hcju/isabelleps huggingface dataset:
Vfrom_Ord_bundle_2: fixes A :: "V"   and i :: "V" assumes "A = A"   and "i = i" shows "Ord (rank i)"
drop_r_p_in_\<PP>: shows "drop r p \<in> \<PP>"
EH\<^sub>N_sum': shows "EH\<^sub>N n = infsetsum f {1..}"
bdd_below_Un: fixes A :: "'b multiset set"   and B :: "'b multiset set" shows "subset_mset.bdd_below (A \<union> B) = (subset_mset.bdd_below A \<and> subset_mset.bdd_below B)"
assms_2: shows "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"
IH: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "set xs = set (take j xs) \<union> set (drop i xs)"
prems_6: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"
has_ell2_norm_C0: fixes \<psi> :: "'a \<Rightarrow> complex" assumes "has_ell2_norm \<psi>" shows "has_ell2_norm (C0 \<psi>)"
DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"   and f' :: "real \<Rightarrow> nat \<Rightarrow> real"   and x0 :: "real"   and a :: "real"   and b :: "real"   and L :: "nat \<Rightarrow> real" assumes "\<And>n. ((\<lambda>x. f x n) has_real_derivative f' x0 n) (at x0)"   and "\<And>x. x \<in> {a<..<b} \<Longrightarrow> summable (f x)"   and "x0 \<in> {a<..<b}"   and "summable (f' x0)"   and "summable L"   and "\<And>n x y. \<lbrakk>x \<in> {a<..<b}; y \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>" shows "((\<lambda>x. suminf (f x)) has_real_derivative suminf (f' x0)) (at x0)"
not_irreducible_one: shows "\<not> irreducible (1::'a)"
map_scale_distrib_right: fixes k :: "'b"   and l :: "'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "(k + l) \<cdot> t = k \<cdot> t + l \<cdot> t"
vector_space_assms_4: fixes x :: "'b" shows "(1::'a) *s x = x"
xivp_3: shows "X \<in> U (X t\<^sub>0) \<rightarrow> S"
boundOutputInduct: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'"   and "x \<sharp> P"   and "\<And>P a x P' b C. \<lbrakk>P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C ([b\<frown>b]P) a x P'"   and "\<And>P a x P' b c C. \<lbrakk>P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'; \<And>C. F C P a x P'; b \<noteq> c\<rbrakk> \<Longrightarrow> F C ([b\<noteq>c]P) a x P'"   and "\<And>P a x P' C. \<lbrakk>P \<longmapsto>\<^sub>l a[x] \<prec>\<^sub>l P'; a \<noteq> x\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a x P'"   and "\<And>P Q a x P' C. \<lbrakk>P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x P'"   and "\<And>P Q a x Q' C. \<lbrakk>Q \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l Q'; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x Q'"   and "\<And>P P' Q a x C. \<lbrakk>P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'; x \<sharp> Q; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P' \<parallel> Q)"   and "\<And>P Q Q' a x C. \<lbrakk>Q \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l Q'; x \<sharp> P; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P \<parallel> Q')"   and "\<And>P P' a x y C. \<lbrakk>P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'; y \<noteq> a; y \<noteq> x; y \<sharp> C; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (<\<nu>y>P) a x (<\<nu>y>P')"   and "\<And>P a x P' C. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'; \<And>C. F C (P \<parallel> !P) a x P'\<rbrakk> \<Longrightarrow> F C (!P) a x P'" shows "F C P a x P'"
"1_5": fixes q :: "nat"   and u :: "char list \<Rightarrow> int" assumes "(WHILE b_ DO c_, s2_) \<Rightarrow> q \<Down> u" shows "u = s3_ \<and> y_ = q"
cmf: shows "continuous_map (subtopology (Euclidean_space m) S) (Euclidean_space n) f"
IH_2: shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, {#}"
