hcju/isabelleps huggingface dataset:
measurable_compose_countable: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and g :: "'d \<Rightarrow> 'c" assumes "\<And>i. f i \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "(\<lambda>x. f (g x) x) \<in> M \<rightarrow>\<^sub>M N"
finite_enum_subset: fixes X :: "'a set"   and Y :: "'a set" assumes "\<And>i. i < card X \<Longrightarrow> enum X i = enum Y i"   and "finite X"   and "finite Y"   and "card X \<le> card Y" shows "X \<subseteq> Y"
rtrancl_converseI: fixes y :: "'a"   and x :: "'a"   and r :: "('a \<times> 'a) set" assumes "(y, x) \<in> r\<^sup>*" shows "(x, y) \<in> (r\<inverse>)\<^sup>*"
diff_disj: fixes x :: "'e"   and A :: "'e set" assumes "x \<notin> A" shows "A - {x} = A"
hyps: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "vc C Q" shows "\<turnstile>\<^sub>2\<^sub>' {pre C Q} strip C {Q}"
M1: shows "Min {j. j < length a \<and> is_nonterminal (a ! j)} \<in> {j. j < length a \<and> is_nonterminal (a ! j)}"
naturality: fixes f :: "'a" assumes "L.arr f" shows "L.comp (L.map (L.cod f)) (L.map f) = L.comp (L.map f) (L.map (L.dom f))"
ring_simprules_26: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> y = \<ominus>\<^bsub>R\<^esub> (x \<otimes>\<^bsub>R\<^esub> y)"
that: fixes \<pi> :: "'b \<Rightarrow> 'b" assumes "\<pi> permutes A"   and "\<And>x. x \<in> A \<Longrightarrow> g (\<pi> x) = f x" shows "thesis"
assign_eval\<^sub>w_load\<^sub>A: fixes x :: "Eg1.var"   and y :: "Eg1.var"   and mds :: "Mode \<Rightarrow> Eg1.var set"   and mem :: "Eg1.var \<Rightarrow> nat" shows "eval_abv\<^sub>A \<langle>x \<leftarrow> aexp.Load y, mds, mem\<rangle>\<^sub>A \<langle>Stop, mds, mem(x := mem y)\<rangle>\<^sub>A"
subcls_direct: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "class G C = Some c"   and "C \<noteq> Object" shows "G\<turnstile>C\<preceq>\<^sub>C super c"
lookup_empty_empty: fixes t :: "('b, 'c) RBT.rbt" shows "(RBT.lookup t = Map.empty) = (t = RBT.empty)"
that: fixes C :: "nat \<Rightarrow> 'a set" assumes "\<And>n. compact (C n)"   and "\<And>n. C n \<subseteq> S"   and "\<And>n. C n \<subseteq> interior (C (Suc n))"   and "\<Union> (range C) = S"   and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> C n" shows "thesis"
PLS_charles_def: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" shows "PLS_charles s as PROB \<equiv> {length as' |as'. as' \<in> valid_plans PROB \<and> exec_plan s as' = exec_plan s as}"
converse_mono: fixes r :: "('c \<times> 'b) set"   and s :: "('c \<times> 'b) set" shows "(r\<inverse> \<subseteq> s\<inverse>) = (r \<subseteq> s)"
one_dim_iso_add: fixes a :: "'b"   and b :: "'b" shows "one_dim_iso (a + b) = one_dim_iso a + one_dim_iso b"
fundfold_comp_chamber_ex_funpow: fixes ss :: "'a permutation list"   and C :: "'a set" assumes "set ss = (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "chamber C" shows "\<exists>n. (fold fst (map Spair ss) ^^ n) ` C = C0"
intros_4: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and f :: "'f" assumes "s \<in> g"   and "\<Gamma>\<turnstile>c \<down> Normal s" shows "\<Gamma>\<turnstile>Guard f g c \<down> Normal s"
aff: fixes X :: "'a set" assumes "X \<in> \<F>" shows "aff_dim X \<le> aff_dim T"
intros_2: fixes d :: "nat"   and c :: "nat" assumes "0 \<le> d"   and "d < c" shows "valid_intv c (Intv d)"
density_RN_deriv: fixes N :: "'a measure" assumes "absolutely_continuous N N"   and "sets N = sets N" shows "density N (RN_deriv N N) = N"
