hcju/isabelleps huggingface dataset:
matrix_scaleR_left: fixes a :: "(('a, 'd) vec, 'c) vec"   and r :: "real"   and b :: "(('a, 'b) vec, 'd) vec" shows "a ** r *\<^sub>R b = r *\<^sub>R (a ** b)"
ide_src: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "B.ide (src\<^sub>B \<mu>)"
h: shows "insertion (nth_default 0 (xs @ x # \<Gamma>)) p = 0 \<and> eval (qe_eq_repeat_helper var A' L F) (xs @ x # \<Gamma>)"
a: shows " M \<longrightarrow>\<^sub>\<beta> M'"
intro: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> (nat \<times> 'b) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list option"   and M_invar :: "'a \<Rightarrow> bool" assumes "Map M_empty M_update M_delete M_lookup M_invar" shows "wgraph_by_map M_empty M_update M_delete M_lookup M_invar"
cases: fixes x :: "('f, 'g, 'd, 'e, 'h) qAbs" obtains xs :: "'d" and x :: "'e" and X :: "('f, 'g, 'd, 'e, 'h) qTerm" where "xa__ = qAbs xs x X"
\<tau>inv_flip: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool" assumes "\<tau>inv trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2" shows "\<tau>inv trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1"
extend_set_Union: fixes A :: "'a set set" shows "extend_set h (\<Union> A) = \<Union> (extend_set h ` A)"
ring_n_pd_tOp_K_tOp: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and x :: "'a"   and y :: "'a" assumes "distinct_pds K n P"   and "x \<in> carrier (O\<^bsub>K P n\<^esub>)"   and "y \<in> carrier (O\<^bsub>K P n\<^esub>)" shows "x \<cdot>\<^sub>r\<^bsub>O\<^bsub>K P n\<^esub>\<^esub> y = x \<cdot>\<^sub>r y"
assms_3: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'"
assms_1: shows "h \<turnstile> ok (f \<bind> g)"
row_transpose: fixes i :: "'c"   and A :: "(('a, 'c) vec, 'b) vec" shows "row i (Finite_Cartesian_Product.transpose A) = column i A"
respects_cut: fixes A :: "('v \<times> 'v) set" shows "respects_cut A (S A)"
summable_LIMSEQ: fixes f :: "nat \<Rightarrow> 'e" assumes "summable f" shows "(\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> suminf f"
abs_eq: fixes x :: "'a list" shows "vec_of_list x = Abs_vec (length x, mk_vec (length x) ((!) x))"
map_id: fixes t :: "('a, 'b) fmap" shows "fmmap id t = t"
assms_2: shows "\<forall>k\<ge>n. coeff p k = (0::'a)"
rev_iterateoi_correct: fixes S :: "'s" assumes "invar S" shows "set_iterator_rev_linord (reverse_iterateoi S) (\<alpha> S)"
comp_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and u :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t"   and "monoidal_language.Arr C u"   and "monoidal_language.Dom C t = monoidal_language.Cod C u" shows "free_monoidal_category.comp C (free_monoidal_category.mkarr C t) (free_monoidal_category.mkarr C u) = free_monoidal_category.mkarr C (t \<^bold>\<cdot> u)"
synth_Int2: fixes x :: "msg"   and A :: "msg set"   and B :: "msg set" assumes "x \<in> synth (A \<inter> B)" shows "x \<in> synth B"
sums_cnj: fixes f :: "nat \<Rightarrow> complex"   and l :: "complex" shows "(\<lambda>x. cnj (f x)) sums cnj l = f sums l"
add_diff_cancel_left': fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) a = b"
2: shows "would_be_Cauchy u \<in> extended_distance_set"
FUA: shows "FUadmit \<sigma> \<phi> UNIV"
assms: fixes Q :: "ccs" shows "(Q \<parallel> \<zero>, Q) \<in> Rel"
comm_inv_char: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<otimes> y = \<one>" shows "inv x = y"
to_ns_sat: fixes cs :: "('a \<times> constraint) list"   and v' :: "(nat, QDelta) mapping" assumes "\<langle>v'\<rangle> \<Turnstile>\<^sub>n\<^sub>s\<^sub>s  (snd ` set (to_ns cs))" shows "\<langle>from_ns v' (map snd (to_ns cs))\<rangle> \<Turnstile>\<^sub>c\<^sub>s (snd ` set cs)"
