hcju/isabelleps huggingface dataset:
seqI': fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "category C"   and "partial_magma.in_hom C f a b"   and "partial_magma.in_hom C g b c" shows "partial_magma.arr C (C g f)"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "S.seq (R g) (R f)"
Inf_less_eq: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and u :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v \<in> A \<Longrightarrow> less_eq v u"   and "A \<noteq> {}" shows "less_eq (Inf A) u"
eq_iff: fixes a :: "'a"   and b :: "'a"   and ko :: "'a key_order" shows "(a = b) = (le_of_key_order ko a b \<and> le_of_key_order ko b a)"
sup_ext: fixes H :: "'c set"   and h :: "'c \<Rightarrow> real"   and c :: "('c \<times> real) set set"   and M :: "('c \<times> real) set set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "graph H h = \<Union> c"   and "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "\<exists>x. x \<in> c" shows "graph F f \<subseteq> graph H h"
sqmatrix_mult_distrib_right: fixes A :: "('c, 'd) sqmatrix"   and B :: "('c, 'd) sqmatrix"   and C :: "('c, 'd) sqmatrix" shows "(A + B) \<cdot> C = A \<cdot> C + B \<cdot> C"
assms: shows "fresh_floatarith (fold_const_fa b) i"
rightInvert: fixes R1 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R2 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R3 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M1 :: "'b"   and M2 :: "'b"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and \<Gamma> :: "('a, 'b) form multiset"   and \<Delta> :: "('a, 'b) form multiset"   and M :: "'b"   and Ms :: "('a, 'b) form list"   and n :: "nat"   and \<Gamma>' :: "('a, 'b) form multiset"   and \<Delta>' :: "('a, 'b) form multiset" assumes "R1 \<subseteq> upRules \<and> R2 \<subseteq> modRules2 \<and> R3 \<subseteq> modRules2 \<and> R = Ax \<union> R1 \<union> p_e R2 M1 M2 \<union> R3 \<and> R' = Ax \<union> R1 \<union> R2 \<union> R3"   and "( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Modal M Ms, n) \<in> derivable (ext R R2 M1 M2)"   and "\<forall>r'\<in>R'. rightPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"   and "M2 \<noteq> M" shows "\<exists>m\<le>n. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable (ext R R2 M1 M2)"
add_diff_cancel_left': fixes a :: "'b"   and b :: "'b" shows "a + b - a = b"
eout_know_k: fixes C :: "specID"   and m :: "Keys" shows "eoutKnowCorrect C (kKS m) = (eout C (kE m) = (m \<in> specKeys C \<or> know C (kKS m)))"
terminal_path_is_path: fixes nodes :: "'a stream"   and inPorts :: "'a \<Rightarrow> 'b fset"   and outPorts :: "'a \<Rightarrow> 'c fset"   and vertices :: "'d fset"   and nodeOf :: "'d \<Rightarrow> 'a"   and edges :: "(('d \<times> 'c) \<times> 'd \<times> 'b) set"   and hyps :: "'a \<Rightarrow> 'c \<Rightarrow> 'b option"   and v :: "'d"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "Scoped_Graph nodes inPorts outPorts vertices nodeOf edges hyps"   and "Scoped_Graph.terminal_path outPorts vertices nodeOf edges hyps v v' pth" shows "Pre_Port_Graph.path edges v v' pth"
to_fun_X: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "X_poly Zp \<bullet> a = a"
pseudo_proj_True_proj_Suc: fixes n :: "nat"   and w :: "bool stream" shows "pseudo_proj_True n (pseudo_proj_True (Suc n) w) = pseudo_proj_True n w"
times_float: fixes x :: "real"   and y :: "real" assumes "x \<in> float"   and "y \<in> float" shows "x * y \<in> float"
