hcju/isabelleps huggingface dataset:
spr_jview_prefix_closed: fixes a :: "'a"   and t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) BEState"   and t' :: "('a, 'es, 'as) BEState Trace"   and s' :: "('a, 'es, 'as) BEState" assumes "spr_jview a (t \<leadsto> s) = spr_jview a (t' \<leadsto> s')" shows "spr_jview a t = spr_jview a t'"
case_2: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b pred \<Rightarrow> 'a"   and f3 :: "'b pred \<Rightarrow> 'b seq \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b pred" shows "(case Insert x21 x22 of Empty \<Rightarrow> f1 | Insert x xa \<Rightarrow> f2 x xa | Join x xa \<Rightarrow> f3 x xa) = f2 x21 x22"
not_square_less_zero: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "\<not> less (times a a) zero"
lookup_single: fixes k :: "'b"   and v :: "'a"   and k' :: "'b" shows "lookup (monomial v k) k' = (v when k = k')"
app_vrange_def: fixes r :: "V" shows "\<R>\<^sub>\<circ> r = set {b. \<exists>a. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
upt_rec: fixes i :: "nat"   and j :: "nat" shows "[i..<j] = (if i < j then i # [Suc i..<j] else [])"
eq_iff: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a = b) = (b \<subseteq># a \<and> a \<subseteq># b)"
inv_into_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "inv_into A f = (\<lambda>x. SOME y. y \<in> A \<and> f y = x)"
simps_2: fixes \<theta> :: "(W.tvar \<times> ty) list"   and x :: "W.var"   and S :: "tyS"   and \<Gamma> :: "(W.var \<times> tyS) list" shows "\<theta><(x, S) # \<Gamma>> = (x, \<theta><S>) # \<theta><\<Gamma>>"
assms_2: shows "G \<in> {GG \<times> I |GG. GG \<in> \<G>}"
lagrange: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G" shows "card (rcosets\<^bsub>G\<^esub> H) * card H = order G"
simps: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list" shows "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems = (length cms\<^sub>1 = length cms\<^sub>2 \<and> length cms\<^sub>1 = length mems \<and> (\<forall>i<length cms\<^sub>1. (\<forall>\<sigma>. dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i \<longrightarrow> (cms\<^sub>1 ! i, subst \<sigma> (fst (mems ! i))) \<approx> (cms\<^sub>2 ! i, subst \<sigma> (snd (mems ! i)))) \<and> (\<forall>x. mem\<^sub>1 x = mem\<^sub>2 x \<or> dma x = High \<longrightarrow> x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)) \<and> (length cms\<^sub>1 = 0 \<and> mem\<^sub>1 =\<^sup>l mem\<^sub>2 \<or> (\<forall>x. \<exists>i<length cms\<^sub>1. x \<notin> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)))"
pbij_lens_axioms_def: fixes x :: "'a \<Longrightarrow> 'b" shows "pbij_lens_axioms x \<equiv> \<forall>\<sigma> v \<rho>. put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v"
semiring_simprules_8: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z)"
dim_mult_mat_vec: fixes A :: "'b mat"   and v :: "'b Matrix.vec" shows "dim_vec (A *\<^sub>v v) = dim_row A"
nc: fixes r :: "'a iarray" assumes "r \<in> set rows" shows "length (IArray.list_of r) = nc"
pmf_pos: fixes p :: "'a pmf"   and x :: "'a" assumes "pmf p x \<noteq> 0" shows "0 < pmf p x"
length_ge_1_conv: fixes l :: "'b list" shows "(Suc 0 \<le> length l) = (l \<noteq> [])"
trd_red_rtrancl: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "(punit.red (set fs))\<^sup>*\<^sup>* p (punit.trd fs p)"
whisker_right: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and K :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and L :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" assumes "functor A B F"   and "natural_transformation B C H K \<tau>"   and "natural_transformation B C K L \<tau>'" shows "vertical_composite.map B C \<tau> \<tau>' \<circ> F = vertical_composite.map A C (\<tau> \<circ> F) (\<tau>' \<circ> F)"
the_inverse_def: fixes g :: "'a" shows "\<ii> g \<equiv> THE h. h \<in> G \<and> binop g h = e \<and> binop h g = e"