diff_delta_lnexp_cf2_def: shows "diff_delta_lnexp_cf2 \<equiv> \<lambda>x. - (x ^ 4) / ((x\<^sup>2 - 6 * x + 12) * (x\<^sup>2 + 6 * x + 12))"
INF_limit_inter: fixes w :: "nat \<Rightarrow> 'b"   and S :: "'b set" assumes "\<exists>\<^sub>\<infinity>n. w n \<in> S"   and "finite (S \<inter> range w)" shows "\<exists>a. a \<in> limit w \<inter> S"
triangular_toD: fixes k :: "nat"   and A :: "'a mat"   and j :: "nat"   and i :: "nat" assumes "triangular_to k A"   and "j < k"   and "j < i"   and "i < dim_row A" shows "A $$ (i, j) = (0::'a)"
valid_in_def: fixes I :: "interp"   and \<phi> :: "fml" shows "valid_in I \<phi> \<equiv> \<forall>\<omega>. \<omega> \<in> fml_sem I \<phi>"
ereal_left_mult_cong: fixes c :: "ereal"   and d :: "ereal"   and a :: "ereal"   and b :: "ereal" assumes "c = d"   and "d \<noteq> 0 \<Longrightarrow> a = b" shows "a * c = b * d"
not_symKeys_priK: fixes b :: "keymode"   and A :: "agent" shows "privateKey b A \<notin> symKeys"
SUP_ereal_minus_left: fixes I :: "'d set"   and c :: "ereal"   and f :: "'d \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "c \<noteq> \<infinity>" shows "(\<Squnion>i\<in>I. f i - c) = \<Squnion> (f ` I) - c"
ucast_NOT: fixes x :: "'c word" shows "UCAST('c \<rightarrow> 'b) (NOT x) = NOT (UCAST('c \<rightarrow> 'b) x) AND mask LENGTH('c)"
invariant_restrict_inD: fixes A :: "('c, 'd) automaton"   and P :: "'c \<Rightarrow> bool"   and QI :: "'d \<Rightarrow> bool" assumes "A \<TTurnstile> P" shows "A \<TTurnstile> (QI \<rightarrow>) P"
map_disc_iff: fixes f :: "'f \<Rightarrow> 'e"   and a :: "'f option" shows "(map_option f a = None) = (a = None)"
ih: fixes ua :: "('s, 'v) tm"   and ta :: "('s, 'v) tm"   and sa :: "('s, 'v) tm" assumes "{#size ua, size ta, size sa#} < {#size u, size t, size s#}"   and "wary ua"   and "wary ta"   and "wary sa"   and "ua >\<^sub>t ta"   and "ta >\<^sub>t sa" shows "ua >\<^sub>t sa"
sfree: fixes i :: "'sf"   and f'1 :: "('sf + 'sz, 'sz) trm" assumes "SFunctions \<sigma> i = Some f'1" shows "dfree f'1"
pre_nts: shows "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)..<j'] nts"
connectedI: fixes U :: "'a set" assumes "\<And>A B. \<lbrakk>open A; open B; A \<inter> U \<noteq> {}; B \<inter> U \<noteq> {}; A \<inter> B \<inter> U = {}; U \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> False" shows "connected U"
Rep_MonoTran_inverse: fixes x :: "'b MonoTran" shows "Abs_MonoTran (Rep_MonoTran x) = x"
simps: fixes i :: "nat"   and j :: "nat" shows "gs1.\<mu> i j = (if j < i then map of_int_hom.vec_hom fs ! i \<bullet> gs1.gso j / \<parallel>gs1.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
length_delete_le: fixes k :: "'c"   and al :: "('c \<times> 'd) list" shows "length (delete k al) \<le> length al"
min_top: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and x :: "'a" assumes "class.order_top less_eq less top" shows "ord.min less_eq top x = x"
infinite_imp_nonempty: fixes S :: "'e set" assumes "infinite S" shows "S \<noteq> {}"
bex_UNIV: fixes P :: "'a \<Rightarrow> bool" shows "Bex UNIV P = Ex P"
assms: fixes c :: "('a, 'b, 'c) cmd" assumes "c \<in> set cl" shows "noWhile c"
has_field_derivative_Lambert_W: fixes x :: "real"   and A :: "real set" assumes "- exp (- 1) < x" shows "(Lambert_W has_real_derivative inverse (x + exp (Lambert_W x))) (at x within A)"
lowner_lub_def: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat" assumes "matrix_seq dim f" shows "matrix_seq.lowner_lub f = (THE M. matrix_seq.lowner_is_lub f M)"