sup_left_commute: fixes sup :: "'ap \<Rightarrow> 'ap \<Rightarrow> 'ap"   and less_eq :: "'ap \<Rightarrow> 'ap \<Rightarrow> bool"   and less :: "'ap \<Rightarrow> 'ap \<Rightarrow> bool"   and x :: "'ap"   and y :: "'ap"   and z :: "'ap" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
push_forward_def: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and X :: "('c \<Rightarrow> real) \<Rightarrow> real" assumes "diff k charts1 charts2 f" shows "diff.push_forward k charts1 charts2 f X = restrict0 (c_manifold.diff_fun_space charts2 k) (\<lambda>g. X (restrict0 (manifold.carrier charts1) (g \<circ> f)))"
strict_mono_o: fixes r :: "'b \<Rightarrow> 'c"   and s :: "'d \<Rightarrow> 'b" assumes "strict_mono r"   and "strict_mono s" shows "strict_mono (r \<circ> s)"
ncoplanar_perm_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet A C D B"
zero_neq_numeral: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "zero \<noteq> numeral.numeral one plus n"
map_simp: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "identity_functor C"   and "partial_magma.arr C f" shows "identity_functor.map C f = f"
sums_divide: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a" assumes "f sums a" shows "(\<lambda>n. f n / c) sums (a / c)"
f_def: shows "f \<equiv> \<lambda>x. 2 *\<^sub>R b + (2 / (1 - b \<bullet> x)) *\<^sub>R (x - b)"
r_dummy_append: fixes f :: "recf"   and xs :: "nat list"   and ys :: "nat list"   and n :: "nat" assumes "recfn (length xs) f"   and "length ys = n" shows "eval (r_dummy n f) (xs @ ys) = eval f xs"
CconcNone: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "dom (Cp a) = {}"   and "p \<noteq> []" shows "Cp (list2FWpolicy (a # p)) x = Cp (list2FWpolicy p) x"
weak_unit_char_2: fixes a :: "'a" shows "weak_unit a = (a \<cong> trg a)"
between_triv_iff: fixes a :: "'b"   and b :: "'b" shows "between (a, a) b = (a = b)"
hyps: assumes "is_tangle_diagram ws" shows "rat_poly.row_length (kauff_mat ws) = 2 ^ nat (domain_wall ws) \<and> length (kauff_mat ws) = 2 ^ nat (codomain_wall ws) \<and> mat (rat_poly.row_length (kauff_mat ws)) (length (kauff_mat ws)) (kauff_mat ws)"
simps: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and a :: "'d"   and v :: "'e"   and ts :: "('d, 'e) bintree list" shows "bt_dfs f (Node a v ts) = f (Node a v ts) # bts_dfs f ts"
initial_unowned_shared: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> ('e \<Rightarrow> 'f option)) list"   and m :: "'g"   and \<S> :: "nat \<Rightarrow> bool option" assumes "initial (ts, m, \<S>)" shows "unowned_shared \<S> ts"
ldropn_all: fixes xs :: "'a llist"   and m :: "nat" assumes "llength xs \<le> enat m" shows "ldropn m xs = LNil"
arr_def: fixes f :: "'c \<times> 'c" shows "CopxC.arr f \<equiv> CopxC.domains f \<noteq> {} \<or> CopxC.codomains f \<noteq> {}"
simps_2: fixes a :: "floatarith"   and bs :: "floatarith list"   and vs :: "real list" shows "interpret_floatariths (a # bs) vs = interpret_floatarith a vs # interpret_floatariths bs vs"
subdiv_distinct_verts0_1: shows "u \<noteq> w"
undual_inject: fixes x :: "'a dual"   and y :: "'a dual" shows "(undual x = undual y) = (x = y)"
s_AppRec: fixes v :: "Syntax.term"   and x :: "Syntax.var"   and \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and uv :: "mode" assumes "value v"   and "atom x \<sharp> (v, \<pi>)" shows "\<lless>\<pi>, App (Rec x e) v\<ggreater> uv\<rightarrow> \<lless>\<pi>, App (e[Rec x e / x]) v\<ggreater>"