WTrtCallNT: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and es :: "char list exp list"   and Ts :: "ty list"   and M :: "char list"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e : NT"   and "P,E,h,sh \<turnstile> es [:] Ts" shows "P,E,h,sh \<turnstile> e\<bullet>M(es) : T"
groupF_equality: fixes f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "\<forall>A\<in>set (groupF f xs). \<forall>a1\<in>set A. \<forall>a2\<in>set A. f a1 = f a2"
poincare_collinear3_between: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc"   and "poincare_collinear {u, v, w}" shows "poincare_between u v w \<or> poincare_between u w v \<or> poincare_between v u w"
in_quotientP_imp_closed: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set"   and x :: "'c"   and y :: "'c" assumes "equivP P \<phi>"   and "(P /// \<phi>) X"   and "x \<in> X"   and "\<phi> x y" shows "y \<in> X"
regular_event_cannot_induce_snapshot: fixes c :: "('p, 's, 'm) configuration"   and p :: "'p"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration" assumes "\<not> ps c p \<noteq> None"   and "c \<turnstile> ev \<mapsto> c'" shows "regular_event ev \<longrightarrow> \<not> ps c' p \<noteq> None"
conjugate_exponent_ennrealI_2: fixes p :: "ennreal"   and q :: "ennreal" assumes "1 / p + 1 / q = 1" shows "1 \<le> q"
observable_def: fixes ob :: "'a \<Rightarrow> 'b"   and P :: "'a set" shows "observable ob P \<equiv> \<forall>s s'. ob s = ob s' \<longrightarrow> s' \<in> P \<longrightarrow> s \<in> P"
eq_\<I>_gpv_relcompp: fixes A :: "'h \<Rightarrow> 'l \<Rightarrow> bool"   and A' :: "'l \<Rightarrow> 'k \<Rightarrow> bool"   and \<I> :: "('i, 'j) \<I>" shows "eq_\<I>_gpv (A OO A') \<I> = eq_\<I>_gpv A \<I> OO eq_\<I>_gpv A' \<I>"
intros: fixes gpv :: "('f, 'd, 'e) gpv"   and \<I> :: "('d, 'e) \<I>" assumes "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> finite_gpv \<I> (c input)" shows "finite_gpv \<I> gpv"
minf_5: fixes t :: "'c" shows "\<exists>z. \<forall>x<z. (x < t) = True"
mono: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and gt' :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "gt \<le> gt'" shows "ext gt \<le> ext gt'"
assert_simplex_ok: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "'a simplex_state"   and j :: "'a"   and s' :: "'a simplex_state" assumes "invariant_simplex cs J s"   and "assert_simplex j s = Inr s'" shows "invariant_simplex cs (insert j J) s'"
leg0_composite_1: shows "HH\<mu>\<nu>H\<pi>\<rho>.dom.leg0 = \<rho>.leg0 \<cdot> \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0\<^sub>0"
zipper_of_tree_def: fixes t :: "'a rose_tree" shows "zipper_of_tree t \<equiv> ([], t)"
filter_eq_val_filter_tm: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "Time_Monad.val (filter_tm P xs) = filter P xs"
infinite_arbitrarily_large: fixes A :: "'q set"   and n :: "nat" assumes "infinite A" shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
inv_stereographic_def: shows "inv_stereographic \<equiv> map_fun rep_complex_homo Abs_riemann_sphere inv_stereographic_hcoords_r3"
lincomb_list_Cons: fixes c :: "nat \<Rightarrow> 'a"   and v :: "'a vec"   and vs :: "'a vec list" shows "m.lincomb_list c (v # vs) = c 0 \<cdot>\<^sub>v v + m.lincomb_list (c \<circ> Suc) vs"
wf_Restr: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wf r" shows "wf (Restr r A)"
lexordp_into_lexordp_eq: fixes less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list" assumes "ord.lexordp less xs ys" shows "ord.lexordp_eq less xs ys"
zhmset_pos_neg_partition: fixes M :: "zhmultiset" shows "M = zhmset_of (hmset_pos M) - zhmset_of (hmset_neg M)"
u_cancellation: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice" assumes "dist_over_sup (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> F\<^sup>#) (x::'b::refinement_lattice) \<sqsubseteq> x"