nprt_le_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq (lattice_ab_group_add.nprt zero inf a) zero"
HEndPhase1_Inv2a_dblock: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv1 s"   and "Inv2a s"   and "Inv2c_inner s p" shows "Inv2a_innermost s' p (dblock s' p)"
same_append_eq: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" shows "(xs \<cdot> ys = xs \<cdot> zs) = (ys = zs)"
strong_sum_closure: fixes x :: "i"   and y :: "i" shows "\<exists>z. (P x z \<and> P y z) \<and> (\<forall>w. P w z \<longrightarrow> O w x \<or> O w y)"
MapsToObj_1: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and X :: "'a"   and Y :: "'a" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "X \<in> obj\<^bsub>C\<^esub>"
insert_absorb: fixes a :: "'g"   and A :: "'g set" assumes "a \<in> A" shows "insert a A = A"
QmpTr0: fixes H :: "'a set"   and N :: "'a set"   and a :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> N "   and "H \<subseteq> N"   and "a \<in> carrier G" shows "(Qm\<^bsub>G H,N\<^esub>) (H \<bullet> a) = N \<bullet> a"
aezfun_common_supp_spanning_set_decomp: fixes G :: "'a set" assumes "finite G" shows "\<exists>gs. distinct gs \<and> set gs = G \<and> (\<forall>x\<in>aezfun_setspan G. \<exists>rs. length rs = length gs \<and> x = sum_list (map2 (\<lambda>r g. r \<delta>\<delta> (0::'a) * ((1::'b) \<delta>\<delta> g)) rs gs))"
Contrapos1: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A IMP B" shows "H \<turnstile> Neg B IMP Neg A"
RedDynCastFail: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and E :: "char list \<Rightarrow> ty option" assumes "hp s a = \<lfloor>(D, S)\<rfloor>"   and "\<not> P \<turnstile> Path D to C unique"   and "\<not> P \<turnstile> Path last Cs to C unique"   and "C \<notin> set Cs" shows "redp P E (Cast C (ref (a, Cs))) s null s"
sublens'_prop1: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and s\<^sub>1 :: "'e"   and s\<^sub>2 :: "'e"   and s\<^sub>3 :: "'d" assumes "vwb_lens X"   and "X \<subseteq>\<^sub>L' Y" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> s\<^sub>1 (get\<^bsub>Y\<^esub> s\<^sub>2)) s\<^sub>3 = put\<^bsub>Y\<^esub> s\<^sub>1 (get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> s\<^sub>2 s\<^sub>3))"
set_l_is: shows "set (concat (map (\<lambda>k. map (Pair k) (\<pi> ! k)) [0..<length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1])) = (\<Union>(k, op)\<in>{(k, op). k \<in> {0..<length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1} \<and> op \<in> set (\<pi> ! k)}. {(k, op)})"
reflects_right_adjoint: fixes g :: "'b" assumes "D.ide g"   and "C.is_right_adjoint (F.right_map g)" shows "D.is_right_adjoint g"
assms: shows "\<bar>err\<bar> \<le> \<bar>pdevs_apply (snd X) n\<bar>"
F_def: shows "F = (\<lambda>x. (f x, g x))"
wfTerminationLess: fixes decisionVars :: "nat set"   and learnL :: "(Literal list list \<times> Literal list list) set"   and F0 :: "Literal list list" assumes "finite decisionVars"   and "wf learnL" shows "wf (terminationLess F0 decisionVars learnL)"
list_update_beyond: fixes xs :: "'d list"   and i :: "nat"   and x :: "'d" assumes "length xs \<le> i" shows "xs[i := x] = xs"
\<phi>_simps_3: shows "trg r\<^sub>0s\<^sub>1.\<phi> = trg s\<^sub>1"
when_def: fixes P :: "bool"   and m :: "'c \<Rightarrow> (unit \<times> 'c) set \<times> bool" shows "when P m \<equiv> if P then m else return ()"
proper_map_compose: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and Z :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "proper_map X Y f"   and "proper_map Y Z g" shows "proper_map X Z (g \<circ> f)"
