hcju/isabelleps huggingface dataset:
Seq_append_app1: fixes s :: "hf"   and k :: "hf"   and l :: "hf"   and s' :: "hf" assumes "Seq s k"   and "l \<^bold>\<in> k" shows "app (seq_append k s s') l = app s l"
tiebidsBridgingLemma: fixes x :: "'c \<times> 'd set"   and N :: "'c set"   and \<Omega> :: "'d set"   and a :: "('c \<times> 'd set) set" assumes "x \<in> N \<times> (Pow \<Omega> - {{}})" shows "summedBidVector (pseudoAllocation a <| (N \<times> finestpart \<Omega>)) N \<Omega> x = tiebidsAlg a N \<Omega> x"
bijective_one_closed: shows "top \<sqinter> id top \<le> top \<and> top \<sqinter> top = top"
Vinv3D: fixes x :: "v_state" assumes "x \<in> Vinv3" shows "ran (decisions x) \<subseteq> locked x"
assms_3: shows "has_greatest_fixpoint (g \<circ> f)"
exp_of_derived_in_carrier: fixes H :: "'c set"   and n :: "nat" assumes "H \<subseteq> carrier H" shows "(derived H ^^ n) H \<subseteq> carrier H"
cnf_of_encode_all_operator_preconditions_structure: fixes \<Pi> :: "'a strips_problem"   and t :: "nat" shows "cnf (encode_all_operator_preconditions \<Pi> (\<Pi>\<^sub>\<O>) t) = (\<Union>(t, op)\<in>{..<t} \<times> set (\<Pi>\<^sub>\<O>). \<Union>v\<in>set (precondition_of op). {{(Operator t (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State t (index (\<Pi>\<^sub>\<V>) v))\<^sup>+}})"
swapAltsP_def: fixes P :: "'c \<Rightarrow> ('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" shows "swapAltsP P a b \<equiv> \<lambda>i. {(u, v). swapAlts a b u \<^bsub>(P i)\<^esub>\<preceq> swapAlts a b v}"
smult_commute_sg_nsg: fixes H :: "'a set"   and N :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> N" shows "H \<diamondop> N = N \<diamondop> H"
mset_set_mset_image: fixes E :: "'b set"   and N :: "nat"   and g :: "'b \<Rightarrow> 'c" assumes "card E = N"   and "finite E" shows "mset_set (g ` E) \<subseteq># image_mset g (mset_set E)"
distinct_39: fixes x4 :: "nat"   and x6 :: "nat" shows "\<not>unicast x4 \<noteq> deliver x6"
cann_4: shows "{i. prime i \<and> 0 \<le> i \<and> i < jj} \<subseteq> set is"
intros_3: fixes c\<^sub>1 :: "com"   and s\<^sub>1 :: "char list \<Rightarrow> int"   and s\<^sub>2 :: "char list \<Rightarrow> int"   and c\<^sub>2 :: "com"   and s\<^sub>3 :: "char list \<Rightarrow> int" assumes "(c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2"   and "(c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3" shows "(c\<^sub>1;; c\<^sub>2, s\<^sub>1) \<Rightarrow> s\<^sub>3"
in_own_span: fixes S :: "'c set" assumes "S \<subseteq> carrier M" shows "S \<subseteq> span S"
naturality: fixes f :: "'d" assumes "D.arr f" shows "\<eta> (D.cod f) \<cdot>\<^sub>D D.map f = (G \<circ> F) f \<cdot>\<^sub>D \<eta> (D.dom f)"
hyps: shows "applied_rule_rev Cp x (removeShadowRules2 ys) = applied_rule_rev Cp x ys"
continuous_within: fixes x :: "'e"   and s :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "continuous (at x within s) f = (f \<longlongrightarrow> f x) (at x within s)"
c: shows "char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1:])"
gen_colossless_restrict_gpv: fixes \<I> :: "('b, 'd) \<I>"   and gpv :: "('e, 'b, 'd) gpv"   and b :: "bool" assumes "\<I> \<turnstile>g gpv \<surd>" shows "gen_lossless_gpv b \<I> (restrict_gpv \<I> gpv) = gen_lossless_gpv b \<I> gpv"
ball_eq_empty: fixes x :: "'c"   and e :: "real" shows "(ball x e = {}) = (e \<le> 0)"
col_image_spec__eq: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and P' :: "'p" assumes "Col A B P"   and "P P' ReflectL A B" shows "P = P'"
