hcju/isabelleps huggingface dataset:
f\<^sub>a': shows "((r \<star> \<epsilon>) \<cdot> \<a>[r, f, f\<^sub>a] \<cdot> (\<rho> \<star> f\<^sub>a)) \<cdot> \<nu> = \<r>\<^sup>-\<^sup>1[r]"
integral_bij_count_space: fixes g :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "bij_betw g A B" shows "LINT x|count_space A. f (g x) = integral\<^sup>L (count_space B) f"
step_pos: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and i :: "nat"   and x :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "i < k"   and "x\<^sub>1 \<le> x" shows "0 < bs ! i * x + (hs ! i) x"
ustar_unfoldl: fixes P :: "'b hrel" shows "P\<^sup>\<star> = II \<sqinter> (P ;; P\<^sup>\<star>)"
distinct_1: fixes x1 :: "'b"   and x2 :: "'b" shows "Le x1 \<noteq> Lt x2"
1: fixes x :: "nat \<Rightarrow> 'a" shows "matricize I (T x) $$ (i, j) = Tensor.lookup (T x) (weave I (digit_encode (nths ds I) i) (digit_encode (nths ds (- I)) j))"
circline_def: fixes A :: "real"   and BC :: "complex"   and D :: "real" shows "circline A BC D = {z. cor A * z * cnj z + cnj BC * z + BC * cnj z + cor D = 0}"
no_output_none_aux: fixes j :: "state stream"   and e :: "((nat \<times> nat) \<times> transition) fset" assumes "\<exists>p r i. j = make_full_observation e None r [] i" shows "alw (output_eq []) j"
cong_dvd_modulus_nat: fixes x :: "nat"   and y :: "nat"   and m :: "nat"   and n :: "nat" assumes "[x = y] (mod m)"   and "n dvd m" shows "[x = y] (mod n)"
finite_def: shows "class.finite TYPE('b) \<equiv> finite UNIV"
order_iff: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" shows "(b \<preceq> a) = (a = linordered_monoid.max a b)"
q_cong: fixes a :: "nat" shows "[a = q + a] (mod q)"
exp_cf5_upper_bound_pos: fixes x :: "real" assumes "0 \<le> x"   and "0 < numer_cf5 (- x)" shows "exp x \<le> exp_cf5 x"
assms_11: shows "dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>"
simps_4: fixes \<nu> :: "'a ltln"   and \<nu>' :: "'a ltln" shows "pure_universal (\<nu> or\<^sub>n \<nu>') = (pure_universal \<nu> \<and> pure_universal \<nu>')"
awalk_ends: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "S.awalk u p v"   and "S.awalk u' p v'" shows "p \<noteq> [] \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
prems_2: shows "all_in_list (x # a # xs) l"
closeI_provable_sound: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat" assumes "sound (SG, C)"   and "sound (closeI SG i, SG ! i)" shows "sound (closeI SG i, C)"
eq: shows "(\<Sum>s\<in>keys p - {t}. lookup (monomial (lookup p s) (monomial (poly_deg p - deg_pm s) x + s)) (monomial (poly_deg p - deg_pm t) x + t)) = (0::'b)"
v11_vintersection: fixes r :: "V"   and s :: "V" assumes "v11 r" shows "v11 (r \<inter>\<^sub>\<circ> s)"
fp_cop_F_range_inv: fixes ds :: "'b set" shows "cop_F_range_inv ds (fp_cop_F ds)"
factorize_polys_map_coprime: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and fss :: "real poly list"   and f :: "real poly"   and g :: "real poly" assumes "factorize_polys ps = (fs, data)"   and "fss = map real_of_rat_poly fs"   and "f \<in> set fss"   and "g \<in> set fss"   and "f \<noteq> g" shows "comm_monoid_mult_class.coprime f g"
extreme_point_of_convex_hull: fixes x :: "'b"   and S :: "'b set" assumes "x extreme_point_of convex hull S" shows "x \<in> S"