l6_3_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> P"   and "B \<noteq> P"   and "\<exists>C. C \<noteq> P \<and> Bet A P C \<and> Bet B P C" shows "Tarski_neutral_dimensionless.Out Bet P A B"
inner_vec_def: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "inner x y = (\<Sum>i\<in>UNIV. inner (x $ i) (y $ i))"
INFM_nat: fixes P :: "nat \<Rightarrow> bool" shows "(\<exists>\<^sub>\<infinity>n. P n) = (\<forall>m. \<exists>n>m. P n)"
LHS_in_hom: shows "\<guillemotleft>(\<mu> \<star> \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr>) \<bullet> \<lparr>Chn = \<mu>_\<nu>\<pi>_\<rho>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<mu> \<star> (\<nu> \<star> \<pi>) \<star> \<rho>)\<rparr> \<bullet> (\<lparr>Chn = \<mu>\<nu>\<pi>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi>)\<rparr> \<star> \<rho>) : ((\<mu> \<star> \<nu>) \<star> \<pi>) \<star> \<rho> \<Rightarrow> \<mu> \<star> \<nu> \<star> \<pi> \<star> \<rho>\<guillemotright>"
enc_word_length: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list" assumes "enc (w, I) = enc (w', I')" shows "length w = length w'"
One_refl: fixes t :: "lam" shows " t \<longrightarrow>\<^sub>1 t"
Hermite_Lindemann_aux1: fixes P :: "int poly set"   and \<beta> :: "int poly \<Rightarrow> int" assumes "finite P"   and "P \<noteq> {}"   and "pairwise algebraic_semidom_class.coprime P"   and "\<And>p. p \<in> P \<Longrightarrow> irreducible p"   and "\<And>p. p \<in> P \<Longrightarrow> 0 < Polynomial.degree p"   and "\<And>p. p \<in> P \<Longrightarrow> \<beta> p \<noteq> 0" shows "(\<Sum>p\<in>P. of_int (\<beta> p) * (\<Sum>\<alpha> | ipoly p \<alpha> = 0. exp \<alpha>)) \<noteq> 0"
add_poly_p_add_mset_sum_0: fixes n :: "int"   and m :: "int"   and A :: "(char list multiset \<times> int) multiset"   and Aa :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset"   and ys :: "char list list" assumes "n + m = 0"   and "add_poly_p\<^sup>*\<^sup>* (A, Aa, {#}) ({#}, {#}, r)" shows "add_poly_p\<^sup>*\<^sup>* (add_mset (mset ys, n) A, add_mset (mset ys, m) Aa, {#}) ({#}, {#}, r)"
rep_eq: fixes x :: "'c"   and xa :: "('a, 'b, 'c) formula" shows "Rep_formula (Formula.Act x xa) = Act\<^sub>\<alpha> x (Rep_formula xa)"
step_fold'_cong: fixes lbls :: "(String.literal, nat) assoc_list"   and lbls' :: "(String.literal, nat) assoc_list"   and pri :: "'a"   and pri' :: "'a"   and pos :: "nat"   and pos' :: "nat"   and steps :: "'b list"   and steps' :: "'b list"   and nxt :: "edgeIndex"   and nxt' :: "edgeIndex"   and onxt :: "edgeIndex option"   and onxt' :: "edgeIndex option"   and iB :: "'c"   and iB' :: "'c"   and g :: "'b \<Rightarrow> (String.literal, nat) assoc_list \<times> 'a \<times> nat \<times> edgeIndex \<times> edgeIndex option \<times> 'c \<Rightarrow> 'd list \<times> edgeIndex \<times> (String.literal, nat) assoc_list"   and g' :: "'b \<Rightarrow> (String.literal, nat) assoc_list \<times> 'a \<times> nat \<times> edgeIndex \<times> edgeIndex option \<times> 'c \<Rightarrow> 'd list \<times> edgeIndex \<times> (String.literal, nat) assoc_list" assumes "lbls = lbls'"   and "pri = pri'"   and "pos = pos'"   and "steps = steps'"   and "nxt = nxt'"   and "onxt = onxt'"   and "iB = iB'"   and "\<And>x d. x \<in> set steps \<Longrightarrow> g x d = g' x d" shows "step_fold' g steps lbls pri pos nxt onxt iB = step_fold' g' steps' lbls' pri' pos' nxt' onxt' iB'"
