hcju/isabelleps huggingface dataset:
Q: shows "index key y n mi ma = Max {i. i < n \<and> offs (enum xs index key n mi ma) 0 ! i \<le> k \<and> 0 < offs_num n xs index key mi ma i}"
dupe_monic_int_def: fixes D :: "int poly"   and H :: "int poly"   and S :: "int poly"   and T :: "int poly"   and U :: "int poly" shows "dupe_monic_int D H S T U = (case pdivmod_monic (Mp (T * U)) D of (q, r) \<Rightarrow> (Mp (S * U + H * q), Mp r))"
FG_advice_b2: fixes Y :: "'b ltln set"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "Y \<subseteq> \<G> \<phi> w"   and "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<mu>" shows "w \<Turnstile>\<^sub>n \<phi>"
Inf_empty: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf {} = top"
rtrancl3p_Nil: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and a' :: "'a" shows "rtrancl3p r a [] a' = (a = a')"
r_cancel_one: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and a :: "'a" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(a \<otimes>\<^bsub>G\<^esub> x = x) = (a = \<one>\<^bsub>G\<^esub>)"
vector_matrix_mult_def: fixes v :: "('c, 'e) vec"   and m :: "(('c, 'd) vec, 'e) vec" shows "v v* m \<equiv> \<chi>j. \<Sum>i\<in>UNIV. m $ i $ j * v $ i"
mult_pos_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less zero b" shows "less zero (times a b)"
help_h: shows "reduction_signs p (qs1 @ qs2) (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (M_mat (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))) = take_indices (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (find_nonzeros_from_input_vec (solve_for_lhs p (qs1 @ qs2) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (M_mat (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))))))"
BD: shows "B \<inter> D \<in> null_sets lborel"
connected_open_polynomial_connected: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "open S"   and "connected S"   and "x \<in> S"   and "y \<in> S" shows "\<exists>g. polynomial_function g \<and> path_image g \<subseteq> S \<and> pathstart g = x \<and> pathfinish g = y"
length_tl: fixes xs :: "'g list" shows "length (tl xs) = length xs - 1"
distinct_2: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "Node x21 x22 x23 x24 x25 \<noteq> Leaf"
cp_thm': fixes p :: "fm"   and i :: "int"   and bs :: "real list"   and d :: "int" assumes "iszlfm p (real_of_int i # bs)"   and "d_\<beta> p 1"   and "d_\<delta> p d"   and "0 < d" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = ((\<exists>j\<in>{1..d}. Ifm (real_of_int j # bs) (minusinf p)) \<or> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (real_of_int i # bs) ` set (\<beta> p). Ifm ((b + real_of_int j) # bs) p))"
border_take_\<ff>: fixes s :: "'a list"   and i :: "nat" shows "border (take (\<ff> s i - 1) s) (take i s)"
extr_Inj: fixes M :: "msg"   and IK :: "msg set"   and bad :: "agent set"   and H :: "chan set" assumes "M \<in> IK" shows "M \<in> extr bad IK H"
ms_closure_basis: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and C :: "'a"   and e :: "real" assumes "ms_closure P xs f (b # basis)"   and "is_expansion C basis" shows "ms_closure P (scale_shift_ms_aux (C, e) xs) (\<lambda>x. eval C x * b x powr e * f x) (b # basis)"
inject_3: fixes x31 :: "poscheckvariant"   and x32 :: "('c, 'd) Term.term"   and x33 :: "('c, 'd) Term.term"   and y31 :: "poscheckvariant"   and y32 :: "('c, 'd) Term.term"   and y33 :: "('c, 'd) Term.term" shows "(\<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t = \<langle>y31: y32 \<doteq> y33\<rangle>\<^sub>s\<^sub>t) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
inf_conj_distrib: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "conj_distrib (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) ((c\<^sub>0::'b::refinement_lattice) \<sqinter> (c\<^sub>1::'b::refinement_lattice)) (d::'b::refinement_lattice) = conj c\<^sub>0 d \<sqinter> conj c\<^sub>1 d"