finite_img_if_fsubst: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "s \<in> fsubst" shows "finite (subst_range s)"
lgc_to_red: fixes Ns :: "('f inference set \<times> ('f \<times> 'l) set) llist" assumes "chain (\<leadsto>LGC) Ns" shows "chain (\<rhd>L) (lmap snd Ns)"
intro: fixes G :: "('d, 'f, 'e, 'g, 'h) galcon_scheme" assumes "partial_order \<X>\<^bsub>G\<^esub>"   and "partial_order \<Y>\<^bsub>G\<^esub>"   and "\<pi>\<^sup>*\<^bsub>G\<^esub> \<in> carrier \<X>\<^bsub>G\<^esub> \<rightarrow> carrier \<Y>\<^bsub>G\<^esub>"   and "\<pi>\<^sub>*\<^bsub>G\<^esub> \<in> carrier \<Y>\<^bsub>G\<^esub> \<rightarrow> carrier \<X>\<^bsub>G\<^esub>" shows "connection G"
verit_minus_simplify_2: fixes a :: "'d" shows "a - (0::'d) = a"
mk_minsky_add1: fixes v :: "nat"   and w :: "nat" assumes "v \<noteq> w" shows "mk_minsky (\<lambda>vs vs'. vs' = (\<lambda>x. if x = v then 0 else if x = w then vs v + vs w else vs x))"
in_rtrancl_UnI: fixes x :: "'c \<times> 'c"   and R :: "('c \<times> 'c) set"   and S :: "('c \<times> 'c) set" assumes "x \<in> R\<^sup>* \<or> x \<in> S\<^sup>*" shows "x \<in> (R \<union> S)\<^sup>*"
simps_2: fixes x :: "'a" shows "the_default x None = x"
emeasure_space_le_1: fixes M :: "'c measure" assumes "subprob_space M" shows "emeasure M (space M) \<le> 1"
l3_defs_5: fixes implem :: "chan \<Rightarrow> msg" assumes "pfslvl3 implem" shows "l3_init \<equiv> {\<lparr>ik = ik_init, secret = {}, progress = Map.empty, signals = \<lambda>x. 0, l3_state.bad = bad_init\<rparr>}"
setmult_subset_G: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "H <#>\<^bsub>add_monoid G\<^esub> K \<subseteq> carrier G"
injectivity: fixes x :: "'b poly"   and y :: "'b poly" assumes "pCons (0::'b) x = pCons (0::'b) y" shows "x = y"
assm: shows "v=v1--v2 \<and> v1=v3--v4"
leading_monomial_tail: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "p = monomial (i.lcf p) (i.lpp p) + i.punit.tail p"
l_neg: fixes x :: "'e" assumes "x \<in> carrier N" shows "\<ominus>\<^bsub>N\<^esub> x \<oplus>\<^bsub>N\<^esub> x = \<zero>\<^bsub>N\<^esub>"
arcs_H: shows "arcs H = arcs G - {a, a'}"
mds\<^sub>A_of_def: fixes mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set" shows "mds\<^sub>A_of mds\<^sub>C \<equiv> \<lambda>m. inv var\<^sub>C_of ` (range var\<^sub>C_of \<inter> mds\<^sub>C m)"
lesseq_a_def: fixes a :: "'a"   and b :: "'a" shows "lesseq_a a b = (a l\<rightarrow> b = (1::'a))"
simps_164: fixes x4 :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (Ref x4) = LID\<^sub>E x4"
union_is_single: fixes M :: "'f multiset"   and N :: "'f multiset"   and a :: "'f" shows "(M + N = {#a#}) = (M = {#a#} \<and> N = {#} \<or> M = {#} \<and> N = {#a#})"
mask_rshift_mult_eq_rshift_lshift: fixes a :: "'b word"   and b :: "nat"   and c :: "nat" shows "(a >> b) * (1 << c) = a >> b << c"
xor_list_left_commute: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "xs [\<oplus>] ys [\<oplus>] zs = ys [\<oplus>] xs [\<oplus>] zs"
dimY: fixes n :: "nat" shows "Y n \<in> carrier_mat m m"
steps_append_single: fixes xs :: "'a list"   and x :: "'a" assumes "steps xs"   and "E (last xs) x"   and "xs \<noteq> []" shows "steps (xs @ [x])"