matchs_eq: fixes fps :: "(char list \<times> pat) list"   and fs :: "(char list \<times> trm) list"   and ts :: "trm list"   and fs' :: "(char list \<times> trm) list" assumes "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts"   and "\<forall>(l, p)\<in>set fps. fs\<langle>l\<rangle>\<^sub>? = fs'\<langle>l\<rangle>\<^sub>?" shows "\<turnstile> fps [\<rhd>] fs' \<Rightarrow> ts"
know1k: fixes C :: "specID"   and m1 :: "Keys" shows "knows C [KS2Expression (kKS m1)] = know C (kKS m1)"
SN_on_imp_on_minimal: fixes r :: "('b \<times> 'b) set"   and x :: "'b" assumes "SN_on r {x}" shows "\<forall>Q. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (z, y) \<in> r \<longrightarrow> y \<notin> Q)"
eps_the_restrict: fixes r :: "Formula.formula Regex.regex"   and A :: "nat set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data option list"   and i :: "nat" assumes "fv_regex r \<subseteq> A" shows "Regex.eps (Formula.sat \<sigma> V (map the (restrict A v))) i r = Regex.eps (Formula.sat \<sigma> V (map the v)) i r"
fls_integral_mult_const_left: fixes c :: "'a"   and f :: "'a fls" shows "fls_integral (fls_const c * f) = fls_const c * fls_integral f"
hyps: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'd option" assumes "ts_inv_ok ts I"   and "thread_oks ts TAS" shows "I \<subseteq>\<^sub>m upd_invs I P TAS"
psimps: fixes data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "gb_schema_dummy_dom (data, D, bs, ps)" shows "gb_schema_dummy data D bs ps = (if ps = [] then (gs @ bs, D) else let sps = sel gs bs ps (snd data); ps0 = ps -- sps; aux = compl gs bs ps0 sps (snd data); remcomps = fst data - count_const_lt_components (fst aux) in if remcomps = 0 then (full_gb (gs @ bs), D) else let (hs, data') = add_indices aux (snd data) in gb_schema_dummy (remcomps, data') (D \<union> (set hs \<times> (set gs \<union> set bs \<union> set hs) \<union> set (ps -- sps) -\<^sub>p set (ap gs bs ps0 hs data'))) (ab gs bs hs data') (ap gs bs ps0 hs data'))"
c_tl_le: fixes u :: "nat" shows "c_tl u \<le> u"
rec_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "rec_latin_type f1 f2 f3 Num = f3"
int_vertices_subset: fixes v :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_path v p v'" shows "int_vertices p \<subseteq> V"
remove_elt_list_set: fixes x :: "'a"   and ys :: "'a list" shows "set (remove_elt_list x ys) = set ys - {x}"
tranclp_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r a y \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r\<^sup>+\<^sup>+ a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
Diff_eq_empty_iff_mset: fixes A :: "'k multiset"   and B :: "'k multiset" shows "(A - B = {#}) = (A \<subseteq># B)"
Seq: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>"
prv_imp_monoR: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (imp \<psi> \<chi>)" shows "bprv (imp (imp \<chi> \<phi>) (imp \<psi> \<phi>))"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "P \<langle>\<rangle>"   and "\<And>hs uu_. P \<langle>hs, uu_, \<langle>\<rangle>\<rangle>"   and "\<And>v va vc vd ve. P \<langle>v, va, \<langle>vc, vd, ve\<rangle>\<rangle>" shows "P a0"
absorb_iff2: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (a \<squnion> b = b)"
uniform_limit_pre_newman: fixes r :: "real"   and s :: "complex" assumes "0 \<le> r"   and "1 / 2 < Re s - r" shows "uniform_limit (cball s r) (\<lambda>n s. \<Sum>p<n. if prime p then B p s else 0) pre_newman sequentially"
assms_3: shows "list_all2 R [i..<j] nts"
minus_one_mult_self: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "times (power.power one times (uminus one) n) (power.power one times (uminus one) n) = one"
C4: shows "card (older_seniors x n) < card {s. \<exists>x. s = senior x n \<and> \<not> sink (token_run s n)}"
fty_eq_def: fixes F1 :: "funty"   and F2 :: "funty" shows "F1 \<simeq> F2 \<equiv> F1 <:: F2 \<and> F2 <:: F1"
f3: fixes A :: "'a set"   and Aa :: "'a set" shows "A - - Aa = Aa - - A"
add_log_eq_powr: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "0 < b"   and "b \<noteq> 1"   and "0 < x" shows "y + log b x = log b (b powr y * x)"