2: shows "exec_plan (fmrestrict_set vs s) (as_proj as vs) = fmrestrict_set vs s'"
conditional_entropy_generic_eq: fixes S :: "'d measure"   and T :: "'e measure"   and Y :: "'a \<Rightarrow> 'e"   and Py :: "'e \<Rightarrow> real"   and X :: "'a \<Rightarrow> 'd"   and Pxy :: "'d \<times> 'e \<Rightarrow> real" assumes "sigma_finite_measure S"   and "sigma_finite_measure T"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>x. x \<in> space T \<Longrightarrow> 0 \<le> Py x"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) (\<lambda>x. ennreal (Pxy x))"   and "\<And>x y. \<lbrakk>x \<in> space S; y \<in> space T\<rbrakk> \<Longrightarrow> 0 \<le> Pxy (x, y)" shows "conditional_entropy b S T X Y = - (LINT (x, y)|(S \<Otimes>\<^sub>M T). Pxy (x, y) * log b (Pxy (x, y) / Py y))"
density_unique2: fixes f :: "'a \<Rightarrow> ennreal"   and f' :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "f' \<in> borel_measurable M"   and "\<And>A. A \<in> sets M \<Longrightarrow> set_nn_integral M A f = set_nn_integral M A f'" shows "AE x in M. f x = f' x"
invar_NodeLeaf: fixes l :: "('c \<times> nat) tree"   and x :: "'c"   and lv :: "nat" shows "invar \<langle>l, (x, lv), \<langle>\<rangle>\<rangle> = (invar l \<and> lv = Suc (lvl l) \<and> lv = Suc 0)"
distinct_3: fixes x11 :: "'d"   and x12 :: "'d"   and x13 :: "('d, 'e, 'f) psi"   and x3 :: "('d, 'e, 'f) psi" shows "RIn_Rep x11 x12 x13 \<noteq> RTau_Rep x3"
valid_as_submap_init_submap_exec_i: fixes s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" shows "fmdom' s \<subseteq> fmdom' (state_succ s a)"
qtname_declclass_def: fixes q :: "qtname" shows "declclass q \<equiv> q"
unat_power_lower32: fixes n :: "nat" assumes "n < word_bits" shows "unat (2 ^ n) = 2 ^ n"
ring_hom_cring_axioms: shows "ring_hom_cring R S h"
simps_3: 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 f :: "char" shows "usubstappt \<sigma> U (Const f) = usappconst \<sigma> U f"
set_rows_carrier: shows "set (rows B) \<subseteq> Rn"
simps_1: fixes carrier :: "'a set"   and more :: "'b" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
rev_ImageI: fixes a :: "'c"   and A :: "'c set"   and b :: "'d"   and r :: "('c \<times> 'd) set" assumes "s2p A a"   and "s2p r (a, b)" shows "s2p (r `` A) b"
module_ow_def: fixes U\<^sub>M :: "'d set"   and plus\<^sub>M :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M :: "'d"   and minus\<^sub>M :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M :: "'d \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "module_ow U\<^sub>M plus\<^sub>M zero\<^sub>M minus\<^sub>M uminus\<^sub>M scale \<equiv> ab_group_add_ow U\<^sub>M plus\<^sub>M zero\<^sub>M minus\<^sub>M uminus\<^sub>M \<and> module_ow_axioms U\<^sub>M plus\<^sub>M scale"
hn_monadic_WHILE_aux: fixes P :: "assn"   and \<Gamma> :: "assn"   and Rs :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and s' :: "'a"   and s :: "'b"   and I :: "'a \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool Heap"   and \<Gamma>b :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and b' :: "'a \<Rightarrow> bool nres"   and f :: "'b \<Rightarrow> 'b Heap"   and \<Gamma>f :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and f' :: "'a \<Rightarrow> 'a nres" assumes "P \<Longrightarrow>\<^sub>t \<Gamma> * hn_ctxt Rs s' s"   and "\<And>s s'. I s' \<Longrightarrow> hn_refine (\<Gamma> * hn_ctxt Rs s' s) (b s) (\<Gamma>b s' s) id_assn (b' s')"   and "\<And>s' s. \<Gamma>b s' s \<Longrightarrow>\<^sub>t \<Gamma> * hn_ctxt Rs s' s"   and "\<And>s' s. I s' \<Longrightarrow> hn_refine (\<Gamma> * hn_ctxt Rs s' s) (f s) (\<Gamma>f s' s) Rs (f' s')"   and "\<And>s' s. \<Gamma>f s' s \<Longrightarrow>\<^sub>t \<Gamma> * hn_ctxt (\<lambda>_ _. true) s' s" shows "hn_refine P (heap_WHILET b f s) (\<Gamma> * hn_invalid Rs s' s) Rs (monadic_WHILEIT I b' f' s')"
state_set_finite: fixes X :: "'a list" shows "finite (state_set X)"