fst_row_add_iterate_PA_preserves_greater_than_n: fixes n :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and P :: "(('a, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'b"   and b :: "'c" assumes "n < nrows A"   and "n < mod_type_class.to_nat a" shows "fst (row_add_iterate_PA (P, A) n i j) $ a $ b = P $ a $ b"
map_bset_bsingleton: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "map_bset f (bsingleton x) = bsingleton (f x)"
partition_wrt_extend: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo' :: "nat"   and hi' :: "nat"   and p :: "nat"   and hi :: "nat"   and lo :: "nat" assumes "isPartition_wrt R xs lo' hi' p"   and "hi < length xs"   and "lo \<le> lo'"   and "lo' \<le> hi"   and "hi' \<le> hi"   and "lo' \<le> p"   and "p \<le> hi'"   and "\<And>i. \<lbrakk>lo \<le> i; i < lo'\<rbrakk> \<Longrightarrow> R (xs ! i) (xs ! p)"   and "\<And>j. \<lbrakk>hi' < j; j \<le> hi\<rbrakk> \<Longrightarrow> R (xs ! p) (xs ! j)" shows "isPartition_wrt R xs lo hi p"
h_def: shows "h = (\<lambda>z. if f z = 0 then 0 else exp (Re (g z) * ln (cmod (f z)) + \<bar>Im (g z)\<bar> * pi))"
simps: fixes p :: "'a \<Rightarrow> bool"   and ws :: "'a list"   and uu :: "'a set" shows "While_Combinator.rtrancl_while_test p (ws, uu) = (ws \<noteq> [] \<and> p (hd ws))"
perm_refl: fixes l :: "'d list" shows "l <~~> l"
der: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
Abs_fnat_hom_Suc: fixes a :: "nat" shows "word_succ (word_of_nat a) = word_of_nat (Suc a)"
cases: fixes a1 :: "'varSort \<times> 'sort"   and a2 :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs a1 a2" obtains xs :: "'varSort" and s :: "'sort" and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" and x :: "'var" where "a1 = (xs, s)" and "a2 = Abs xs x X" and "isInBar (xs, s)" and "wls s X"
no_functionality_subsumed: fixes t1 :: "transition"   and t2 :: "transition"   and c :: "nat \<Rightarrow>f value option" assumes "Label t1 = Label t2"   and "Arity t1 = Arity t2"   and "\<nexists>i. can_take_transition t1 i c" shows "subsumes t2 c t1"
interpret_ode_d_fa: fixes xs :: "floatarith list"   and vs :: "real list"   and ds :: "floatarith list"   and n :: "nat" assumes "einterpret xs vs \<in> Csafe"   and "length ds = DIM('d)"   and "length xs = DIM('d)" shows "einterpret (ode_d_fa n xs ds) vs = blinfun_apply (ode_d n (einterpret xs vs) (einterpret ds vs)) (einterpret ds vs)"
simps: fixes s :: "name"   and x :: "tm"   and y :: "tm"   and z :: "tm" assumes "atom s \<sharp> (x, y, z)" shows "HaddP x y z = SyntaxN.Ex s (SeqHaddP (Var s) x y z)"
override_on_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
min_int_poly_prime_elem: fixes x :: "'b" shows "prime_elem (min_int_poly x)"
set_map_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf" shows "set_spmf (map_spmf f p) = f ` set_spmf p"
N_le_MMn: assumes "m \<noteq> 0" shows "N \<le> nat M * nat M * n"
absorb_iff2: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (inf a b = b)"
ltl_G_stabilize: fixes \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" assumes "finite \<G>"   and "Only_G \<G>" obtains i :: "nat" where "\<And>\<chi> j. \<chi> \<in> \<G> \<Longrightarrow> suffix i w \<Turnstile> \<chi> = suffix (i + j) w \<Turnstile> \<chi>"
l': shows "l ! i = vec_to_lpoly (0\<^sub>v (dim_col A) @\<^sub>v row A\<^sup>T (i - dim_vec b))"