convergent_prod_iff_shift: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "convergent_prod (\<lambda>i. f (i + n)) = convergent_prod f"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G H \<tau>"
assms_2: shows "\<alpha> \<in> {0<..<pi}"
splus_mono: fixes v :: "'t"   and w :: "'t"   and t :: "'a" assumes "v \<preceq>\<^sub>t w" shows "t \<oplus> v \<preceq>\<^sub>t t \<oplus> w"
map_closed: fixes a :: "'b" assumes "a \<in> \<FF> (fst y)" shows "\<rho> (fst y) z a \<in> \<FF> z"
insert_dom_image: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "i < Suc n"   and "j < Suc n"   and "f ` {0..<n} = {0..<Suc n} - {j}" shows "insert_dom f i j ` {0..<Suc n} = {0..<Suc n}"
ExcessNotAt_def: fixes g :: "graph"   and v_opt :: "nat option" shows "ExcessNotAt g v_opt \<equiv> let ps = ExcessTable g (vertices g) in case v_opt of None \<Rightarrow> ExcessNotAtRec ps g | Some v \<Rightarrow> ExcessNotAtRec (deleteAround g v ps) g"
Cauchy_Schwarz_ineq2: fixes x :: "'a"   and y :: "'a" shows "cmod (cinner x y) \<le> norm x * norm y"
sums: shows "(\<lambda>k. x ^ n * (x ^ k / fact (n + k))) sums (exp x - approx)"
\<sigma>_def: shows "\<sigma> \<equiv> \<pi> \<circ> inv \<phi>"
axioms_1: fixes H :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subring H R" shows "subgroup H (add_monoid R)"
some_eq: shows "(\<lambda>v'. \<chi>i. v' *\<^sub>R ODE_sem I ODE (sol s) $ i) = (\<lambda>v'. v' *\<^sub>R ODE_sem I ODE (sol s))"
primroot_unique: fixes u :: "'b list"   and r :: "'b list" assumes "u \<in>\<^sub>p r *" shows "\<rho> u = r"
added_ids_Deliver_Add_same_collapse: fixes i :: "'c"   and j :: "'c"   and e :: "'b" shows "added_ids [Deliver (i, Add j e)] e = [j]"
div: shows "\<R>_div (smap (\<lambda>u. [u]\<^sub>\<R>) xs)"
simps_1: fixes y :: "'a"   and x :: "'a" shows "local.efficient_power y x 0 = y"
tanh_real_zero_iff: fixes x :: "real" shows "(tanh x = 0) = (x = 0)"
subprob_not_empty: fixes p :: "'c spmf" shows "space (measure_spmf p) \<noteq> {}"
infinite_coin_toss_space_axioms: shows "infinite_coin_toss_space p M"
absolute_neighbourhood_extensor_imp_ANR: fixes S :: "'a set" assumes "\<And>f U T. \<lbrakk>continuous_on T f; f ` T \<subseteq> S; closedin (top_of_set U) T\<rbrakk> \<Longrightarrow> \<exists>V g. T \<subseteq> V \<and> openin (top_of_set U) V \<and> continuous_on V g \<and> g ` V \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)" shows "ANR S"
geq: shows "convex hull \<Union> (K ` I) \<subseteq> K0"
v11_vlrestriction_vcomp: fixes f :: "V"   and A :: "V"   and g :: "V" assumes "v11 (f \<restriction>\<^sup>l\<^sub>\<circ> A)"   and "v11 (g \<restriction>\<^sup>l\<^sub>\<circ> f `\<^sub>\<circ> A)" shows "v11 ((g \<circ>\<^sub>\<circ> f) \<restriction>\<^sup>l\<^sub>\<circ> A)"
assms_1: shows "h \<circ> l = l' \<circ> h"
simps_9: fixes f1 :: "'d"   and f2 :: "'d"   and f3 :: "'e \<Rightarrow> 'e ifex \<Rightarrow> 'e ifex \<Rightarrow> 'd" shows "(case Falseif of Trueif \<Rightarrow> f1 | Falseif \<Rightarrow> f2 | IF x xa xb \<Rightarrow> f3 x xa xb) = f2"
step_Normal_elim_cases_8: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and g :: "'e set"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Guard f g c # cs, css, Normal s) \<rightarrow> u" obtains "u = (c # cs, css, Normal s)" and "s \<in> g"   | "u = (cs, css, Fault f)" and "s \<notin> g"
eq_sum: fixes I :: "'a set"   and p :: "'a \<Rightarrow> 'b" assumes "finite I" shows "sum' p I = sum p I"
INT_greatest: fixes A :: "'b set"   and C :: "'c set"   and B :: "'b \<Rightarrow> 'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x" shows "C \<subseteq> \<Inter> (B ` A)"