SystemVSubst2_def: shows "SystemVSubst2 = \<lparr>SFunctions = Map.empty, SPredicates = \<lambda>p. if p = vid1 then Some (Geq ($f (Inl fid1) local.empty) (Const 0)) else None, SContexts = Map.empty, SPrograms = \<lambda>a. if a = vid1 then Some (EvolveODE (OProd (OSing vid1 ($f fid1 local.empty)) (OSing vid2 (Var vid1))) TT) else None, SODEs = Map.empty\<rparr>"
that: fixes h :: "real" assumes "h \<in> T"   and "h \<in> existence_ivl0 x0"   and "(flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1"   and "\<And>h'. h' \<in> {0<--<h} \<Longrightarrow> (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX" shows "thesis"
ring_hom_cring_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "ring_hom_cring R S h \<equiv> cring R \<and> cring S \<and> ring_hom_cring_axioms R S h"
iota_def: fixes F :: "('e, 'f) Group_scheme" shows "\<iota>\<^bsub>F\<^esub> = (\<lambda>x\<in>carrier F. x)"
gt_half_sum: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a b" shows "less (divide (plus a b) (plus one one)) b"
ap_congL: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and f :: "'a itrm"   and f' :: "'a itrm"   and x :: "'a itrm" assumes "itrm_cong R f f'" shows "itrm_cong R (f \<diamondop> x) (f' \<diamondop> x)"
upto_enum_nat: fixes n :: "nat"   and m :: "nat" shows "[n .e. m] = [n..<Suc m]"
rejects_trace_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and p :: "value list"   and t :: "(String.literal \<times> value list \<times> value list) list" shows "rejects_trace e s r ((l, i, p) # t) = (\<forall>(s', T)|\<in>|possible_steps e s r l i. evaluate_outputs T i r \<noteq> map Some p \<or> rejects_trace e s' (evaluate_updates T i r) t)"
normal_form_1: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "(punit.red (punit.reduced_GB F))\<^sup>*\<^sup>* p (normal_form F p)"
bigo_bounded: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x. (0::'d) \<le> f x"   and "\<forall>x. f x \<le> g x" shows "f \<in> O(g)"
reconstruct_poly_def: fixes c :: "'a"   and roots :: "'a list" shows "reconstruct_poly c roots = Polynomial.smult c (\<Prod>a\<leftarrow>roots. [:- a, 1::'a:])"
less_divide_eq_1_neg: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less one (divide b a) = less b a"
size_partition: fixes p :: "'b"   and t :: "'b tree"   and l' :: "'b tree"   and r' :: "'b tree" assumes "Splay_Heap.partition p t = (l', r')" shows "size t = size l' + size r'"
FOREACHi_def: fixes \<Phi> :: "'e set \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'e set" shows "FOREACH\<^bsup>\<Phi>\<^esup> S \<equiv> FOREACH\<^sub>C\<^bsup>\<Phi>\<^esup> S (\<lambda>_. True)"
comp_null_1: fixes f :: "'a arrow_of_spans_data" shows "V.null \<bullet> f = V.null"
shiftD: fixes x :: "hf"   and f :: "hf"   and delta :: "hf" assumes "x \<^bold>\<in> shift f delta" shows "\<exists>u. u \<^bold>\<in> f \<and> x = \<langle>delta @+ hfst u, hsnd u\<rangle>"
int_lt_Suc_le: fixes a :: "real"   and b :: "real" assumes "a \<in> \<int>"   and "b \<in> \<int>"   and "a < b + 1" shows "a \<le> b"
intro: fixes G :: "('c, 'd) ring_scheme" assumes "Group.comm_monoid (add_monoid G)" shows "abelian_monoid G"
dom_ord: fixes x :: "'a"   and y :: "'a" shows "(ad (ad x) \<le> ad (ad y)) = (ad (ad x) \<cdot> ad (ad y) = ad (ad x))"
inj_list_encode: fixes A :: "nat list set" shows "inj_on list_encode A"
isomorphic_implies_hpar_3: fixes f :: "'a"   and f' :: "'a" assumes "B.isomorphic f f'" shows "src\<^sub>B f = src\<^sub>B f'"
VV_impl2: fixes v :: "'a"   and p :: "Player" assumes "v \<in> VV p**" shows "v \<notin> VV p"
one_or_two_sides_aux: fixes C :: "'p"   and A :: "'p"   and B :: "'p"   and D :: "'p"   and X :: "'p" assumes "\<not> Col C A B"   and "\<not> Col D A B"   and "Col A C X"   and "Col B D X" shows "A B TS C D \<or> A B OS C D"
implies_not_eq: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "irreflexive A less"   and "less x y"   and "x \<in> A" shows "x \<noteq> y"