monoseq_le: fixes a :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "monoseq a"   and "a \<longlonglongrightarrow> x" shows "(\<forall>n. a n \<le> x) \<and> (\<forall>m n. m \<le> n \<longrightarrow> a m \<le> a n) \<or> (\<forall>n. x \<le> a n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> a n \<le> a m)"
dens_ctxt_measure_empty_bind: fixes \<rho> :: "nat \<Rightarrow> val"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> 'a measure"   and N :: "'a measure" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "f \<in> state_measure V' \<Gamma> \<rightarrow>\<^sub>M subprob_algebra N" shows "dens_ctxt_measure ({}, V', \<Gamma>, \<lambda>_. 1) \<rho> \<bind> f = f \<rho>"
hd_conv_nth: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "hd xs = xs ! 0"
finite_imp_nat_seg_image_inj_on: fixes A :: "'a set" assumes "finite A" shows "\<exists>n f. A = f ` {i. i < n} \<and> inj_on f {i. i < n}"
intro: fixes M :: "'a set set"   and G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and M1 :: "'a set" assumes "M \<in> {s. s \<subseteq> carrier G \<and> card s = p ^ a} // {(N1, N2). N1 \<in> {s. s \<subseteq> carrier G \<and> card s = p ^ a} \<and> N2 \<in> {s. s \<subseteq> carrier G \<and> card s = p ^ a} \<and> (\<exists>g\<in>carrier G. N1 = N2 #>\<^bsub>G\<^esub> g)}"   and "\<not> p ^ Suc (multiplicity p m) dvd card M"   and "M1 \<in> M" shows "sylow_central_axioms G p a m M1 M"
compose_conv: fixes xs :: "('d \<times> 'e) list"   and ys :: "('e \<times> 'c) list"   and k :: "'d" shows "map_of (AList.compose xs ys) k = (map_of ys \<circ>\<^sub>m map_of xs) k"
system_automaton_alt_def: shows "s1.system_automaton = stutter_extend s1.system_automaton'"
tLength_tMap: fixes f :: "'b \<Rightarrow> 'a"   and t :: "'b Trace" shows "tLength (tMap f t) = tLength t"
mmonitorable_regex_def: fixes b :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex" shows "mmonitorable_regex b g r = (safe_regex b g r \<and> regex.pred_regex Formula.future_bounded r)"
Imp: fixes n :: "nat"   and p :: "pf"   and q :: "pf" shows "dfa_of_pf n (Imp p q) = imp_dfa (dfa_of_pf n p) (dfa_of_pf n q)"
snull_bottom_iff: fixes xs :: "[:'a:]" shows "(snull\<cdot>xs = \<bottom>) = (xs = \<bottom>)"
dim1: shows "\<forall>n. matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>\<^sub>1) n \<in> carrier_mat d d"
mkAlgAuto_def: fixes aOps :: "('k, 'j, 'i list) MapOps"   and tOps :: "('l, 'j \<times> 'h, 'j) MapOps"   and simObs :: "'g \<Rightarrow> 'j \<Rightarrow> 'h"   and simInit :: "'g \<Rightarrow> 'h \<Rightarrow> 'j"   and simTrans :: "'g \<Rightarrow> 'j \<Rightarrow> 'j list"   and simAction :: "'g \<Rightarrow> 'j \<Rightarrow> 'i list"   and frontier :: "'g \<Rightarrow> 'j list" shows "mkAlgAuto aOps tOps simObs simInit simTrans simAction frontier \<equiv> \<lambda>a. let auto = alg_dfs aOps tOps (simObs a) (simTrans a) (simAction a) (frontier a) in \<lparr>pInit = simInit a, pTrans = \<lambda>obs ec. the (lookup tOps (aTrans auto) (ec, obs)), pAct = \<lambda>ec. the (lookup aOps (aActs auto) ec)\<rparr>"
eq_Nil_null: fixes xs :: "'b list" shows "(xs = []) = List.null xs"
that_1: assumes "k \<notin> K"   and "m k = Some x" shows "thesis"