degh1: shows "MPoly_Type.degree (Const (real (i + 1)) * Var x ^ i) x = MPoly_Type.degree (Const (real (i + 1))) x + MPoly_Type.degree (Var x ^ i) x"
comp_top_linorder_matrix: fixes f :: "'d \<times> 'd \<Rightarrow> 'c"   and i :: "'d"   and j :: "'d" shows "(f \<odot> mtop) (i, j) = sup_monoid.sum (\<lambda>k. f (i, k)) {k. True}"
const_name_def: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.const_name unconst t = (case unconst t of Some name \<Rightarrow> name)"
inv_unique: fixes y :: "'a list"   and x :: "'a list"   and y' :: "'a list" assumes "y \<oplus>\<^bsub>K [X]\<^esub> x = \<zero>\<^bsub>K [X]\<^esub>"   and "x \<oplus>\<^bsub>K [X]\<^esub> y' = \<zero>\<^bsub>K [X]\<^esub>"   and "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])"   and "y' \<in> carrier (K [X])" shows "y = y'"
preserves_cod: fixes f :: "'a" assumes "B.arr f" shows "B.cod (B.\<ll> f) = B.map (B.cod f)"
assms: shows "\<forall>rs\<in>ran \<Gamma>. no_call_to c rs"
UN_least: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'd set"   and C :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C" shows "\<Union> (B ` A) \<subseteq> C"
strict_coboundedI2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c < b" shows "c < a \<squnion> b"
select_convs_1: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and more :: "'a" shows "progress \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, \<dots> = more\<rparr> = progress"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_before_State_Idle_conv: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a" assumes "Suc 0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c"   and "\<box> t1 [0\<dots>,k - Suc 0] \<oplus> t0. \<not> (State_Idle localState output_fun trans_fun (localState (s t1)) \<and> output_fun (s t1) \<noteq> NoMsg)" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) = (\<not> State_Idle localState output_fun trans_fun (localState (s t1)). t1 \<U> t2 [0\<dots>] \<oplus> t0. output_fun (s t2) = m \<and> (\<circle> t3 t2 [0\<dots>]. output_fun (s t4) = NoMsg. t4 \<U> t5 [0\<dots>] \<oplus> t3. output_fun (s t5) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t5))))"
distinct_21: fixes x2 :: "('e, 'f) Term.term"   and x71 :: "'f list"   and x72 :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and x73 :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list" shows "receive\<langle>x2\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
vend_fail_def: shows "vend_fail \<equiv> \<lparr>Label = STR ''vend'', Arity = 0, Guards = [Lt (V (R 2)) (L (Num 100))], Outputs = [], Updates = [(1, V (R 1)), (2, V (R 2))]\<rparr>"
vfield_vsingleton: fixes a :: "V"   and b :: "V" shows "\<F>\<^sub>\<circ> (set {\<langle>a, b\<rangle>}) = set {a, b}"
split_max_order: fixes k :: "nat"   and t :: "'a btree" assumes "order k t"   and "t \<noteq> Leaf"   and "nonempty_lasttreebal t" shows "almost_order k (fst (split_max k t))"
const_poly_dvd_iff_dvd_content: fixes c :: "'b"   and p :: "'b poly" shows "([:c:] dvd p) = (c dvd content p)"
Let: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and m :: "nat"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and p :: "String.literal"   and n :: "nat"   and R :: "event_data list set"   and \<psi> :: "('msaux, 'muaux) mformula"   and \<psi>' :: "Formula.formula"   and b :: "nat" assumes "wf_mformula \<sigma> j P V m UNIV \<phi> \<phi>'"   and "wf_mformula \<sigma> j (P(p \<mapsto> Monitor.progress \<sigma> P \<phi>' j)) (V(p \<mapsto> \<lambda>i. {v. length v = m \<and> Formula.sat \<sigma> V v i \<phi>'})) n R \<psi> \<psi>'"   and "{0..<m} \<subseteq> fv \<phi>'"   and "b \<le> m"   and "m = Formula.nfv \<phi>'" shows "wf_mformula \<sigma> j P V n R (MLet p m \<phi> \<psi>) (formula.Let p \<phi>' \<psi>')"