not_choice_simps_8: fixes l :: "'d"   and fdata :: "'a \<Rightarrow> nat"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_choice ({l}deliver(fdata) . p)"
iso_can: fixes f :: "'c local.term set" assumes "can f" shows "local.iso f"
monoD: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.preord le"   and "preord.mono le f"   and "le x y" shows "le (f x) (f y)"
preserves_inverse_arrows: fixes f :: "'a"   and g :: "'a" assumes "A.inverse_arrows f g" shows "B.inverse_arrows (F f) (F g)"
is_symbol_def: fixes s :: "symbol" shows "is_symbol s = (is_terminal s \<or> is_nonterminal s)"
cons_to_append: fixes a :: "'b"   and b :: "'b list" shows "a # b = [a] @ b"
not_less_zero: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
hyps: shows "rotate (l * \<^bold>|u\<^bold>|) (u \<^sup>@ k) = u \<^sup>@ k"
induct: fixes P :: "(natural \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> natural \<Rightarrow> bool"   and a0 :: "natural \<Rightarrow> (bool \<times> term list) option"   and a1 :: "natural"   and a2 :: "natural" assumes "\<And>f d i. (\<not> d < i \<Longrightarrow> P f d (i + 1)) \<Longrightarrow> P f d i" shows "P a0 a1 a2"
eigen_value_root_charpoly: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and k :: "'b" shows "eigen_value A k = (poly (charpoly A) k = (0::'b))"
rel_propE: fixes c :: "nat \<Rightarrow> nat"   and s :: "nat" assumes "c \<in> {0..<k} \<rightarrow> {0..l}"   and "s < k"   and "\<forall>j\<le>s. c j < l" shows "col (f c) = col (f (\<lambda>i. if i \<le> s then 0 else c i))"
inner_simps_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<bullet> (y - z) = x \<bullet> y - x \<bullet> z"
similar_mat_witD_2: fixes n :: "nat"   and A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "Q * P = 1\<^sub>m n"
AGM_Contraction_axioms: shows "AGM_Contraction Cn (\<div>)"
rep_eq: fixes x :: "('b, 'c) mapping" shows "Mapping.lookup x = mapping.rep x"
transpos_def: fixes i :: "nat"   and j :: "nat" shows "transpos i j = (\<lambda>k. if k = i then j else if k = j then i else k)"
height_balL: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "avl l"   and "avl r"   and "height l = height r + 2" shows "height (balL l a r) \<in> {height r + 2, height r + 3}"
A: shows "g_image_filter f m = iterate_to_map m2.empty m2.update_dj (set_iterator_image_filter f (m1.iteratei m))"
no_overlaps_lroft_hlp2: fixes amr :: "('b \<times> 'c \<times> 'd \<times> 'e) list"   and fm :: "'c \<Rightarrow> 'f list"   and fs :: "'e \<Rightarrow> 'd \<Rightarrow> 'g" assumes "distinct (map fst amr)"   and "\<And>r. distinct (fm r)" shows "distinct (concat (map (\<lambda>(p, r, c, a). map (\<lambda>b. (p, b, fs a c)) (fm r)) amr))"
domin_1: shows "\<forall>n. (flow0 x \<circ> s) n \<in> dom"
INF_lower2: fixes i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd \<times> 'e"   and u :: "'d \<times> 'e" assumes "s2p A i"   and "less_eq_prod (f i) u" shows "less_eq_prod (Inf_prod (\<P> f A)) u"
run_fail_option: fixes return :: "'e option \<Rightarrow> 'd" shows "run_option (fail_option return) = return None"
iter_hcomp_empty: fixes p :: "'b \<Rightarrow> interface" shows "iter_hcomp {||} p = \<epsilon>"
TID_self_zero: fixes x :: "'c Gromov_completion" shows "turn_into_distance extended_predist x x = 0"
top_meet: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "\<top>\<^bsub>L\<^esub> \<sqinter>\<^bsub>L\<^esub> x = x"
conditional_mutual_information_eq: fixes Z :: "'a \<Rightarrow> 'd"   and Pz :: "'d \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'e"   and Pyz :: "'e \<times> 'd \<Rightarrow> real"   and X :: "'a \<Rightarrow> 'f"   and Pxz :: "'f \<times> 'd \<Rightarrow> real"   and Pxyz :: "'f \<times> 'e \<times> 'd \<Rightarrow> real" assumes "simple_distributed M Z Pz"   and "simple_distributed M (\<lambda>x. (Y x, Z x)) Pyz"   and "simple_distributed M (\<lambda>x. (X x, Z x)) Pxz"   and "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Pxyz" shows "\<I>( X ; Y | Z ) = (\<Sum>(x, y, z)\<in>(\<lambda>x. (X x, Y x, Z x)) ` space M. Pxyz (x, y, z) * log b (Pxyz (x, y, z) / (Pxz (x, z) * (Pyz (y, z) / Pz z))))"