simps: fixes l :: "rat"   and r :: "rat"   and l' :: "rat"   and r' :: "rat" shows "sub_interval (l, r) (l', r') = (l' \<le> l \<and> r \<le> r')"
OV_inv3_inductive_1: shows "init obsv_TS \<subseteq> OV_inv3"
mem_memS_eq: fixes x :: "'a"   and l :: "'a list" shows "x mem l \<equiv> memS x l"
line_left_inter_2: fixes P :: "Points"   and m :: "Lines"   and l :: "Lines" assumes "P \<in> line_ext m"   and "P \<notin> line_ext l" shows "line (Pascal_Property.inter l m) P = m"
Iagree_def: fixes I :: "('a, 'b, 'c) interp"   and J :: "('a, 'b, 'c) interp"   and V :: "('a + 'b + 'c) set" shows "Iagree I J V \<equiv> \<forall>i\<in>V. (\<forall>x. i = Inl x \<longrightarrow> Functions I x = Functions J x) \<and> (\<forall>x. i = Inr (Inl x) \<longrightarrow> Contexts I x = Contexts J x) \<and> (\<forall>x. i = Inr (Inr x) \<longrightarrow> Predicates I x = Predicates J x) \<and> (\<forall>x. i = Inr (Inr x) \<longrightarrow> Programs I x = Programs J x) \<and> (\<forall>x. i = Inr (Inr x) \<longrightarrow> ODEs I x = ODEs J x) \<and> (\<forall>x. i = Inr (Inr x) \<longrightarrow> ODEBV I x = ODEBV J x)"
step_Stuck_prop: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Stuck" shows "s' = Stuck"
term_variants_pred_const_cases_2: fixes P :: "'c \<Rightarrow> 'c list"   and a :: "'c"   and t :: "('c, 'd) Term.term" assumes "P a = []" shows "term_variants_pred P (Fun a []) t = (t = Fun a [])"
item_rule_finished_item: fixes \<alpha> :: "symbol list" shows "item_rule (finished_item \<alpha>) = (\<SS>, \<alpha>)"
pv_decomp: shows "awalk_verts u p = xs @ y # ys @ y # zs"
ennreal_right_diff_distrib: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" assumes "a \<noteq> \<top>" shows "a * (b - c) = a * b - a * c"
mult_stalk_def: shows "mult_stalk \<equiv> mult_rel"
fls_X_intpow_div_fls_X_intpow_semiring1: fixes i :: "int"   and j :: "int" assumes "inverse (1::'a) = (1::'a)" shows "fls_X_intpow i / fls_X_intpow j = fls_X_intpow (i - j)"
continuous_within_tendsto_compose': fixes a :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "continuous (at a within s) f"   and "\<And>n. x n \<in> s"   and "(x \<longlongrightarrow> a) F" shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
hyps_5: shows "AbstForm v 0 (q_All allw) allvw"
converse_subset_swap: fixes r :: "('b \<times> 'c) set"   and s :: "('c \<times> 'b) set" shows "(r \<subseteq> s\<inverse>) = (r\<inverse> \<subseteq> s)"
v_pj_sj: shows "v mod p_j = [:s_j:]"
real_of_3_def: shows "real_of_3 \<equiv> (rep_real_alg_3 ---> id) real_of_2"
one_outcome_card_support_1: fixes outcomes :: "'b set" assumes "card outcomes = 1" shows "\<forall>l\<in>lotteries_on outcomes. card (set_pmf l) = 1"
UNIV_not_singleton_converse: assumes "\<And>x. UNIV \<noteq> {x}" shows "\<exists>x y. x \<noteq> y"
leg_simps_3: shows "C.arr \<mu>'.cod.leg1"
C''_2: shows "qReachable c' (Proc - {q}) c''"
simps_3: shows "is_disj \<bottom> = is_lit_plus \<bottom>"
elims_3: fixes x :: "nat \<times> nat list"   and xa :: "abc_inst list" assumes "\<not> abc_final x xa" obtains s :: "nat" and lm :: "nat list" and p :: "abc_inst list" where "x = (s, lm)" and "xa = p" and "s \<noteq> length p"
r_add_prim: shows "prim_recfn 2 r_add"
even_conv_AND: fixes i :: "int" shows "even i = (i AND 1 = 0)"