elims: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and xa :: "'c"   and xb :: "'a \<times> 'd"   and y :: "'a \<times> 'b" assumes "backtrack_cs x xa xb = y" obtains backt :: "'c \<Rightarrow> 'd \<Rightarrow> 'b" and c :: "'c" and cs :: "'a" and s :: "'d" where "x = backt" and "xa = c" and "xb = (cs, s)" and "y = (cs, backt c s)"
Q_invar_when_discharged2: fixes u :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and Q :: "nat list" assumes "QD_invar u f Q"   and "excess f u \<noteq> (0::'capacity)" shows "Q_invar f (Q @ [u])"
zip_null_iff: fixes xs :: "'b list"   and k :: "nat"   and ys :: "'c list" assumes "length xs = k"   and "length ys = k"   and "zip xs ys = []" shows "xs = [] \<and> ys = []"
funionE: fixes c :: "'b"   and A :: "'b fset"   and B :: "'b fset" assumes "c |\<in>| A |\<union>| B" obtains "c |\<in>| A"   | "c |\<in>| B"
key_def: fixes \<eta> :: "nat" shows "key \<eta> \<equiv> spmf_of_set (nlists UNIV \<eta>)"
power_add: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m + n) = times (power.power one times a m) (power.power one times a n)"
simps: fixes I :: "('sf, 'sc, 'sz) interp"   and S :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" shows "seq_sem I S = fml_sem I (seq2fml S)"
mult2_altI: fixes X :: "'b multiset"   and X1 :: "'b multiset"   and X2 :: "'b multiset"   and Y :: "'b multiset"   and Y1 :: "'b multiset"   and Y2 :: "'b multiset"   and ns :: "('b \<times> 'b) set"   and b :: "bool"   and s :: "('b \<times> 'b) set" assumes "X = X1 + X2"   and "Y = Y1 + Y2"   and "(X1, Y1) \<in> multpw ns"   and "b \<or> Y2 \<noteq> {#}"   and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)" shows "(X, Y) \<in> mult2_alt b ns s"
scalar_prod_smult_left: fixes w :: "'c vec"   and v :: "'c vec"   and k :: "'c" assumes "dim_vec w = dim_vec v" shows "(k \<cdot>\<^sub>v w) \<bullet> v = k * (w \<bullet> v)"
getNeg_set: fixes a :: "'a"   and x :: "'a negation_type list" shows "(Neg a \<in> set x) = (a \<in> set (getNeg x))"
is_red_monomial_iff: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and v :: "'t"   and c :: "'c" shows "is_red F (monomial c v) = (c \<noteq> (0::'c) \<and> (\<exists>f\<in>F. f \<noteq> 0 \<and> lt f adds\<^sub>t v))"
star_of_add: fixes x :: "'a"   and y :: "'a" shows "star_of (x + y) = star_of x + star_of y"
refine: fixes s0 :: "nat"   and l :: "nat dlist list" shows "RETURN (pick_sum_code s0 l) \<le> pick_sum_impl s0 l"
nn_integral_monotone_convergence_simple: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "incseq f"   and "\<And>i. simple_function M (f i)" shows "(SUP i. integral\<^sup>S M (f i)) = \<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M"
valid_cone_decomp_subset_Polys: fixes X :: "'x set"   and 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" assumes "valid_decomp X ps"   and "cone_decomp T ps" shows "T \<subseteq> P[X]"
Limsup_mono: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x \<le> g x" shows "Limsup F f \<le> Limsup F g"
proj2_Col_permute_1: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2" assumes "proj2_Col a b c" shows "proj2_Col a c b"
L_def: fixes x :: "real"   and y :: "real"   and n :: "nat" assumes "x \<in> {a<..<b}"   and "y \<in> {a<..<b}" shows "\<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>"
last_ladder_\<gamma>: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and a :: "symbol list" assumes "is_ladder D L"   and "ladder_last_n L = length D" shows "ladder_\<gamma> a D L (length L - Suc 0) = Derive a D"
simps_1: fixes t :: "'b bro" shows "tree (N1 t) = t"