left_adjoint_extends_to_adjoint_pair: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "chosen_right_adjoints V H \<a> \<i> src trg"   and "\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g" shows "bicategory.adjoint_pair V H \<a> \<i> src trg f (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg f)"
Ide_implies_Can: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Ide V src trg t" shows "bicategorical_language.Can V src trg t"
pointermap_sane_getmkD: fixes m :: "'a pointermap"   and a :: "'a"   and p :: "nat"   and u :: "'a pointermap" assumes "pointermap_sane m"   and "pointermap_getmk a m = (p, u)" shows "pointermap_sane u \<and> pointermap_p_valid p u"
inf_sup_aci_5: fixes inf :: "'l \<Rightarrow> 'l \<Rightarrow> 'l"   and less_eq :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and less :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and sup :: "'l \<Rightarrow> 'l \<Rightarrow> 'l"   and x :: "'l"   and y :: "'l" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
mem_UnitsD: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "monoid M composition unit"   and "u \<in> monoid.Units M composition unit" shows "monoid.invertible M composition unit u \<and> u \<in> M"
HaddP_SUCC1: fixes x :: "tm"   and y :: "tm"   and z :: "tm" shows "{HaddP x y z} \<turnstile> HaddP (SUCC x) y (SUCC z)"
ord0_linear_lemma: fixes m :: "ord0"   and n :: "ord0" shows "m < n \<or> n < m \<or> m \<le> n \<and> n \<le> m"
simps: fixes a :: "('s, 'v) tm" shows "wary a = ((\<exists>\<zeta>. a = Hd \<zeta>) \<or> (\<exists>s t. a = App s t \<and> wary s \<and> wary t \<and> enat (num_args s) < arity_hd (head s)))"
strict_implies_not_eq: fixes a :: "nat list"   and b :: "nat list" assumes "a <\<^sub>v b" shows "a \<noteq> b"
h5: shows "MPoly_Type.coeff (lowerPoly i j (liftPoly i j (p1 + p2))) = MPoly_Type.coeff (lowerPoly i j (liftPoly i j p1)) + MPoly_Type.coeff (lowerPoly i j (liftPoly i j p2))"
EMPTY: shows "GEN_OP empty2 {} (\<langle>Rk'\<rangle>Rs2)"
similar_triangle_aa: fixes b1 :: "'b"   and c1 :: "'b"   and b2 :: "'c"   and c2 :: "'c"   and a1 :: "'b"   and a2 :: "'c" assumes "b1 \<noteq> c1"   and "b2 \<noteq> c2"   and "angle a1 b1 c1 = angle a2 b2 c2"   and "angle b1 c1 a1 = angle b2 c2 a2" shows "angle b1 a1 c1 = angle b2 a2 c2"
bsplit: shows "ball x e = b1 \<union> b2 \<union> b3"
wbisim23: shows "delay_bisimulation_diverge trsys2 trsys3 bisim23 tlsim23 \<tau>move2 \<tau>move3"
negligible_powerI: fixes k :: "real" assumes "\<bar>k\<bar> < 1" shows "negligible ((^) k)"
rts0: shows "\<forall>p t i. correct p t \<and> t \<le> te p (i + 1) \<longrightarrow> t - te p i \<le> rmax"
induct: fixes P :: "'a option \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a option"   and a1 :: "'a list"   and a2 :: "'a list" assumes "\<And>ao. P ao [] []"   and "\<And>ao y ys. P (Some y) [] ys \<Longrightarrow> P ao [] (y # ys)"   and "\<And>ao x xs. P (Some x) xs [] \<Longrightarrow> P ao (x # xs) []"   and "\<And>ao x xs y ys. \<lbrakk>x < y \<Longrightarrow> P (Some x) xs (y # ys); \<lbrakk>\<not> x < y; y < x\<rbrakk> \<Longrightarrow> P (Some y) (x # xs) ys\<rbrakk> \<Longrightarrow> P ao (x # xs) (y # ys)" shows "P a0 a1 a2"
count_mset_set_leq': fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "count (mset_set A) x \<le> Suc 0"
Upd_eq_restr_stackD2: fixes V :: "Vars.var set"   and S' :: "stack_elem list"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "restr_stack V S' = Upd x # S" shows "x \<in> V"
simps: fixes i :: "nat" shows "fs.gs.gso i = map of_int_hom.vec_hom fs ! i + gs.M.sumlist (map (\<lambda>j. - fs.gs.\<mu> i j \<cdot>\<^sub>v fs.gs.gso j) [0..<i])"
small_insert: fixes X :: "'a set"   and a :: "'a" assumes "small X" shows "small (insert a X)"