Lim_measure_incseq: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "incseq A"   and "emeasure M (\<Union> (range A)) \<noteq> \<infinity>" shows "(\<lambda>i. Sigma_Algebra.measure M (A i)) \<longlonglongrightarrow> Sigma_Algebra.measure M (\<Union> (range A))"
ipath_to_rtrancl: fixes E :: "('e \<times> 'e) set"   and r :: "nat \<Rightarrow> 'e"   and i1 :: "nat"   and i2 :: "nat" assumes "ipath E r"   and "i1 \<le> i2" shows "(r i1, r i2) \<in> E\<^sup>*"
NE_intT: fixes \<sigma> :: "'tp" shows "\<exists>a. intT \<sigma> a"
vsubstAbs_ident: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and zs :: "'varSort"   and z :: "'var" assumes "goodAbs A" shows "A $[z // z]_zs = A"
vconverse_atD: fixes b :: "V"   and a :: "V" assumes "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>" shows "r\<lparr>a\<rparr> = b"
max_0R: fixes n :: "nat" shows "max n 0 = n"
pow_is_concat_replicate: fixes u :: "'b list"   and n :: "nat" shows "u \<^sup>@ n = concat (replicate n u)"
swap_self: fixes a :: "'d"   and f :: "'d \<Rightarrow> 'e" shows "Fun.swap a a f = f"
p_prod_oner: fixes X :: "'a list set" shows "p_prod X p_one = X"
hyps_3: shows "bn \<alpha> \<sharp>* (P \<parallel> (Q \<parallel> R))"
one_ow_def: fixes U :: "'b set"   and one :: "'b" shows "one_ow U one \<equiv> neutral_ow U one"
principal_means_premiss: fixes r :: "Gp sequent list \<times> Gp sequent"   and A :: "Gp form"   and B :: "Gp form" assumes "rightPrincipal r (A \<and>* B)"   and "r \<in> g3cp" shows "( \<Empt> \<Rightarrow>* \<LM> A  \<RM>) \<in> set (fst r)"
lebesgue_integral_point_measure_finite: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a" shows "integral\<^sup>L (point_measure A (\<lambda>x. ennreal (f x))) g = (\<Sum>a\<in>A. f a *\<^sub>R g a)"
dom_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.dom (C g f) = C.dom f"
proj_Cons_3: fixes m :: "'a"   and n :: "'a"   and a :: "'b"   and A :: "('a strand_label \<times> 'b) list" assumes "m \<noteq> n" shows "proj n ((ln m, a) # A) = proj n A"
better: shows "T\<^sub>p init qs (take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat) + 1 = T\<^sub>p init qs Strat"
abs_eq: shows "max_sls = Abs_semilattice_set max"
UP_is_equivalence_pseudofunctor: shows "equivalence_pseudofunctor (\<cdot>) (\<star>) \<a> \<i> src trg (\<cdot>\<^sub>S) (\<star>\<^sub>S) S.\<a> S.\<i> S.src S.trg S.UP S.cmp\<^sub>U\<^sub>P"
fun_sum_list_commute_canc: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list" assumes "\<And>x y. f (x + y) = f x + f y" shows "f (sum_list xs) = sum_list (map f xs)"
gcd_assoc: 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 "somegcd G (somegcd G a b) c \<sim> somegcd G a (somegcd G b c)"
less_numeral_extra_3: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less zero zero"
weak_ranking_aux_Union: fixes A :: "'a set" assumes "A \<subseteq> carrier" shows "\<Union> (set (weak_ranking_aux A)) = A"
indep_var_distribution_eq: fixes S :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and T :: "'b measure"   and Y :: "'a \<Rightarrow> 'b" shows "indep_var S X T Y = (random_variable S X \<and> random_variable T Y \<and> distr M S X \<Otimes>\<^sub>M distr M T Y = distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
wprepare_loop_goon_on_rightbmost_Bk_False: fixes lm :: "nat list"   and m :: "nat"   and b :: "cell list"   and a :: "cell"   and lista :: "cell list" assumes "lm \<noteq> []"   and "wprepare_loop_start_on_rightmost m lm (b, Bk # a # lista)" shows "wprepare_loop_goon_on_rightmost m lm (Bk # b, a # lista)"
assms_2: assumes "\<And>r. \<lbrakk>r dvd p; r dvd q\<rbrakk> \<Longrightarrow> r dvd (1::'a)" shows "thesis"
n: shows "(derived G ^^ n) I = {\<one>\<^bsub>G\<^esub>}"
int: shows "(\<lambda>x. prod (f x) (g' x)) integrable_on {a..b}"
simps: fixes f :: "'op"   and ts :: "'op rose_tree list" shows "U_sum (T f ts) = U f (map state ts) + sum_list (map U_sum ts)"