pair: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm" assumes "A >> A'"   and "B >> B'" shows "SimplyTyped.Pair A B >> SimplyTyped.Pair A' B'"
ccw_scaleR1: fixes xr :: "real \<times> real"   and P :: "real \<times> real"   and e :: "real" assumes "det3 0 xr P \<noteq> 0"   and "0 < e"   and "ccw 0 xr P" shows "ccw 0 (e *\<^sub>R xr) P"
effect_tapE: fixes f :: "heap \<Rightarrow> 'a"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "effect (tap f) h h' r" obtains "h' = h" and "r = f h"
cons: shows "\<forall>x y e. x \<in> vertices G' \<longrightarrow> y \<in> vertices G' \<longrightarrow> (g (f x), g (f y)) \<in> :map_graph_fn G g:\<lbrakk>e\<rbrakk> \<longrightarrow> (x, y) \<in> :G':\<lbrakk>e\<rbrakk>"
xor_nat_def: fixes m :: "nat"   and n :: "nat" shows "m XOR n = nat (int m XOR int n)"
valid_Act_fresh: fixes \<alpha> :: "'act"   and P :: "'state"   and x :: "('a, 'pred, 'act) formula" assumes "bn \<alpha> \<sharp>* P" shows "P \<Turnstile> Act \<alpha> x = (\<exists>P'. P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<and> P' \<Turnstile> x)"
chart_eucl_in_atlas: fixes k :: "enat" shows "chart_eucl \<in> manifold_eucl.atlas k"
undirected_def: fixes G :: "'a graph" shows "undirected G = \<lparr>nodes = nodes G, edges = edges G \<union> {(b, a). (a, b) \<in> edges G}\<rparr>"
weak_reduction_bisimulation_and_closures_6: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_bisimulation ((symcl (Rel\<^sup>=))\<^sup>+) Cal"
InvariantsAfterInitialization: fixes F0 :: "Literal list list" shows "let state' = initialize F0 initialState in InvariantConsistent (getM state') \<and> InvariantUniq (getM state') \<and> InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state') \<and> InvariantWatchListsUniq (getWatchList state') \<and> InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state') \<and> InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state') \<and> InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state') \<and> InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state') \<and> InvariantNoDecisionsWhenConflict (getF state') (getM state') (currentLevel (getM state')) \<and> InvariantNoDecisionsWhenUnit (getF state') (getM state') (currentLevel (getM state')) \<and> InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state')) \<and> InvariantUniqQ (getQ state') \<and> InvariantVarsM (getM state') F0 {} \<and> InvariantVarsQ (getQ state') F0 {} \<and> InvariantVarsF (getF state') F0 {} \<and> (getConflictFlag state' \<or> getQ state' = []) \<and> currentLevel (getM state') = 0"
queue_invar_def: fixes q :: "('e, 'f) SkewBinomialTree list" shows "queue_invar q \<equiv> \<forall>t\<in>set q. tree_invar t"
XX: shows "\<forall>X\<in>XX. well_related_set X (\<sqsubseteq>) \<and> (\<forall>x\<in>X. \<forall>y\<in>\<Union> XX. \<not> x \<sqsubseteq> y \<longrightarrow> y \<in> X)"
simps_1: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'b" shows "foldl f a [] = a"
assms: shows "\<forall>l\<in>set (g1 G). set (g2 l N) = f2 (set l) N"
less_infI2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and x :: "'b"   and a :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less b x" shows "less (inf a b) x"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "bicategory V H \<a> \<i> src trg" shows "self_evaluation_map V H \<a> \<i> src trg"