F2_def: shows "F2 = filter (\<lambda>f. \<not> (\<exists>v\<in>set V1. f \<in> set (facesAt g v))) (faces g)"
c: fixes j :: "'a" assumes "j \<in> Basis" shows "c j \<noteq> 0"
real_cos_eq: fixes x :: "real" shows "Re (cos (complex_of_real x)) = cos x"
r_right_minus_eq: fixes a :: "'a"   and b :: "'a" shows "(a - b = (0::'a)) = (a = b)"
fst_image_times: fixes A :: "'b set"   and B :: "'c set" shows "fst ` (A \<times> B) = (if B = {} then {} else A)"
well_valued0_butlast: fixes t :: "'b tree"   and as :: "'b list" assumes "well_shaped t"   and "well_valued0 as t"   and "r t < length as" shows "well_valued0 (butlast as) t"
BrnNFT_elim: fixes ii :: "nat"   and cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list" assumes "ii \<in> BrnNFT cl dl" obtains n :: "nat" and i :: "nat" where "n \<in> theNFTBoth cl dl" and "i < brn (cl ! n)" and "ii = brnL cl n + i"
assms: shows "\<forall>c. f c = (0::'b) \<longrightarrow> c = (0::'a)"
smooth_on_snd: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "k-smooth_on S f"   and "open S" shows "k-smooth_on S (\<lambda>x. snd (f x))"
lunit'_simps_4: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.dom (S.lunit' f) = f"
mr_def: shows "mr \<equiv> Max {r. poly p r = 0 \<and> r < ub}"
set_selsort: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "set (linorder_list0.selsort le xs) = set xs"
dvd_mult_cancel_right: fixes 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 a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.idom times one plus zero minus uminus" shows "dvd.dvd times (times a c) (times b c) = (c = zero \<or> dvd.dvd times a b)"
sum_of_2_squares_nat_sufficient: fixes n :: "nat" assumes "0 < n"   and "\<And>p. \<lbrakk>p \<in># prime_factorization n; [p = 3] (mod 4)\<rbrakk> \<Longrightarrow> even (multiplicity p n)" shows "sum_of_2_squares_nat n"
elims: fixes x :: "'c ltl list"   and y :: "'c ltl" assumes "Or x = y" obtains "x = []" and "y = false"   | xa :: "'c ltl" and xs :: "'c ltl list" where "x = xa # xs" and "y = foldl LTLOr xa xs"
preservation1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<le> x \<cdot> y \<cdot> x" shows "x \<cdot> y\<^sup>\<dagger> \<le> (x \<cdot> y + z)\<^sup>\<dagger> \<cdot> x"
less_2: fixes i :: "nat" shows "dickson_less_v d m (seq (Suc i)) (seq i)"
continuous_on_product_then_coordinatewise: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and i :: "'b" assumes "continuous_on S f" shows "continuous_on S (\<lambda>x. f x i)"
ipurge_fail_aux_t_inv_3_def: fixes P :: "'b process"   and I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and X :: "'b set"   and Y :: "('b, 'c) ipurge_rec" shows "ipurge_fail_aux_t_inv_3 P I D xs X Y \<equiv> secure P I D \<longrightarrow> (xs, X) \<in> failures P \<longrightarrow> ((case ListOp Y of None \<Rightarrow> [] | Some ys \<Rightarrow> ys) @ List Y, Set Y) \<in> failures P"
ex_progress_ge: fixes \<sigma> :: "'a trace"   and x :: "nat" shows "\<exists>\<pi>. prefix_of \<pi> \<sigma> \<and> x \<le> progress \<pi>"
real_of_int_div4: fixes n :: "int"   and x :: "int" shows "real_of_int (n div x) \<le> real_of_int n / real_of_int x"
simps_17: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a"   and x2 :: "'b" shows "(case Abrupt x2 of Normal x \<Rightarrow> f1 x | Abrupt x \<Rightarrow> f2 x | Fault x \<Rightarrow> f3 x | Stuck \<Rightarrow> f4) = f2 x2"