store_buffer_execution_result_sequential_consistent: fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and x :: "'c"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and x' :: "'c"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(ts\<^sub>s\<^sub>b, m, x) \<Rightarrow>\<^sub>s\<^sub>b\<^sup>* (ts\<^sub>s\<^sub>b', m', x')"   and "empty_store_buffers ts\<^sub>s\<^sub>b'"   and "ts\<^sub>s\<^sub>b \<sim>\<^sub>d ts "   and "initial\<^sub>d ts \<S> valid"   and "safe_reach_direct safe_free_flowing (ts, m, \<S>)" shows "\<exists>ts' \<S>'. (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>') \<and> ts\<^sub>s\<^sub>b' \<sim>\<^sub>d ts' "
game_def: fixes \<A> :: "(unit, bool list + bool list \<times> 'hash, 'hash + unit) gpv" shows "UPF.game \<A> = upf_key_gen \<bind> (\<lambda>key. exec_gpv (UPF.oracle key) \<A> (False, {}) \<bind> (\<lambda>(uu_, flag', L'). return_spmf flag'))"
prod_in_hom: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and d :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> c\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> d\<guillemotright>" shows "\<guillemotleft>HF'.prod f g : HF'.prod a b \<rightarrow> HF'.prod c d\<guillemotright>"
B_Rep: fixes b :: "'bl"   and b' :: "'br" shows "B b b' = (b = Rep_b b')"
\<rho>_def: shows "\<rho> \<equiv> run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w"
map\<^sub>0_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (\<rho>\<^sub>0 a) = G.map\<^sub>0 a"
inf_right_idem: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) y = inf x y"
that: fixes q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "q \<in> set (comp_min_basis xs)"   and "lt q adds\<^sub>t lt p" shows "thesis"
group_hom_def: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f" shows "group_hom G H h \<equiv> Group.group G \<and> Group.group H \<and> group_hom_axioms G H h"
measure_spmf_cong: fixes A :: "'h set"   and p :: "'h spmf"   and B :: "'h set" assumes "A \<inter> set_spmf p = B \<inter> set_spmf p" shows "Sigma_Algebra.measure (measure_spmf p) A = Sigma_Algebra.measure (measure_spmf p) B"
echelon_foldl_condition5: fixes mb :: "'d"   and k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec" assumes "\<not> is_zero_row_upt_k mb k A"   and "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) = nrows A" shows "nrows A = Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) A))"
pfinite_comp_converter: fixes \<I>1 :: "('y, 'z) \<I>"   and \<I>2 :: "('aa, 'ab) \<I>"   and conv1 :: "('y, 'z, 'aa, 'ab) converter"   and \<I>3 :: "('ac, 'ad) \<I>"   and conv2 :: "('aa, 'ab, 'ac, 'ad) converter" assumes "pfinite_converter \<I>1 \<I>2 conv1"   and "pfinite_converter \<I>2 \<I>3 conv2"   and "\<I>1, \<I>2 \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>2, \<I>3 \<turnstile>\<^sub>C conv2 \<surd>" shows "pfinite_converter \<I>1 \<I>3 (conv1 \<odot> conv2)"
IH: fixes y :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and ys :: "'a list" assumes "y < n"   and "y \<le> length xs"   and "bal y xs = (t, ys)" shows "wbalanced t"
pos_add_strict: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "{#} \<subset># a"   and "b \<subset># c" shows "b \<subset># a + c"
foldr_map: fixes g :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and a :: "'c" shows "foldr g (map f xs) a = foldr (g \<circ> f) xs a"
MaxR_def: fixes r :: "('d \<times> 'd) set" shows "MaxR r \<equiv> Linear_order r"
assms_2: shows "\<forall>\<^sub>F w in at z. g w = f w"
SD_pref_profileD_2: fixes i :: "'a"   and q :: "'b pmf"   and p :: "'b pmf" assumes "i \<in> agents"   and "SD (R i) q p" shows "q \<in> lotteries_on alts"
has_integral_iff: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and S :: "'b set" shows "(f has_integral i) S = (f integrable_on S \<and> integral S f = i)"