parCasesF': fixes P :: "pi"   and Q :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<parallel> Q \<longmapsto> \<alpha> \<prec> P'" obtains P :: "pi" and \<alpha>' :: "freeRes" and P'a :: "pi" and Qa :: "pi" where "Pa__ \<parallel> Q = Pb__ \<parallel> Qa__" and "\<alpha> \<prec> P' = \<alpha>'__ \<prec> P'a__ \<parallel> Qa__" and "Pb__ \<longmapsto> \<alpha>'__ \<prec> P'a__"   | Qa :: "pi" and \<alpha>' :: "freeRes" and Q' :: "pi" and P :: "pi" where "Pa__ \<parallel> Q = Pc__ \<parallel> Qaa__" and "\<alpha> \<prec> P' = \<alpha>' \<prec> Pc__ \<parallel> Q'__" and "Qaa__ \<longmapsto> \<alpha>' \<prec> Q'__"   | P :: "pi" and a :: "name" and b :: "name" and P'a :: "pi" and Qa :: "pi" and Q' :: "pi" where "Pa__ \<parallel> Q = Pd__ \<parallel> Qab__" and "\<alpha> \<prec> P' = \<tau> \<prec> P'aa__ \<parallel> Q'a__" and "Pd__ \<longmapsto> a__<b__> \<prec> P'aa__" and "Qab__ \<longmapsto> a__[b__] \<prec> Q'a__"   | P :: "pi" and a :: "name" and b :: "name" and P'a :: "pi" and Qa :: "pi" and Q' :: "pi" where "Pa__ \<parallel> Q = Pe__ \<parallel> Qac__" and "\<alpha> \<prec> P' = \<tau> \<prec> P'ab__ \<parallel> Q'b__" and "Pe__ \<longmapsto> aa__[b] \<prec> P'ab__" and "Qac__ \<longmapsto> aa__<b> \<prec> Q'b__"   | P :: "pi" and a :: "name" and x :: "name" and P'a :: "pi" and Qa :: "pi" and Q' :: "pi" where "Pa__ \<parallel> Q = Pf__ \<parallel> Qad__" and "\<alpha> \<prec> P' = \<tau> \<prec> <\<nu>x__>(P'ac__ \<parallel> Q'c__)" and "Pf__ \<longmapsto> ab__<x__> \<prec> P'ac__" and "Qad__ \<longmapsto> ab__<\<nu>x__> \<prec> Q'c__" and "x__ \<sharp> Pf__" and "x__ \<sharp> Qad__" and "x__ \<noteq> ab__"   | P :: "pi" and a :: "name" and x :: "name" and P'a :: "pi" and Qa :: "pi" and Q' :: "pi" where "Pa__ \<parallel> Q = P \<parallel> Qa" and "\<alpha> \<prec> P' = \<tau> \<prec> <\<nu>x>(P'a \<parallel> Q')" and "P \<longmapsto> a<\<nu>x> \<prec> P'a" and "Qa \<longmapsto> a<x> \<prec> Q'" and "x \<sharp> P" and "x \<sharp> Qa" and "x \<noteq> a"
inductChainSimps: fixes xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "xvec \<sharp>* inputChain xvec N P"
norm_vector_1: fixes x :: "('c, 1) vec" shows "norm x = norm (x $ 1)"
lim: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and J :: "nat set"   and X :: "(nat \<Rightarrow> 'b) set" assumes "Ionescu_Tulcea P M"   and "finite J"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emeasure (projective_family.lim UNIV (Ionescu_Tulcea.CI P M) M) (prod_emb UNIV M J X) = emeasure (Ionescu_Tulcea.CI P M J) X"
distinct_union: fixes xs :: "'a list"   and ys :: "'a list" shows "distinct (List.union xs ys) = distinct ys"
xx_1: shows "x \<in> set (if iiface r = ifaceAny then [None] else List.maps (\<lambda>i. if match_iface (iiface r) i then [Some i] else []) ifs)"
reparam_def: fixes \<gamma>1 :: "real \<Rightarrow> 'a"   and \<gamma>2 :: "'b \<Rightarrow> 'a" shows "reparam \<gamma>1 \<gamma>2 \<equiv> \<exists>\<phi>. (\<forall>x\<in>{0..1}. \<gamma>1 x = (\<gamma>2 \<circ> \<phi>) x) \<and> \<phi> piecewise_C1_differentiable_on {0..1} \<and> \<phi> 0 = (0::'b) \<and> \<phi> 1 = (1::'b) \<and> bij_betw \<phi> {0..1} {0::'b..1::'b} \<and> \<phi> -` {0::'b..1::'b} \<subseteq> {0..1} \<and> (\<forall>x\<in>{0::'b..1::'b}. finite (\<phi> -` {x}))"
assms_2: shows "dom \<pi>\<^sub>\<UU> = dom \<pi>\<^sub>\<A>"
drop_Suc_Cons: fixes n :: "nat"   and x :: "'e"   and xs :: "'e list" shows "drop (Suc n) (x # xs) = drop n xs"