fmrel_iff: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'f \<rightharpoonup>\<^sub>f 'd"   and n :: "'f \<rightharpoonup>\<^sub>f 'e" shows "fmrel R m n = (\<forall>x. rel_option R (fmlookup m x) (fmlookup n x))"
f: shows "f = (\<lambda>x. card (no_better_than x carrier relation))"
vars_subset_if_in_strand_ik2: fixes t :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
eta_reds_appR: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* t" shows "u $ s \<rightarrow>\<^sub>\<eta>\<^sup>* u $ t"
elims: fixes x :: "'a SDG_node"   and y :: "'a" assumes "parent_node x = y" obtains n :: "'a" where "x = CFG_node n" and "y = n"   | m :: "'a" and xa :: "nat" where "x = Formal_in (m__, xa__)" and "y = m__"   | m :: "'a" and xa :: "nat" where "x = Formal_out (ma__, xaa__)" and "y = ma__"   | m :: "'a" and xa :: "nat" where "x = Actual_in (mb__, xab__)" and "y = mb__"   | m :: "'a" and xa :: "nat" where "x = Actual_out (m, xa)" and "y = m"
intro: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> less x y \<or> x = y \<or> less y x" shows "semiconnex A less"
countably_additiveI: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" assumes "\<And>A. \<lbrakk>range A \<subseteq> M; disjoint_family A; \<Union> (range A) \<in> M\<rbrakk> \<Longrightarrow> (\<Sum>i. f (A i)) = f (\<Union> (range A))" shows "countably_additive M f"
M_in_subprob: fixes M :: "'b pmf" shows "measure_pmf M \<in> space (subprob_algebra (measure_pmf M))"
det_dependent_rows: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "vec.dependent (rows A)" shows "det A = (0::'a)"
hf_of_bool_def: fixes b :: "bool" shows "hf_of b \<equiv> if b then 1 else 0"
basic3: shows "\<forall>u v. from u to v = positionVector v \<ominus> positionVector u"
ep_pair_sprod: shows "ep_pair sprod_emb sprod_prj"
IH_2: assumes "inv_max_hi r"   and "Sorted_Less.sorted (Tree2.inorder r)" shows "search r x = has_overlap (Tree2.set_tree r) x"
times_Nat_eval: fixes x :: "nat discr"   and y :: "nat discr" shows "box\<cdot>x * box\<cdot>y = box\<cdot>(x * y)"
d_ne: fixes k :: "nat" shows "d k \<noteq> {}"
intensional_simps_7: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "P \<noteq> Q \<equiv> P = \<not> Q"
rel_star_contr: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'b) set" shows "X\<^sup>* ; Y = (\<Union>i. rel_d.power X i ; Y)"
card_gt_1D: fixes A :: "'a set" assumes "1 < card A" shows "\<exists>x y. x \<in> A \<and> y \<in> A \<and> x \<noteq> y"
multisetUnionLessMono1: fixes r :: "('b \<times> 'b) set"   and a1 :: "'b multiset"   and a2 :: "'b multiset"   and b :: "'b multiset" assumes "trans r"   and "(a1, a2) \<in> mult r" shows "(a1 + b, a2 + b) \<in> mult r"
simps_29: fixes f1 :: "b_e \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "cl \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> e list \<Rightarrow> e list \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> v list \<Rightarrow> e list \<Rightarrow> 'a"   and x51 :: "nat"   and x52 :: "nat"   and x53 :: "v list"   and x54 :: "e list" shows "(case Local x51 x52 x53 x54 of $x \<Rightarrow> f1 x | Trap \<Rightarrow> f2 | Callcl x \<Rightarrow> f3 x | Label x xa xb \<Rightarrow> f4 x xa xb | Local x xa xb xc \<Rightarrow> f5 x xa xb xc) = f5 x51 x52 x53 x54"
has_derivative_transform_within_ivl: fixes b :: "real"   and a :: "real"   and s :: "real set"   and f :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b"   and x :: "real"   and f' :: "real \<Rightarrow> 'b" assumes "0 < b"   and "\<forall>x\<in>{a..b} - s. f x = g x"   and "x \<in> {a..b} - s"   and "(f has_derivative f') (at x within {a..b} - s)" shows "(g has_derivative f') (at x within {a..b} - s)"