cong_minus_minus_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo (uminus b) (uminus c) a = unique_euclidean_semiring.cong modulo b c a"
remove_elem: fixes A :: "'d set"   and B :: "'d set set"   and b :: "'d set" assumes "Congruence.partition A B"   and "b \<in> B" shows "Congruence.partition (A - b) (B - {b})"
del_arc_simps_2: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "arcs (pre_digraph.del_arc G a) = arcs G - {a}"
dual_Sup: 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"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and post :: "'c \<Rightarrow> 'c"   and X :: "'c set" assumes "class.complete_mbt_algebra Inf Sup inf less_eq less sup bot top one times dual dual_star omega star post" shows "dual (Sup X) = Inf (dual ` X)"
simps_3: fixes xs :: "char list list"   and n :: "int"   and ys :: "char list list"   and m :: "int"   and p :: "(char list list \<times> int) list" shows "merge_coeffs ((xs, n) # (ys, m) # p) = (if xs = ys then if n + m \<noteq> 0 then merge_coeffs ((xs, n + m) # p) else merge_coeffs p else (xs, n) # merge_coeffs ((ys, m) # p))"
chine_in_hom: shows "\<guillemotleft>\<nu>.chine : \<nu>.dom.apex \<rightarrow>\<^sub>C \<nu>.cod.apex\<guillemotright>"
relax_finished_id: fixes v :: "'v"   and u :: "'v" assumes "v \<in> S" shows "relax_outgoing u D v = D v"
emeasure_PiM_exponential_Ioi_finite: fixes J :: "'d set"   and I :: "'d set"   and R :: "'d \<Rightarrow> real"   and x :: "real" assumes "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> I \<Longrightarrow> 0 < R i"   and "0 \<le> x" shows "emeasure (Pi\<^sub>M I (\<lambda>i. exponential (R i))) (prod_emb I (\<lambda>i. exponential (R i)) J (J \<rightarrow>\<^sub>E {x<..})) = ennreal (exp (- x * sum R J))"
deriv_closed: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp" shows "deriv f a \<in> carrier Zp"
separable: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "x \<noteq> y" shows "\<exists>f\<in>R. f x \<noteq> f y"
simps_3: fixes uu :: "'b" shows "is_lit_plus (Atom uu) = True"
assms_5: shows "qf1 x1 y1 = 0"
cyclic_groupI2: fixes a :: "'a" assumes "a \<in> carrier G" shows "Finite_And_Cyclic_Groups.cyclic_group (G\<lparr>carrier := generate G {a}\<rparr>) a"
arctan_mult_le: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 \<le> x"   and "x \<le> y"   and "y * z \<le> arctan y" shows "x * z \<le> arctan x"
td_ext_unat: shows "td_ext unat word_of_nat (unats LENGTH('a)) (\<lambda>i. i mod 2 ^ LENGTH('a))"
wf_mbuf2'_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and j :: "nat"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and buf :: "event_data option list set list \<times> event_data option list set list" shows "wf_mbuf2' \<sigma> P V j n R \<phi> \<psi> buf = wf_mbuf2 (min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)) (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j) (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>)) buf"
equivI: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "refl_on A r"   and "sym r"   and "trans r" shows "equiv A r"
inject: fixes nat :: "nat"   and nat' :: "nat" shows "(enat nat = enat nat') = (nat = nat')"
igSubstCls_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.igSubstCls Delta MOD \<equiv> FixSyn.igSubstIGVar1 Delta MOD \<and> FixSyn.igSubstIGVar2 Delta MOD \<and> FixSyn.igSubstIGAbs Delta MOD \<and> FixSyn.igSubstIGOp Delta MOD"
hull_rel_relcomp_trivialI: fixes x :: "'b"   and y :: "'c"   and R :: "('b \<times> 'c) set" assumes "(x, y) \<in> R" shows "(x, y) \<in> hull_rel O R"
3: shows "continuous_on S (\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
