rat" shows "(snd \<circ> quotient_of) (x + y) \<le> (snd \<circ> quotient_of) x * (snd \<circ> quotient_of) y"
Edges_Cons: fixes x :: "nat"   and xs :: "nat list" shows "Edges (x # xs) = (if xs = [] then {} else Edges xs \<union> {(x, hd xs)})"
assms_1: shows "fst a \<subseteq>\<^sub>f s1"
Snoc_step1_SnocD: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and y :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "step1 r (ys @ [y]) (xs @ [x])" shows "step1 r ys xs \<and> y = x \<or> ys = xs \<and> r y x"
glb_above: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and z :: "'c"   and x :: "'c" assumes "class.po below"   and "po.is_glb below S z" shows "below x (po.glb below S) = po.is_lb below S x"
1: fixes M :: "('c \<times> 'b) list"   and J :: "'b" assumes "distinct (map snd M)"   and "J \<in> snd ` set M" shows "(the (map_of (map (\<lambda>(a, b). (b, a)) M) J), J) \<in> set M"
encode_positive_transition_frame_axiom_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat"   and v :: "'a" shows "encode_positive_transition_frame_axiom \<Pi> t v \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O>; adding_operators = filter (\<lambda>op. ListMem v (add_effects_of op)) ops in Atom (State t (index vs v)) \<^bold>\<or> (\<^bold>\<not> (Atom (State (Suc t) (index vs v))) \<^bold>\<or> \<^bold>\<Or>map (\<lambda>op. Atom (Operator t (index ops op))) adding_operators)"
dom_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o cardSuc (cmax |UNIV| natLeq)" shows "local.dom (mkIde A) = mkIde A"
ccExp_subst: fixes x :: "Vars.var"   and S :: "Vars.var set"   and y :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "x \<notin> S"   and "y \<notin> S" shows "\<G>\<^bsub>a\<^esub> e[y::=x] G|` S \<sqsubseteq> \<G>\<^bsub>a\<^esub> e G|` S"
assms_2: shows "effective_redundancy_criterion \<Gamma> Rf Ri"
rel_mset_replicate_mset_right: fixes A :: "'a multiset"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and n :: "nat" assumes "\<And>x. x \<in># A \<Longrightarrow> R x y"   and "size A = n" shows "rel_mset R A (replicate_mset n y)"
Inf_UNIV: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf UNIV = bot"
discrete_topology_unique: fixes U :: "'b set"   and X :: "'b topology" shows "(discrete_topology U = X) = (topspace X = U \<and> (\<forall>x\<in>U. openin X {x}))"
isPath_fwd_cases: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "cf.isPath s p t" obtains "p = []" and "t = s"   | p' :: "(nat \<times> nat) list" and u :: "nat" where "p = (s, u) # p'" and "(s, u) \<in> cf.E" and "cf.isPath u p' t"
id_is_mor_pr_rngs: shows "morphism_presheaves_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF> \<rho> b add_str mult_str zero_str one_str (\<lambda>U. identity (\<FF> U))"
axioms_1: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j, 'k) sa_rec_scheme"   and B :: "('i, 'j, 'l) sa_rec_scheme" assumes "lsimulation R A B" shows "sa A"
"2_2": shows "(return_time p2 has_derivative dR2) (at (poincare_map p1 x0))"
order_widen: fixes m :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" assumes "wf_prog m P" shows "order (subtype P)"
trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst': fixes a :: "('c, 'd) stateful_strand_step"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "\<not> is_NegChecks a" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
liftAll_def: fixes phi :: "'d \<Rightarrow> bool"   and inp :: "'e \<Rightarrow> 'd option" shows "liftAll phi inp \<equiv> \<forall>i v. inp i = Some v \<longrightarrow> phi v"
equalityE: fixes A :: "'a set"   and B :: "'a set" assumes "A = B" obtains "A \<subseteq> B" and "B \<subseteq> A"
PE_result_rel_def: shows "PE_result_rel \<equiv> \<lambda>i. case i of 0 \<Rightarrow> ValFF | Suc n \<Rightarrow> ValTT"
BoxOr: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool"   and sigma :: "behavior" assumes "sigma \<Turnstile> \<box>F \<or> \<box>G" shows "sigma \<Turnstile> \<box>(F \<or> G)"
uminus_add_minus_mat: fixes l :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and r :: "'a mat" assumes "l \<in> carrier_mat nr nc"   and "r \<in> carrier_mat nr nc" shows "- (l + r) = - l - r"
iterate_step_empty_stack: fixes t :: "nat"   and rv :: "nat option" shows "iterate t step ([], rv) = ([], rv)"