ty: shows "\<Gamma> \<turnstile> App t1 t2 : T"
inj_on_contraD: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'c" assumes "inj_on f A"   and "x \<noteq> y"   and "s2p A x"   and "s2p A y" shows "f x \<noteq> f y"
at_set_avoiding2: fixes xs :: "atom set"   and c :: "'e"   and x :: "'f" assumes "finite xs"   and "finite (supp c)"   and "finite (supp x)"   and "xs \<sharp>* x" shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
mset_eq_perm: fixes xs :: "'a list"   and ys :: "'a list" shows "(mset xs = mset ys) = (xs <~~> ys)"
foundation16: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<delta> X) = (X \<tau> \<noteq> UML_Types.bot_class.bot \<and> X \<tau> \<noteq> null)"
poly_hyp: shows "0 < poly (prod_list (retrieve_polys qs I)) x \<or> poly (prod_list (retrieve_polys qs I)) x < 0"
asAbs_qSwapAbs_swapAbs: fixes qA :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "qGoodAbs qA" shows "asAbs (qA $[[x1 \<and> x2]]_xs) = asAbs qA $[x1 \<and> x2]_xs"
beta: fixes s :: "dB"   and s' :: "dB"   and t :: "dB"   and t' :: "dB" assumes "s => s'"   and "t => t'" shows "Abs s \<degree> t => s'[t'/0]"
multLessResolve: fixes l :: "Literal"   and C :: "Literal list"   and reason :: "Literal list"   and M :: "(Literal \<times> bool) list" assumes "opposite l el C"   and "isReason reason l (elements M)" shows "(resolve C reason (opposite l), C) \<in> multLess M"
substitute_simps_2: fixes x :: "'a"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set"   and t :: "'a ttree" shows "(x # xs \<in> paths (substitute f T t)) = (possible t x \<and> xs \<in> paths (substitute (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x)))"
adjoint_adjoint: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f" shows "adjoint (adjoint f) = f"
TT_def: shows "TT = Geq (Number 0) (Number 0)"
one_is_normal: shows "{\<zero>} \<lhd> add_monoid R"
cfs_one: fixes n :: "nat" shows "\<one>\<^bsub>P\<^esub> n = (if n = 0 then \<one> else \<zero>)"
Mprefix_refines_Mndet: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process" shows "mndet A P \<le> Mprefix A P"
simps_5: fixes b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "flatten (Cond b c\<^sub>1 c\<^sub>2) = [Cond b c\<^sub>1 c\<^sub>2]"
when_1: fixes P :: "bool"   and zero :: "'b"   and a :: "'b" assumes "P" shows "zero.when zero a P = a"
null_char: shows "B.VVV.null = B.VxVxV.null"
sbintrunc_n_minus1: fixes n :: "nat" shows "signed_take_bit n (- 1) = - 1"
bf_ifex_eq: fixes af :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and at :: "'a ifex"   and bf :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and bt :: "'a ifex" assumes "(af, at) \<in> bf_ifex_rel"   and "(bf, bt) \<in> bf_ifex_rel" shows "(af = bf) = (at = bt)"
sorted_mid_iff: fixes xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "sorted (xs @ y # ys) = (sorted (xs @ [y]) \<and> sorted (y # ys))"
A: fixes Z' :: "'a set" shows "(xs @ zs, Z') \<in> failures P \<longrightarrow> (xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z') \<in> failures P"
\<sigma>_def: shows "\<sigma> = (INF s\<in>B. s \<bullet> (1::'a))"
filter_syzygy_basis_def: fixes n :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "filter_syzygy_basis n bs = filter (\<lambda>b. component_of_term ` keys b \<subseteq> {0..<n}) bs"
vars_snd_rcv_union: fixes x :: "('d, 'c) strand_step" shows "vars\<^sub>s\<^sub>t\<^sub>p x = fv\<^sub>s\<^sub>n\<^sub>d x \<union> fv\<^sub>r\<^sub>c\<^sub>v x \<union> fv\<^sub>e\<^sub>q assign x \<union> fv\<^sub>e\<^sub>q check x \<union> fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q x \<union> set (bvars\<^sub>s\<^sub>t\<^sub>p x)"
eval_Dom_ide: fixes g :: "('a E.term, 'a) arr" assumes "ide g" shows "EVAL (Dom g) = Map g"
preservation_of_barbs_and_closures_3: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_preserves_binary_pred (Rel\<^sup>*) (\<lambda>a. hasBarb a CWB)"
simps_1: fixes M :: "nat" shows "tensors_from_net (Input M) = Matrix.vec M (unit_vec M)"