subst_exi_num: fixes \<phi> :: "'fmla"   and x :: "'var"   and y :: "'var"   and n :: "'trm" assumes "\<phi> \<in> fmla"   and "x \<in> var"   and "y \<in> var"   and "n \<in> num"   and "x \<noteq> y" shows "subst (exi x \<phi>) n y = exi x (subst \<phi> n y)"
bpp_closed: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "assoc_bpp A f"   and "x \<in> addition_set f A"   and "y \<in> addition_set f A" shows "x \<^sub>f+ y \<in> addition_set f A"
strategyproofness_2: shows "(pmf (sds R25) d + pmf (sds R25) c < pmf (sds R28) d + pmf (sds R28) c \<or> pmf (sds R25) d + (pmf (sds R25) c + pmf (sds R25) a) < pmf (sds R28) d + (pmf (sds R28) c + pmf (sds R28) a)) \<or> pmf (sds R25) d + pmf (sds R25) c = pmf (sds R28) d + pmf (sds R28) c \<and> pmf (sds R25) d + (pmf (sds R25) c + pmf (sds R25) a) = pmf (sds R28) d + (pmf (sds R28) c + pmf (sds R28) a)"
degree_sum_less: fixes n :: "nat"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "0 < n"   and "\<And>x. x \<in> A \<Longrightarrow> degree (f x) < n" shows "degree (sum f A) < n"
load_size: fixes m :: "mem"   and n :: "nat"   and off :: "nat"   and l :: "nat" shows "(load m n off l = None) = (mem_size m < off + n + l)"
map_add_None: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b" shows "((m ++ n) k = None) = (n k = None \<and> m k = None)"
simps_8: fixes f :: "'b \<Rightarrow> 'a" shows "map f [] = []"
Inr_not_Inl: fixes b :: "'e"   and a :: "'d" shows "Inr b \<noteq> Inl a"
lub_spmf_def: fixes Y :: "'c spmf set" shows "lub_spmf Y = embed_spmf (\<lambda>x. enn2real (\<Squnion>p\<in>Y. ennreal (spmf p x)))"
parallel_converter2_id_id: fixes \<I>1 :: "('h, 'j) \<I>"   and \<I>2 :: "('i, 'k) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2,\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<turnstile>\<^sub>C 1\<^sub>C |\<^sub>= 1\<^sub>C \<sim> 1\<^sub>C"
cases: fixes a1 :: "'f"   and a2 :: "'g list"   and a3 :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set" assumes "(a1, a2, a3) \<in> trcl t" obtains c :: "'f" where "a1 = cb__" and "a2 = []" and "a3 = cb__"   | c :: "'f" and a :: "'g" and c' :: "'f" and w :: "'g list" and c'' :: "'f" where "a1 = c" and "a2 = a # w" and "a3 = c''" and "(c, a, c') \<in> t" and "(c', w, c'') \<in> trcl t"
assms_3: shows "p \<le> \<^bold>|w\<^bold>|"
intersection_number_def: fixes b1 :: "'b set"   and b2 :: "'b set" shows "b1 |\<inter>| b2 \<equiv> int (card (b1 \<inter> b2))"
gen_ge_dim: fixes A :: "'c set" assumes "finite A"   and "A \<subseteq> carrier V"   and "V.module.gen_set A" shows "V.dim \<le> card A"
borel_measurable_iff_halfspace_le: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. f w \<bullet> i \<le> a} \<in> sets M)"
count_inI: fixes M :: "'d multiset"   and x :: "'d" assumes "count M x = 0 \<Longrightarrow> False" shows "x \<in># M"
lossless_R1: fixes m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool" shows "lossless_spmf (R1_14_OT m1 m2)"
mult_def: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "Matrix_Tensor.mult id f \<equiv> ((\<forall>a b. f a b = f b a) \<and> (\<forall>a b c. f (f a b) c = f a (f b c))) \<and> (\<forall>x. f id x = x) \<and> (\<forall>x. f x id = x)"