subspace_is_vs: fixes W :: "'e set" assumes "subspace K W W" shows "vectorspace K (W.vs W)"
steps_map_equiv': fixes a :: "'a"   and b :: "'b"   and as :: "'a list" assumes "a \<sim> b"   and "QA a"   and "QB b" shows "A.steps (a # as) = B.steps (b # map f as)"
nonempty_has_size: fixes S :: "'a multiset" shows "(S \<noteq> \<Empt>) = (0 < size S)"
set_map: fixes a :: "'c"   and b :: "'c" assumes "C.ide a"   and "C.ide b" shows "S.set (local.map (b, a)) = local.set (b, a)"
dom_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "partial_magma.dom (free_monoidal_category.comp C) (free_monoidal_category.mkarr C t) = free_monoidal_category.mkarr C (monoidal_language.Dom C t)"
absolutely_integrable_norm: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on S" shows "norm \<circ> f absolutely_integrable_on S"
GSubmodule_GSpan_single: fixes m :: "'v" assumes "m \<in> V" shows "RModule FG (\<cdot>) (GSpan [m]) \<and> GSpan [m] \<subseteq> V"
pt_fresh_right_ineq: fixes pi :: "('b \<times> 'b) list"   and a :: "'c"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "cp TYPE('a) TYPE('b) TYPE('c)" shows "pi \<bullet> a \<sharp> x = a \<sharp> rev pi \<bullet> x"
mono: fixes f :: "(indi \<Rightarrow> (alt \<times> alt) set) \<Rightarrow> alt"   and P :: "indi \<Rightarrow> (alt \<times> alt) set"   and P' :: "indi \<Rightarrow> (alt \<times> alt) set" assumes "GS f"   and "P \<in> Prof"   and "P' \<in> Prof"   and "\<forall>i a. a <\<^bsub>P i\<^esub> f P \<longrightarrow> a <\<^bsub>P' i\<^esub> f P" shows "f P' = f P"
stateful_list_policy_to_list_graph_def: fixes \<T> :: "'b stateful_list_policy" shows "stateful_list_policy_to_list_graph \<T> = \<lparr>nodesL = hostsL \<T>, edgesL = flows_fixL \<T> @ filter (\<lambda>e. e \<notin> set (flows_fixL \<T>)) (flows_stateL \<T>) @ filter (\<lambda>e. e \<notin> set (flows_fixL \<T>)) (backlinks (flows_stateL \<T>))\<rparr>"
the_trivial_SomeE: fixes v :: "'a"   and vs :: "'a list"   and v' :: "'a" assumes "the_trivial v vs = Some v'" obtains "v \<noteq> v'" and "set vs = {v'}"   | "v \<noteq> v'" and "set vs = {v, v'}"
set1_FGcontra_def: fixes x :: "('p, 'q, 'r, 'q, 's, 't, ('u, 'v, 'w, 'x, 'q, 'y, 'z) G, 'u, 'aa, 'ab) F" shows "set1_FGcontra x = set1_F x"
fold_min_eqI1: fixes ys :: "'a list"   and y :: "'a" assumes "fold min_for ys y \<notin> set ys" shows "fold min_for ys y = y"
set_less_eq_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubseteq> B) = (if A \<in> infinite_complement_partition then A \<sqsubseteq>''' B \<or> B \<notin> infinite_complement_partition else B \<notin> infinite_complement_partition \<and> - B \<sqsubseteq>''' - A)"
borel_measurable_neq: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
coeff_smult: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and p :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R"   and "p \<in> carrier (UP R)" shows "coeff (UP R) (a \<odot>\<^bsub>UP R\<^esub> p) n = a \<otimes>\<^bsub>R\<^esub> coeff (UP R) p n"
zcf_eval: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "zcf f = to_fun f \<zero>"
rrep_def: shows "rrep \<equiv> \<lambda>(hops, dip, dsn, oip, sip). Rrep hops dip dsn oip sip"
abs_conv_abscissa_perzeta: fixes q :: "real" shows "abs_conv_abscissa (fds_perzeta q) \<le> 1"
fundchamber_S_chamber: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "chamber (s `\<rightarrow> C0)"
dom_dom: fixes f :: "(bool, bool list) concrete_category.arr" assumes "arr f" shows "local.dom (local.dom f) = local.dom f"
surjective: fixes r :: "('c, 'd, 'e) sas_plus_operator_scheme" shows "r = \<lparr>sas_plus_operator.precondition_of = sas_plus_operator.precondition_of r, effect_of = effect_of r, \<dots> = sas_plus_operator.more r\<rparr>"
Sup_set_def: fixes A :: "'c set set" shows "\<Union> A = {x. \<Squnion> ((\<in>) x ` A)}"
okClocks_def: fixes p :: "nat"   and q :: "nat"   and i :: "nat" shows "okClocks p q i = (\<forall>t. 0 \<le> t \<and> t < max (te p i) (te q i) \<and> correct p t \<and> correct q t \<longrightarrow> \<bar>VC p t - VC q t\<bar> \<le> \<delta>)"
of_int_mult: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w * z) = times (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"