copy_first_matrix_def: fixes nr :: "nat"   and nc :: "nat" shows "copy_first_matrix nr nc = mat nr nc (\<lambda>(r, c). if c = 0 then 1 else 0)"
WriterT_def: shows "WriterT \<equiv> \<Lambda> app. writerT_abs\<cdot>app"
lowerPowers_eq: fixes b :: "nat"   and i :: "nat"   and p :: "nat \<Rightarrow>\<^sub>0 'b"   and x :: "nat" shows "lookup (lowerPowers b i p) x = (if b \<le> x \<and> x < b + i then 0::'b else lookup p (dec_above b i x))"
nn_integral_measure_pmf: fixes p :: "'b pmf"   and f :: "'b \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf p) f = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
cwiseext_singleton: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" shows "cwiseext gt [y] [x] = gt y x"
prems_6: shows "\<forall>y1 y2 z1 z2. y1 = y2 \<and> z1 = z2 \<longrightarrow> g1 (f1 (w1 x1) y1) z1 = g2 (f2 (w2 x2) y2) z2"
abs_eq: fixes x :: "16 word" shows "uint64_of_uint16 (Abs_uint16 x) = Abs_uint64 (ucast x)"
in_joinI: fixes n :: "nat"   and A :: "nat set"   and X :: "'b option list set"   and B :: "nat set"   and Y :: "'b option list set"   and b :: "bool"   and v :: "'b option list" assumes "table n A X"   and "table n B Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A"   and "wf_tuple n (A \<union> B) v"   and "restrict A v \<in> X"   and "b \<Longrightarrow> restrict B v \<in> Y"   and "\<not> b \<Longrightarrow> restrict B v \<notin> Y" shows "v \<in> join X b Y"
intros_5: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and x :: "'a itrm"   and y :: "'a itrm" assumes "itrm_cong R x y" shows "itrm_cong R y x"
fin_digraphI: assumes "finite (pverts G)"   and "finite (parcs G)" shows "fin_digraph (with_proj G)"
hom_minus: fixes x :: "'b"   and y :: "'b" shows "[:x - y:] = [:x:] - [:y:]"
nvi_4: fixes i :: "nat"   and w :: "'a vec" assumes "i < length us"   and "w \<in> set (ws @ us) - {us ! i}" shows "normal_vectors ws ! i \<bullet> w = (0::'a)"
v'_def: fixes X :: "'b set"   and v :: "'b \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'b" assumes "Beta_Regions' X v n not_in_X" shows "Beta_Regions'.v' X v n not_in_X \<equiv> \<lambda>i. if i \<le> n then THE c. c \<in> X \<and> v c = i else not_in_X"
len_Ls_Bs: shows "length (Ls @ Bs) = n"
simps_5: fixes f1 :: "'a"   and f2 :: "'b tree \<Rightarrow> 'b \<Rightarrow> 'b tree \<Rightarrow> 'a"   and x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "(case \<langle>x21, x22, x23\<rangle> of \<langle>\<rangle> \<Rightarrow> f1 | \<langle>x, xa, xb\<rangle> \<Rightarrow> f2 x xa xb) = f2 x21 x22 x23"
co_agree_sem: fixes s :: "real" shows "Vagree (fst (mk_v I ODE (sol 0, b) (sol s)), s *\<^sub>R (\<chi>i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0)) (fst (mk_v I ODE (sol 0, b) (sol s)), s *\<^sub>R snd (mk_v I ODE (sol 0, b) (sol s))) (semBV I ODE)"
dim_UNIV: fixes a :: "'c" shows "manifold_eucl.tangent_space.dim \<infinity> a (manifold_eucl.tangent_space \<infinity> a) = card (directional_derivative \<infinity> a ` Basis)"
axioms_2: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" assumes "merge_on A h bo m" shows "merge_on_axioms A h bo m"
qelim_def: fixes qe :: "'a list \<Rightarrow> 'a fm"   and as :: "'a list" shows "qelim qe as = (let qf = qe (filter depends\<^sub>0 as); indep = concat (map (\<lambda>a. if \<not> depends\<^sub>0 a then [Atom (decr a)] else []) as) in and qf (list_conj indep))"
Sup_mono: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and A :: "'b::type set"   and B :: "'b::type set" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "\<And>a::'b::type. a \<in> (A::'b::type set) \<Longrightarrow> \<exists>b::'b::type\<in>B::'b::type set. (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) a b" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Sup::'b::type set \<Rightarrow> 'b::type) (A::'b::type set)) (Sup (B::'b::type set))"
assms_2: shows "prov_triple (p2, c2, q2)"
conforms_StdXcpt: fixes xn :: "xname"   and s :: "st"   and G :: "prog"   and L :: "lenv" shows "(Some (Xcpt (Std xn)), s)\<Colon>\<preceq>(G, L) = Norm s\<Colon>\<preceq>(G, L)"