solve_invalidate: fixes \<xi> :: "state" assumes "\<forall>ip\<in>dom (dests \<xi>). ip \<in> kD (rt \<xi>) \<and> sqn (rt \<xi>) ip \<le> the (dests \<xi> ip)" shows "\<forall>ip. sqn (rt \<xi>) ip \<le> sqn (invalidate (rt \<xi>) (dests \<xi>)) ip"
a1: shows "\<forall>x. \<exists>y. u x = s x + y"
simps_46: fixes x31 :: "('a, 'b) fm"   and x32 :: "('a, 'b) fm" shows "params_fm (x31 \<^bold>\<longrightarrow> x32) = params_fm x31 \<union> params_fm x32"
commute_dest: fixes R :: "('a \<times> 'b) set"   and Tc :: "('b \<times> 'b) set"   and Ta :: "('a \<times> 'a) set"   and sa :: "'a"   and sc :: "'b"   and sc' :: "'b" assumes "R O Tc \<subseteq> Ta O R"   and "(sa, sc) \<in> R"   and "(sc, sc') \<in> Tc" shows "\<exists>sa'. (sa, sa') \<in> Ta \<and> (sa', sc') \<in> R"
path_split: fixes v :: "'v"   and v' :: "'v"   and pth1 :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list"   and e :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort"   and pth2 :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" shows "path v v' (pth1 @ [e] @ pth2) = (path v (edge_end e) (pth1 @ [e]) \<and> path (edge_end e) v' pth2)"
upper_split_interval2: fixes X :: "'a interval"   and m :: "'a" shows "upper (snd (split_interval X m)) = max (upper X) m"
sasp_op_to_strips_injective: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and op\<^sub>1 :: "('a, 'b) sas_plus_operator"   and op\<^sub>2 :: "('a, 'b) sas_plus_operator" assumes "\<phi>\<^sub>O \<Psi> op\<^sub>1 = \<phi>\<^sub>O \<Psi> op\<^sub>2" shows "op\<^sub>1 = op\<^sub>2"
assms_2: shows "agree (fmrestrict_set vs s1) s2"
map_sel: fixes a :: "'c option"   and f :: "'c \<Rightarrow> 'd" assumes "a \<noteq> None" shows "the (map_option f a) = f (the a)"
A: shows "x - (1::'a) / (2::'a) < of_int (round x)"
assms: shows "\<forall>\<^sub>F x in at_top. f x \<in> {l x..u x}"
has_integral_le: fixes f :: "'c \<Rightarrow> real"   and i :: "real"   and S :: "'c set"   and g :: "'c \<Rightarrow> real"   and j :: "real" assumes "(f has_integral i) S"   and "(g has_integral j) S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x" shows "i \<le> j"
S4: shows "x \<le> y \<and> y \<le> x"
mult_monomials_def: shows "mult_monomials = (\<lambda>(x, a) (y, b). (mult_monoms x y, a * b))"
surjective: fixes r :: "('b, 'c) dfa_scheme" shows "r = \<lparr>dfa.states = dfa.states r, init = dfa.init r, final = dfa.final r, nxt = dfa.nxt r, \<dots> = dfa.more r\<rparr>"
tendsto_power: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and n :: "nat" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
col_def: fixes A :: "'b mat"   and j :: "nat" shows "col A j = Matrix.vec (dim_row A) (\<lambda>i. A $$ (i, j))"
plus_Box_def: fixes x :: "'b Box"   and y :: "'b Box" shows "x + y \<equiv> bliftM2 (\<Lambda> a b. a + b)\<cdot>x\<cdot>y"
\<theta>_simps_4: shows "local.dom \<theta>' = p\<^sub>0 \<star> w'"
exhaust: fixes y :: "ev_consume_t" obtains "y = EV_CONSUME_ALL"   | "y = EV_CONSUME_ONE"
min_list_Min: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "min_list xs = Min (set xs)"
finite_dom_lookup: fixes t :: "('b, 'c) RBT.rbt" shows "finite (dom (RBT.lookup t))"