fw_outermost_mono: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "i \<le> n"   and "j \<le> n" shows "fw m n 0 i j i j \<le> m i j"
max_add_distrib_right: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x + subset_mset.max y z = subset_mset.max (x + y) (x + z)"
stream_of_llist_llist_of_stream: fixes xs :: "'c stream" shows "stream_of_llist (llist_of_stream xs) = xs"
weighted_majority_nonagent2: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and y :: "'alt"   and x :: "'alt" assumes "is_pref_profile R"   and "y \<notin> alts" shows "weighted_majority R x y = 0"
indiff_set_Max_wrt_among: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and carrier :: "'c set"   and A :: "'c set"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "pref_profile_wf agents alts R"   and "finite carrier"   and "A \<subseteq> carrier"   and "A \<noteq> {}"   and "total_preorder_on carrier S" shows "R'.indiff_set S (Max_wrt_among S A)"
Mp_factorization_m: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "q.factorization_m (q.Mp f) cfs = q.factorization_m f cfs"
assms_1: fixes n :: "nat"   and m :: "nat" shows "AE x in M. u (n + m) x \<le> u n x + u m ((T ^^ n) x)"
runiq_paste2: fixes Q :: "('a \<times> 'b) set"   and P :: "('a \<times> 'b) set" assumes "runiq Q"   and "runiq P" shows "runiq (P +< Q)"
hom_graph_unique: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b"   and a' :: "'b" assumes "hom_graph f S x a"   and "hom_graph f S x a'" shows "a = a'"
strictly_more_defined_def: fixes p :: "'c \<Rightarrow> 'd decision option"   and q :: "'c \<Rightarrow> 'd decision option" shows "strictly_more_defined p q = (dom q \<subset> dom p)"
Dmonoid_axioms: shows "Dmonoid (\<circ>) id"
Init: fixes s :: "'c"   and F :: "'c program" assumes "s \<in> Init F" shows "s \<in> reachable F"
one_closed: shows "\<one>\<^bsub>K\<^esub> \<in> carrier K"
add_less_cancel_left0: fixes x :: "V"   and z :: "V" shows "(x \<subset>\<^sub>\<circ> x + z) = ([]\<^sub>\<circ> \<subset>\<^sub>\<circ> z)"
rep_eq: fixes x :: "'f cset" shows "cBex x = Bex (rcset x)"
lift_spmf_return: fixes x :: "'d" shows "lift_spmf (return_spmf x) = Generative_Probabilistic_Value.Done x"
of_complex_1: shows "of_complex 1 = (1::'b)"
inner_axis: fixes x :: "('c, 'd) vec"   and i :: "'d"   and y :: "'c" shows "x \<bullet> axis i y = x $ i \<bullet> y"
has_field_derivative_csqrt: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "(csqrt has_field_derivative inverse (2 * csqrt z)) (at z)"
f: fixes i :: "nat" assumes "i \<notin> S" shows "f i = i"
elims: fixes x :: "'b \<times> 'b"   and xa :: "'b"   and xb :: "('b \<times> 'b) list"   and y :: "('b \<times> 'b) list" assumes "co_path x xa xb = y" obtains uu_ :: "'b \<times> 'b" and uv_ :: "'b" where "x = uu_" and "xa = uv_" and "xb = []" and "y = []"   | uw_ :: "'b \<times> 'b" and ux_ :: "'b" and e :: "'b \<times> 'b" where "x = uw_" and "xa = ux_" and "xb = [e]" and "y = [e]"   | u :: "'b" and v :: "'b" and w :: "'b" and e1 :: "'b \<times> 'b" and e2 :: "'b \<times> 'b" and es :: "('b \<times> 'b) list" where "x = (u, v)" and "xa = w" and "xb = e1 # e2 # es" and "y = (if e1 = (u, w) \<and> e2 = (w, v) then (u, v) # co_path (u, v) w es else if e1 = (v, w) \<and> e2 = (w, u) then (v, u) # co_path (u, v) w es else e1 # co_path (u, v) w (e2 # es))"
xX_simp1: shows "backflows X - (backflows (X - {x}) - E) = backflows {x}"