intros_1: fixes e :: "e list"   and e' :: "e list"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "\<lparr>e\<rparr> \<leadsto> \<lparr>e'\<rparr>" shows "\<lparr>s;vs;e\<rparr> \<leadsto>_ i \<lparr>s;vs;e'\<rparr>"
insert_map_upd_same: fixes i :: "nat"   and t :: "'c"   and f :: "nat \<Rightarrow> 'c"   and s :: "'c" shows "(insert_map i t f)(i := s) = insert_map i s f"
while_option_commute: fixes b :: "'c \<Rightarrow> bool"   and b' :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and c :: "'c \<Rightarrow> 'c"   and c' :: "'d \<Rightarrow> 'd"   and s :: "'c" assumes "\<And>s. b s = b' (f s)"   and "\<And>s. b s \<Longrightarrow> f (c s) = c' (f s)" shows "map_option f (while_option b c s) = while_option b' c' (f s)"
measure_empty: fixes M :: "'b measure" shows "Sigma_Algebra.measure M {} = 0"
dim_basis: fixes A :: "'a vec set" assumes "finite A"   and "A.basis A" shows "A.dim = card A"
SG_def: shows "S = (\<lambda>i. sum (G i) {..i})"
possibleD_colors: fixes n :: "nat"   and p :: "'person"   and w :: "'person \<Rightarrow> 'color"   and w' :: "'person \<Rightarrow> 'color"   and p' :: "'person" assumes "local.possible n p w w'"   and "p' \<noteq> p" shows "w' p' = w p'"
cxt_stmt_seq: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and E :: "('Var, 'AExp, 'BExp) Stmt list"   and c' :: "('Var, 'AExp, 'BExp) Stmt" shows "c ;; cxt_to_stmt E c' = cxt_to_stmt (c' # E) c"
subspace_vimage: fixes S :: "'c set" assumes "m2.subspace S" shows "m1.subspace (f -` S)"
le_diff_conv: fixes j :: "nat"   and k :: "nat"   and i :: "nat" shows "(j - k \<le> i) = (j \<le> i + k)"
assms_2: shows "h \<turnstile> g \<rightarrow>\<^sub>e e"
simps_5: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "nat" shows "(case Pkt x11 x12 of Pkt x xa \<Rightarrow> f1 x xa | Newpkt x xa \<Rightarrow> f2 x xa) = f1 x11 x12"
image_poly_eval_ideal: fixes a :: "'d \<Rightarrow> 'c"   and F :: "(('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "poly_eval a ` ideal F = ideal (poly_eval a ` F)"
bal_delete: fixes t :: "'b tree234"   and x :: "'b" assumes "bal t" shows "bal (delete x t)"
bij_iff: fixes f :: "'a \<Rightarrow> 'b" shows "bij f = (\<forall>x. \<exists>!y. f y = x)"
integrable_P: fixes f :: "nat set set \<Rightarrow> real" shows "integrable P f"
inv_solve_left': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(\<ominus> b \<oplus> c = a) = (c = b \<oplus> a)"
zeta_partial_sum_bigo_pos: fixes s :: "real" assumes "0 < s"   and "s \<noteq> 1" shows "(\<lambda>n. (\<Sum>k = 1..n. real k powr - s) - real n powr (1 - s) / (1 - s) - Re (zeta (complex_of_real s))) \<in> O(\<lambda>x. real x powr - s)"
aristo_conversion2: fixes A :: "'b set"   and B :: "'b set" assumes "A I B" shows "B I A"
Integer_arith_strict_2: fixes x :: "Integer" shows "x + \<bottom> = \<bottom>"
is_projection_on_id_minus: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set" assumes "is_projection_on \<pi> M"   and "closed_csubspace M" shows "is_projection_on (id - \<pi>) (orthogonal_complement M)"
basis_exists: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and V :: "'e set" assumes "vector_space scale" obtains B :: "'e set" where "B \<subseteq> V" and "\<not> module.dependent scale B" and "V \<subseteq> module.span scale B" and "card B = vector_space.dim scale V"
cnorm_eq_square: fixes x :: "'b"   and a :: "real" shows "(norm x = a) = (0 \<le> a \<and> \<langle>x, x\<rangle> = complex_of_real (a\<^sup>2))"
LIMSEQ_imp_Cauchy: fixes "open" :: "'b set \<Rightarrow> bool"   and uniformity :: "('b \<times> 'b) filter"   and X :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "class.uniform_space open uniformity"   and "filterlim X (topological_space.nhds open x) sequentially" shows "uniform_space.Cauchy uniformity X"