varTrue: fixes b :: "BExpr"   and t :: "Var \<Rightarrow> Val"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and r :: "Var \<Rightarrow> Val"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val" assumes "evalB b t"   and "\<Phi> (r, t)"   and "\<forall>w. A r w \<longrightarrow> (b, A, \<Phi>, s, w) \<in> var" shows "(b, A, \<Phi>, s, t) \<in> var"
left_neutral: fixes a :: "'b::bounded_semilattice_inf_top" shows "\<top> \<sqinter> (a::'b::bounded_semilattice_inf_top) = a"
assms_1: fixes A :: "'a set" assumes "A \<in> sets M" shows "\<integral>\<^sup>+ x. indicator A x * g x \<partial>M = \<integral>\<^sup>+ x. indicator A (T x) * f x \<partial>M"
association_class_end_det: fixes classes :: "'a fset"   and associations :: "String.literal \<rightharpoonup>\<^sub>f String.literal \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and association_classes :: "'a \<rightharpoonup>\<^sub>f String.literal"   and \<A> :: "'a"   and role :: "String.literal"   and end\<^sub>1 :: "'a \<times> nat \<times> enat \<times> bool \<times> bool"   and end\<^sub>2 :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" assumes "object_model classes associations"   and "unique_association_class_end association_classes associations \<A> role end\<^sub>1"   and "unique_association_class_end association_classes associations \<A> role end\<^sub>2" shows "end\<^sub>1 = end\<^sub>2"
conv_radius_def: fixes f :: "nat \<Rightarrow> 'b" shows "conv_radius f = inverse (limsup (\<lambda>n. ereal (root n (norm (f n)))))"
elims_3: fixes x :: "(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 xa :: "char list \<Rightarrow> ty option"   and xb :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and xc :: "expr list"   and xd :: "ty list" assumes "\<not> x,xa,xb \<turnstile> xc [:]\<^bsub>NT\<^esub> xd" obtains 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 E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and e :: "expr" and es :: "expr list" and T :: "ty" and Ts :: "ty list" where "x = Pa__" and "xa = Ea__" and "xb = h__" and "xc = e # es" and "xd = T # Ts" and "\<not> (Pa__,Ea__,h__ \<turnstile> e :\<^bsub>NT\<^esub> T \<and> Pa__,Ea__,h__ \<turnstile> es [:]\<^bsub>NT\<^esub> Ts)"   | 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 E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and v :: "expr" and va :: "expr list" where "x = Pb__" and "xa = Eb__" and "xb = ha__" and "xc = v__ # va__" and "xd = []"   | 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 E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and v :: "ty" and va :: "ty list" where "x = P" and "xa = E" and "xb = h" and "xc = []" and "xd = v # va"
tranclp_trancl: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "r\<^sup>+\<^sup>+ = (\<lambda>x y. (x, y) \<in> {(a, b). r a b}\<^sup>+)"
preds_finite: fixes g :: "'g"   and \<phi> :: "'val" shows "finite (trueArgs g \<phi>)"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> carrier (add_monoid G); y \<in> carrier (add_monoid G)\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>add_monoid G\<^esub> y = y \<otimes>\<^bsub>add_monoid G\<^esub> x" shows "Group.comm_monoid (add_monoid G)"
bezout_int: fixes x :: "int"   and y :: "int" shows "\<exists>u v. u * x + v * y = gcd x y"
fold_insert_remove: fixes A :: "'a set"   and z :: "'b"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z (A - {x}))"
lm054: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "argmax f A = A \<inter> {x. f x = Max (f ` A)}"
reads_consistent_takeWhile: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and P :: "'b memref \<Rightarrow> bool" assumes "reads_consistent pending_write \<O> m sb" shows "reads_consistent pending_write \<O> m (takeWhile P sb)"
open_ereal_lessThan: fixes a :: "ereal" shows "open {..<a}"
ring_nsum_zero: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j = \<zero>" shows "\<Sigma>\<^sub>e A f n = \<zero>"