closed_Compl: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open"   and "open S" shows "topological_space.closed open (- S)"
dtor_unfold: fixes s :: "'d \<Rightarrow> ('c, 'd) coF0"   and b :: "'d" shows "dtor_coF (dtor_unfold_coF s b) = map_pre_coF id (dtor_unfold_coF s) (s b)"
simplex_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b set" assumes "ChamberComplexRetraction X f"   and "x \<in> X" shows "f ` x \<in> X"
eRSV_deltaF: fixes R :: "('i \<Rightarrow> 'd) admS dual"   and e :: "'d \<rightarrow> 'd"   and S :: "('i \<Rightarrow> 'd) admS" assumes "eRSV e R S" shows "eRSV (\<delta>\<cdot>e) (dual (F (dual S, undual R))) (F (R, S))"
gcd_zero_iff': fixes a :: "'b"   and b :: "'b" shows "((0::'b) = some_gcd a b) = (a = (0::'b) \<and> b = (0::'b))"
is_poly_def: fixes ops :: "'b arith_ops_record"   and xs :: "'b list" shows "is_poly ops xs = (list_all (arith_ops_record.DP ops) xs \<and> no_trailing ((=) (arith_ops_record.zero ops)) xs)"
list_lcm_pos_3: fixes xs :: "int list" assumes "0 \<notin> set xs" shows "0 < lcm_list xs"
pCons_0_as_mult: fixes p :: "'b poly" shows "pCons (0::'b) p = [:0::'b, 1::'b:] * p"
simps_2: fixes uu :: "('b \<times> color) tree"   and uv :: "'b"   and c :: "color"   and uw :: "('b \<times> color) tree" shows "color \<langle>uu, (uv, c), uw\<rangle> = c"
PiE_eq_iff: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd set"   and F' :: "'c \<Rightarrow> 'd set" shows "(Pi\<^sub>E I F = Pi\<^sub>E I F') = ((\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
invariantI: fixes A :: "('e set \<times> 'e set \<times> 'e set) \<times> 'd set \<times> ('d \<times> 'e \<times> 'd) set"   and P :: "'d \<Rightarrow> bool" assumes "\<And>s. s \<in> starts_of A \<Longrightarrow> P s"   and "\<And>s t a. \<lbrakk>reachable A s; P s\<rbrakk> \<Longrightarrow> (s, a, t) \<in> trans_of A \<longrightarrow> P t" shows "invariant A P"
advantage_nonneg: fixes \<A> :: "(bool, '\<alpha>, '\<gamma>) gpv" shows "0 \<le> local.prf.advantage \<A>"
enabled_stream_trans: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and xs :: "('s \<times> ('c \<Rightarrow> real)) cfg stream" assumes "cfg \<in> MDP.valid_cfg"   and "MDP.MC.enabled cfg xs" shows "stream_trans (smap (snd \<circ> state) xs)"
assms_4: fixes s :: "'s"   and cfg :: "'s cfg"   and t :: "'s" assumes "P s cfg"   and "t \<in> set_pmf (action cfg)" shows "P t (cont cfg t)"
\<delta>\<^sub>1_def_3: shows "(\<delta>\<^sub>1', p\<^sub>1') = find_closest_bf_code p (p\<^sub>2 # ps)"
b_def: fixes i :: "nat" shows "b i = a (i + m) / a m"
Glb_lower: fixes A :: "'a set"   and i :: "'i"   and a :: "'a" assumes "A \<subseteq> L i"   and "a \<in> A" shows "Glb i A \<le> a"
If_cong: fixes b :: "tr"   and b' :: "tr"   and t :: "'a"   and t' :: "'a"   and e :: "'a"   and e' :: "'a" assumes "b = b'"   and "b = TT \<Longrightarrow> t = t'"   and "b = FF \<Longrightarrow> e = e'" shows "If b then t else e = If b' then t' else e'"
fmfilter_fmmap_keys: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and m :: "('c, 'e) fmap" shows "fmfilter P (fmmap_keys f m) = fmmap_keys f (fmfilter P m)"