int_cases2': fixes z :: "int" assumes "z \<noteq> 0" obtains n :: "nat" where "0 \<noteq> int n__" and "z = int n__"   | n :: "nat" where "0 \<noteq> - int n" and "z = - int n"
weak_partial_order_axioms_def: fixes L :: "('a, 'b) gorder_scheme" shows "weak_partial_order_axioms L \<equiv> ((\<forall>x. x \<in> carrier L \<longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> x) \<and> (\<forall>x y. x \<sqsubseteq>\<^bsub>L\<^esub> y \<longrightarrow> y \<sqsubseteq>\<^bsub>L\<^esub> x \<longrightarrow> x \<in> carrier L \<longrightarrow> y \<in> carrier L \<longrightarrow> x .=\<^bsub>L\<^esub> y)) \<and> (\<forall>x y z. x \<sqsubseteq>\<^bsub>L\<^esub> y \<longrightarrow> y \<sqsubseteq>\<^bsub>L\<^esub> z \<longrightarrow> x \<in> carrier L \<longrightarrow> y \<in> carrier L \<longrightarrow> z \<in> carrier L \<longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> z) \<and> (\<forall>x y z w. x .=\<^bsub>L\<^esub> y \<longrightarrow> z .=\<^bsub>L\<^esub> w \<longrightarrow> x \<in> carrier L \<longrightarrow> y \<in> carrier L \<longrightarrow> z \<in> carrier L \<longrightarrow> w \<in> carrier L \<longrightarrow> (x \<sqsubseteq>\<^bsub>L\<^esub> z) = (y \<sqsubseteq>\<^bsub>L\<^esub> w))"
w_def: shows "w = (\<lambda>n x. u n x - v x)"
infty_rel_internal_def: fixes R :: "('a \<times> 'b) set" shows "infty_rel R \<equiv> {(Num a, Num a') |a a'. (a, a') \<in> R} \<union> {(infty.Infty, infty.Infty)}"
dropWhile_eq_self_iff: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "(dropWhile P xs = xs) = (xs = [] \<or> \<not> P (hd xs))"
geometric_sums: fixes c :: "'a" assumes "norm c < 1" shows "(^) c sums ((1::'a) / ((1::'a) - c))"
inner_gt_zero_iff: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and x :: "'e" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(0 < inner x x) = (x \<noteq> zero)"
assms_2: shows "U \<in> FIN_wrt \<psi>"
shunt_mapping: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "mapping z" shows "(x \<le> z * y) = (z\<^sup>T * x \<le> y)"
simps_7: fixes \<pi> :: "'b prm"   and P :: "'b ptrm" shows "\<pi> \<bullet> PSnd P = PSnd (\<pi> \<bullet> P)"
nodes_height_lower_bound: fixes k :: "nat"   and t :: "'b btree" assumes "order k t"   and "bal t" shows "bound k (height t) \<le> nodes t * k"
cons_exec_def: fixes e :: "'b \<times> 'c list"   and p :: "'c" shows "cons_exec e p \<equiv> (fst e, Cons p (snd e))"
ipath_to_rtrancl: fixes E :: "('a \<times> 'a) set"   and r :: "nat \<Rightarrow> 'a"   and i1 :: "nat"   and i2 :: "nat" assumes "ipath E r"   and "i1 \<le> i2" shows "(r i1, r i2) \<in> E\<^sup>*"
pdivmod_monic_def: fixes f :: "'a poly"   and g :: "'a poly" shows "pdivmod_monic f g \<equiv> let cg = coeffs g; cf = coeffs f; (q, r) = divmod_poly_one_main_list [] (rev cf) (rev cg) (1 + length cf - length cg) in (poly_of_list q, poly_of_list (rev r))"
invariant_cop_FD: fixes ds :: "'d set"   and P :: "'d set \<Rightarrow> 'x set \<Rightarrow> bool"   and fp :: "'x set" assumes "invariant ds P"   and "P ds fp" shows "P ds (cop_F ds fp)"
transfer_FOREACHc_plain: fixes iterate :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'d set"   and fi :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "set_iterator iterate s"   and "\<And>x \<sigma>. RETURN (fi x \<sigma>) \<le> f x \<sigma>" shows "RETURN (iterate c fi \<sigma>) \<le> FOREACH\<^sub>C s c f \<sigma>"
cont_apply: fixes t :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "cont t"   and "\<And>x. cont (f x)"   and "\<And>y. cont (\<lambda>x. f x y)" shows "cont (\<lambda>x. f x (t x))"