elims_1: fixes x :: "'b extN"   and xa :: "'b extN"   and y :: "bool" assumes "(x \<le> xa) = y" obtains uu_ :: "'b extN" where "xa__ = extN.Bot" and "xa = uu_" and "ya__"   | uv_ :: "'b" where "xa__ = extN.Val uv_" and "xa = extN.Bot" and "\<not> ya__"   | x :: "'b" and y :: "'b" where "xa__ = extN.Val xb__" and "xa = extN.Val yb__" and "ya__ = (xb__ \<le> yb__)"   | uw_ :: "'b" and ux_ :: "nat" where "xa__ = extN.Val uw_" and "xa = extN.N ux_" and "ya__"   | uy_ :: "'b" where "xa__ = extN.Val uy_" and "xa = extN.Top" and "ya__"   | uz_ :: "nat" where "xa__ = extN.N uz_" and "xa = extN.Bot" and "\<not> ya__"   | va_ :: "nat" and vb_ :: "'b" where "xa__ = extN.N va_" and "xa = extN.Val vb_" and "\<not> ya__"   | x :: "nat" and y :: "nat" where "xa__ = extN.N x" and "xa = extN.N y" and "ya__ = (x \<le> y)"   | vc_ :: "nat" where "xa__ = extN.N vc_" and "xa = extN.Top" and "ya__"   | "xa__ = extN.Top" and "xa = extN.Bot" and "\<not> ya__"   | vd_ :: "'b" where "xa__ = extN.Top" and "xa = extN.Val vd_" and "\<not> ya__"   | ve_ :: "nat" where "xa__ = extN.Top" and "xa = extN.N ve_" and "\<not> ya__"   | "xa__ = extN.Top" and "xa = extN.Top" and "ya__"
pi_image: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and it' :: "('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "proper_it it it'" shows "proper_it (set_iterator_image g it) (set_iterator_image g it')"
trans: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "c1.le b a"   and "c1.le c b" shows "c1.le c a"
CURRY_def: fixes R :: "(('g \<times> 'h \<Rightarrow> 'i) \<times> ('j \<times> 'k \<Rightarrow> 'l)) set" shows "CURRY R \<equiv> {(f, g). (uncurry f, uncurry g) \<in> R}"
gbinomial_int_pochhammer: fixes a :: "int"   and k :: "nat" shows "a gchoose k = (- 1) ^ k * pochhammer (- a) k div fact k"
Diff: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and a :: "'b set"   and b :: "'b set" assumes "ring_of_sets \<Omega> M"   and "a \<in> M"   and "b \<in> M" shows "a - b \<in> M"
power_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq a b"   and "less_eq zero a" shows "less_eq (power.power one times a n) (power.power one times b n)"
write_cpu_tt_err_mode: fixes s :: "'b sparc_state"   and w :: "word8" shows "err_mode_val s = err_mode_val (snd (fst (write_cpu_tt w s)))"
that_1: fixes s :: "'a set" assumes "s \<in> S" shows "higher_differentiable_on s f n"
cases: fixes x :: "'a \<times> 'b \<times> 'a" obtains q :: "'a" and \<nu> :: "'b" and q' :: "'a" where "x = (q, \<nu>, q')"
State_works: fixes t :: "nat"   and k :: "nat" assumes "valid_state_var (State t k)" shows "dimacs_to_var (var_to_dimacs (State t k)) = State t k"
f'_def: fixes x1 :: "'a"   and x2 :: "'b" shows "f' x1 x2 = f (x1, x2)"
strict_implies_order: fixes a :: "'a"   and b :: "'a" assumes "b \<succ> a" shows "b \<succeq> a"
not_zero_ia_A: shows "\<not> is_zero_row ia A"
Suc_1: fixes k :: "nat"   and m :: "nat" assumes "k < K" shows "dyad_rec_dom (b, l, r, real m / 2 ^ k)"
assms_4: shows "fmrestrict_set vs1 fma \<subseteq>\<^sub>f s"
delete_edges_simp2: fixes G :: "'b graph"   and E :: "('b \<times> 'b) set" shows "delete_edges G E = \<lparr>nodes = nodes G, edges = edges G - E\<rparr>"
OO_def: fixes R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
Variable: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "map_of \<Gamma> x = Some e"   and "delete x \<Gamma> : e \<Down>\<^bsub>x # L\<^esub> \<Delta> : z" shows "\<Gamma> : Var x \<Down>\<^bsub>L\<^esub> (x, z) # \<Delta> : z"
ord_of_nat_succ_vempty: shows "ZFC_in_HOL.succ 0 = 1\<^sub>\<nat>"
on_curvep_iff_on_curve: fixes a :: "'b"   and b :: "'b"   and p :: "'b \<times> 'b \<times> 'b" shows "on_curvep a b p = on_curve a b (make_affine p)"
Syntax_with_Numerals_axioms: shows "Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num"
ctr_2: fixes A :: "'c llist set" assumes "\<not> (\<forall>x\<in>A. lnull x)" shows "lSup A = LCons (THE x. x \<in> lhd ` (A \<inter> {xs. \<not> lnull xs})) (lSup (ltl ` (A \<inter> {xs. \<not> lnull xs})))"