evaln_max3: fixes G :: "prog"   and s1 :: "state"   and t1 :: "Term.term"   and n1 :: "nat"   and w1 :: "vals"   and s1' :: "state"   and s2 :: "state"   and t2 :: "Term.term"   and n2 :: "nat"   and w2 :: "vals"   and s2' :: "state"   and s3 :: "state"   and t3 :: "Term.term"   and n3 :: "nat"   and w3 :: "vals"   and s3' :: "state" assumes "G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>n1\<rightarrow> (w1, s1')"   and "G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>n2\<rightarrow> (w2, s2')"   and "G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>n3\<rightarrow> (w3, s3')" shows "G\<turnstile>s1 \<midarrow>t1\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> (w1, s1') \<and> G\<turnstile>s2 \<midarrow>t2\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> (w2, s2') \<and> G\<turnstile>s3 \<midarrow>t3\<succ>\<midarrow>max (max n1 n2) n3\<rightarrow> (w3, s3')"
"case": fixes f :: "(oref \<Rightarrow> obj option) \<Rightarrow> (lname \<Rightarrow> val option) \<Rightarrow> 'a"   and x1 :: "oref \<Rightarrow> obj option"   and x2 :: "lname \<Rightarrow> val option" shows "(case st x1 x2 of st x xa \<Rightarrow> f x xa) = f x1 x2"
assertion_disjunctive: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "x \<in> disjunctive.disjunctive sup sup times"
read_only_unowned_axioms: shows "read_only_unowned \<S> ts"
line_def: fixes P :: "Points"   and Q :: "Points" shows "line P Q \<equiv> SOME l. incid P l \<and> incid Q l"
app_vdomainE: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" obtains b :: "V" where "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
arc_set_balanced_all: fixes u :: "'a"   and v :: "'a" shows "arc_set_balanced u (arcs G) v = (if u = v then \<forall>w\<in>verts G. in_degree G w = out_degree G w else (\<forall>w\<in>verts G. w \<noteq> u \<and> w \<noteq> v \<longrightarrow> in_degree G w = out_degree G w) \<and> in_degree G u + 1 = out_degree G u \<and> out_degree G v + 1 = in_degree G v)"
denote_measure_def: fixes n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and DS :: "(complex mat \<Rightarrow> complex mat) list"   and \<rho> :: "complex mat" shows "denote_measure n M DS \<rho> = matrix_sum d (\<lambda>k. (DS ! k) (M k * \<rho> * adjoint (M k))) n"
simps_1: fixes a :: "('U, 'C, 'Out) Noninterference.aprop"   and p :: "pvar"   and env :: "pvar \<Rightarrow> nat" shows "sem (Atom a p) env = atom a (env p)"
Gram_Schmidt_cons: fixes a :: "('a, 'b) vec"   and xs :: "('a, 'b) vec list" shows "Gram_Schmidt (a # xs) = Gram_Schmidt_step a (Gram_Schmidt xs)"
poly_abelian_monoid_def: shows "local.poly_abelian_monoid = \<lparr>carrier = UNIV, monoid.mult = (*), one = 1::'b, zero = 0::'c, add = (+), module.smult = Polynomial.smult\<rparr>"
4: fixes v :: "nat \<times> 'a" assumes "v \<in> greachable A w V u" shows "finite (gsuccessors A w V v)"
n_def: shows "n = card (GAUSS.E q (int p))"
set_less_eq_aux_infinite_complement_partitionD: fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubseteq>' B" shows "A \<in> infinite_complement_partition \<and> B \<in> infinite_complement_partition"
hyps: shows "P \<parallel> Q \<longmapsto> \<alpha> \<prec> R'"
lex_append_rightI: fixes xs :: "'b list"   and ys :: "'b list"   and r :: "('b \<times> 'b) set"   and vs :: "'b list"   and us :: "'b list" assumes "(xs, ys) \<in> lex r"   and "length vs = length us" shows "(xs @ us, ys @ vs) \<in> lex r"
valid_strategy_updates_set_strong: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "G'.strategy p \<sigma>"   and "G'.strategy p \<sigma>'" shows "G'.strategy p (override_on \<sigma> \<sigma>' A)"
comp_arr_ide: fixes a :: "Arr"   and f :: "Arr" assumes "ide a"   and "seq f a" shows "local.comp f a = f"