invar_def: fixes t :: "'a tree" shows "S2.invar t = (True \<and> Sorted_Less.sorted (inorder t))"
f1: shows " |x\<^sup>\<star>] ( |x] y + ad y) = ad ( |x\<^sup>\<star>\<rangle> A x (ad y))"
set_rotate1: fixes xs :: "'b list" shows "ran\<^sub>l (rotate1 xs) = ran\<^sub>l xs"
assms_2: fixes w :: "'a stream" assumes "R w" shows "ev \<phi> w \<and> ev R (stl w)"
f\<^sub>1: shows "f\<^sub>1 = fst (split_face f u v vs)"
f_shrink_last_Nil: fixes k :: "nat" shows "[] \<div>\<^sub>l k = []"
content_division: fixes \<D> :: "'a set set"   and S :: "'a set" assumes "\<D> division_of S" shows "sum (Sigma_Algebra.measure lebesgue) \<D> = Sigma_Algebra.measure lebesgue S"
lActive_equality: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat" assumes "active c (t i)"   and "\<And>x. active c (t x) \<Longrightarrow> x \<le> i" shows "dynamic_component.lActive active c t = i"
Iod_def: fixes D :: "('c, 'd) Order_scheme"   and T :: "'c set" shows "Iod D T = D\<lparr>carrier := T, rel := {(a, b). (a, b) \<in> rel D \<and> a \<in> T \<and> b \<in> T}\<rparr>"
list_to_vec_def: fixes xs :: "'c list" shows "list_to_vec xs = (\<chi>i. xs ! mod_type_class.to_nat i)"
walk_2: fixes G :: "('e, 'f) Graph_scheme"   and v :: "'e"   and w :: "'e" assumes "Graph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w" shows "Graph.walk G [v, w]"
simps_10: fixes v1 :: "'b"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (V v1) (Times v va) = True"
Assign: fixes a :: "aexp"   and s :: "char list \<Rightarrow> val"   and v :: "val"   and x :: "char list" assumes "taval a s v" shows "(x ::= a, s) \<rightarrow> (SKIP, s(x := v))"
zero_sheaf_spec_extensional: fixes U :: "'a set set" shows "zero_sheaf_spec U \<in> extensional U"
4: shows "bprv (imp (cnj (PP \<langle>PP \<langle>\<phi>G\<rangle>\<rangle>) (PP \<langle>neg (PP \<langle>\<phi>G\<rangle>)\<rangle>)) (PP \<langle>fls\<rangle>))"
assms_2: shows "\<exists>es. char_poly B = (\<Prod>a\<leftarrow>es. [:- a, 1::'a:])"
UN_bool_eq: fixes A :: "bool \<Rightarrow> 'a set" shows "\<Union> (range A) = A True \<union> A False"
univ_def: fixes f :: "'c \<Rightarrow> 'b"   and X :: "'c set" shows "univ f X \<equiv> f (SOME x. x \<in> X)"
distinct_1: fixes x1 :: "('a, 'b) Term.term"   and x2 :: "('a, 'b) Term.term" shows "send\<langle>x1\<rangle>\<^sub>s\<^sub>t \<noteq> receive\<langle>x2\<rangle>\<^sub>s\<^sub>t"
ln_le_harm: fixes n :: "nat" shows "ln (real n + 1) \<le> harm n"
ctor1_o_fold1: shows "ctor1 \<circ> fold1 (F1map id ctor1 ctor2) (F2map id ctor1 ctor2) = id"
from_to_Gromov_completion: fixes x :: "'a" shows "from_Gromov_completion (to_Gromov_completion x) = x"
simps_2: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and r :: "real" shows "usubstappt \<sigma> U (Number r) = Aterm (Number r)"
ins_xlate: fixes t :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree"   and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "map BsSkewBinomialHeapStruc.bsmapt (BsSkewBinomialHeapStruc.ins t q) = SkewBinomialHeapStruc.ins (BsSkewBinomialHeapStruc.bsmapt t) (map BsSkewBinomialHeapStruc.bsmapt q)"
inverse_permutation_of_list_id: fixes x :: "'b"   and xs :: "('b \<times> 'b) list" assumes "x \<notin> set (map snd xs)" shows "inverse_permutation_of_list xs x = x"
countable_acc_point_imp_seq_compact: fixes s :: "'b set" assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. \<forall>U. x \<in> U \<and> open U \<longrightarrow> infinite (U \<inter> t))" shows "seq_compact s"
prems_3: shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"