psubset_card_mono: fixes B :: "'d set"   and A :: "'d set" assumes "finite B"   and "A \<subset> B" shows "card A < card B"
bind_cong_simp: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> 'c measure"   and g :: "'b \<Rightarrow> 'c measure" assumes "M = N"   and "\<And>x. x \<in> space M =simp=> f x = g x" shows "M \<bind> f = N \<bind> g"
lift_Suc_mono_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less (f n) (f (Suc n))"   and "n < n'" shows "less (f n) (f n')"
product_topology_base_alt: fixes I :: "'c set"   and X :: "'c \<Rightarrow> 'd topology" shows "finite intersection_of (\<lambda>F. \<exists>i U. F = {f. f i \<in> U} \<and> i \<in> I \<and> openin (X i) U) relative_to (\<Pi>\<^sub>E i\<in>I. topspace (X i)) = (\<lambda>F. \<exists>U. F = Pi\<^sub>E I U \<and> finite {i \<in> I. U i \<noteq> topspace (X i)} \<and> (\<forall>i\<in>I. openin (X i) (U i)))"
Q: fixes h :: "'a \<Rightarrow> 'b" assumes "continuous_map X Y h"   and "P h" shows "Q h"
tensor_preserves_ide: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "ide (a \<otimes> b)"
sel: fixes x3 :: "'a" shows "vertex.inner \<langle>x3\<rangle> = x3"
dpf_def: shows "dpf = discounted_value r (\<lambda>m. pyf) matur"
univ: shows "UNIV = Abs_bit1 ` {0..<1 + 2 * int CARD('a)}"
main: fixes lc :: "int" assumes "0 < lc"   and "lc < p" shows "comm_monoid_mult_class.coprime lc p"
generalized_zipping_lemma4: fixes \<Gamma>1 :: "'e Gamma"   and \<Gamma>2 :: "'e Gamma"   and \<rho>1 :: "'e V_rec \<Rightarrow> 'e set"   and \<rho>2 :: "'e V_rec \<Rightarrow> 'e set" assumes "\<nabla>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub>"   and "\<Delta>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub>"   and "\<Upsilon>\<^bsub>\<Gamma>1\<^esub> \<subseteq> E\<^bsub>ES1\<^esub>"   and "\<nabla>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub>"   and "\<Delta>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub>"   and "\<Upsilon>\<^bsub>\<Gamma>2\<^esub> \<subseteq> E\<^bsub>ES2\<^esub>"   and "BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub>"   and "BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub>"   and "EventSystems.total ES1 (C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub>)"   and "EventSystems.total ES2 (C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub>)"   and "FCIA \<rho>1 \<Gamma>1 \<V>1 Tr\<^bsub>ES1\<^esub>"   and "FCIA \<rho>2 \<Gamma>2 \<V>2 Tr\<^bsub>ES2\<^esub>"   and "V\<^bsub>\<V>1\<^esub> \<inter> V\<^bsub>\<V>2\<^esub> \<subseteq> \<nabla>\<^bsub>\<Gamma>1\<^esub> \<union> \<nabla>\<^bsub>\<Gamma>2\<^esub>"   and "C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub> \<subseteq> \<Upsilon>\<^bsub>\<Gamma>1\<^esub>"   and "C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub> \<subseteq> \<Upsilon>\<^bsub>\<Gamma>2\<^esub>"   and "N\<^bsub>\<V>1\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {}"   and "N\<^bsub>\<V>2\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {}" shows "\<forall>\<tau> lambda t1 t2. set \<tau> \<subseteq> E\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> set lambda \<subseteq> V\<^bsub>\<V>\<^esub> \<and> set t1 \<subseteq> E\<^bsub>ES1\<^esub> \<and> set t2 \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES1\<^esub> @ t1 \<in> Tr\<^bsub>ES1\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES2\<^esub> @ t2 \<in> Tr\<^bsub>ES2\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES1\<^esub> = t1 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES2\<^esub> = t2 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> t1 \<upharpoonleft> C\<^bsub>\<V>1\<^esub> = [] \<and> t2 \<upharpoonleft> C\<^bsub>\<V>2\<^esub> = [] \<longrightarrow> (\<exists>t. \<tau> @ t \<in> Tr\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> t \<upharpoonleft> V\<^bsub>\<V>\<^esub> = lambda \<and> t \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
sqrt_ge_absD: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> sqrt y" shows "x\<^sup>2 \<le> y"