bounds_of_interval_inject: fixes x :: "'b interval"   and y :: "'b interval" shows "(bounds_of_interval x = bounds_of_interval y) = (x = y)"
subst_subst: fixes t :: "('a, 'd) Term.term"   and \<sigma> :: "'d \<Rightarrow> ('a, 'c) Term.term"   and \<tau> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "t \<cdot> \<sigma> \<cdot> \<tau> = t \<cdot> \<sigma> \<circ>\<^sub>s \<tau>"
obj_ty_CInst: fixes C :: "qtname"   and vs :: "vn \<Rightarrow> val option" shows "obj_ty \<lparr>tag = CInst C, values = vs\<rparr> = Class C"
Ex1_Normal_form: fixes f :: "int \<Rightarrow> int" shows "(f \<in> CofGroups.Ex1) = (\<exists>k. \<forall>n. f n = n + k)"
cone_decomp_shift_list: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and x :: "'x" assumes "valid_decomp X ps"   and "cone_decomp T ps"   and "(h, U) \<in> set ps"   and "x \<in> U" shows "cone_decomp T (shift_list (h, U) x ps)"
assms_4: shows "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}"
scale_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and a :: "'d"   and v :: "'e \<times> 'f" assumes "module_prod s1 s2" shows "module_prod.scale s1 s2 a v = (s1 a (fst v), s2 a (snd v))"
C_def: shows "C = P1 * A1 @\<^sub>c P1 * A2 * Q1"
partition_on_mset_distinct_disjoint: fixes A :: "'a multiset"   and P :: "'a multiset multiset"   and p1 :: "'a multiset"   and p2 :: "'a multiset" assumes "partition_on_mset A P"   and "distinct_mset A"   and "p1 \<in># P"   and "p2 \<in># remove1_mset p1 P" shows "p1 \<inter># p2 = {#}"
select_correct_factor_main: fixes f :: "nat \<Rightarrow> rat \<times> rat"   and x :: "real"   and i :: "nat"   and a :: "'a"   and todo :: "(int poly \<times> root_info) list"   and old :: "(int poly \<times> root_info) list"   and l :: "rat"   and r :: "rat"   and n :: "nat"   and q :: "int poly"   and ri_fin :: "root_info"   and l_fin :: "rat"   and r_fin :: "rat" assumes "converges_to f x"   and "local.at_step f i a"   and "local.select_correct_factor_main a todo old l r n = ((q, ri_fin), l_fin, r_fin)"   and "bnd_get a = (l, r)"   and "\<And>q ri. (q, ri) \<in> set todo \<union> set old \<Longrightarrow> root_info_cond ri q"   and "\<And>q ri. (q, ri) \<in> set todo \<union> set old \<Longrightarrow> q \<noteq> 0"   and "\<exists>q. q \<in> fst ` set todo \<union> fst ` set old \<and> poly (real_of_int_poly q) x = 0"   and "distinct (map fst (todo @ old))"   and "\<And>q ri. (q, ri) \<in> set old \<Longrightarrow> root_info.l_r ri l r \<noteq> 0"   and "\<And>x. \<exists>q. q \<in> fst ` set todo \<union> fst ` set old \<and> poly (real_of_int_poly q) x = 0 \<Longrightarrow> \<exists>!q. q \<in> fst ` set todo \<union> fst ` set old \<and> poly (real_of_int_poly q) x = 0"   and "n = (\<Sum>(q, ri)\<leftarrow>old. root_info.l_r ri l r)" shows "(\<exists>!x. root_cond (q, l_fin, r_fin) x) \<and> (q, ri_fin) \<in> set todo \<union> set old \<and> x = real_of_1 (q, l_fin, r_fin)"
count_inject: fixes x :: "'k multiset"   and y :: "'k multiset" shows "(count x = count y) = (x = y)"
ip_constant: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, uu_). ip \<xi> = i)"
Max_le_Min_conv_singleton: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "(Max A \<le> Min A) = (\<exists>x. A = {x})"
monotoneI: fixes orda :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and ordb :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h" assumes "\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)" shows "monotone orda ordb f"