hcju/isabelleps huggingface dataset:
mat_to_polys_nth: fixes i :: "nat"   and A :: "'d mat"   and ts :: "'e list" assumes "i < dim_row A" shows "mat_to_polys ts A ! i = row_to_poly ts (row A i)"
assms_1: shows "C \<in> stfx.carrier_stalk"
guha_equal: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and ft :: "('a, 'c) flow_entry_match list"   and p :: "'b"   and d :: "'c option" assumes "no_overlaps \<gamma> ft" shows "(OF_priority_match \<gamma> ft p = option_to_ftb d) = guha_table_semantics \<gamma> ft p d"
vec_eq_iff: fixes x :: "('c, 'd) Finite_Cartesian_Product.vec"   and y :: "('c, 'd) Finite_Cartesian_Product.vec" shows "(x = y) = (\<forall>i. x $h i = y $h i)"
P_Pf: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "let PPf = \<lambda>t1 t2. psubst Pf [(t1, yy), (t2, xx)] in bprv (eqv (subst P \<langle>\<phi>\<rangle> xx) (exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>)))"
1: shows "interp_ins op_list = (xxs @ [x] @ xys) @ y # xzs"
hi_un_eq_up: shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) neg un = up [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> Brouwer_degree2 (n - Suc 0) neg"
linear_exp_bound: shows "\<exists>p. \<forall>x. b ^ x * of_nat x \<le> p"
shuffle_coefficients_def: fixes xs :: "(char list list \<times> int) list" shows "shuffle_coefficients xs = SPEC (\<lambda>ys. map (\<lambda>(a, b). (mset a, b)) (rev xs) = map (\<lambda>(a, b). (mset a, b)) ys \<and> (\<forall>xs\<in>mononoms ys. sorted_wrt var_order xs))"
commute: fixes a :: "'k"   and b :: "'k" shows "min a b = min b a"
well_def_signs_step: fixes qs1 :: "real poly list"   and qs2 :: "real poly list"   and signs1 :: "rat list list"   and signs2 :: "rat list list" assumes "0 < length qs1"   and "0 < length qs2"   and "well_def_signs (length qs1) signs1"   and "well_def_signs (length qs2) signs2" shows "well_def_signs (length (qs1 @ qs2)) (signs_smash signs1 signs2)"
tt'C: shows "{t, t'} \<subseteq> SPR.jkbpC"
assms: shows "(if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'a) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'a)) + 1 else 0) \<le> rank A"
phis_finite: fixes g :: "'g" shows "finite (dom (Mapping.lookup (p_g g s1_phis g)))"
ListSum_ExcessNotAtRecList: fixes E :: "nat \<Rightarrow> nat"   and vs :: "nat list"   and ps :: "(nat \<times> nat) list"   and g :: "graph" assumes "isTable E vs ps" shows "ExcessNotAtRec ps g = ListSum (ExcessNotAtRecList ps g) E"
walls_betw_def: fixes C :: "'a set"   and D :: "'a set" shows "walls_betw C D \<equiv> {H \<in> \<Union>(f, g)\<in>foldpairs. {{f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}}. separated_by H C D}"
R11_def: shows "R11 \<equiv> {(s, s'). s = med11 s'}"
edges_of_walk_def: fixes xs :: "'a list" shows "H.edges_of_walk xs = {uu_. \<exists>v w xs_pre xs_post. uu_ = (v, w) \<and> xs = xs_pre @ v # w # xs_post}"
diff_subset_eq_self: fixes M :: "'k multiset"   and N :: "'k multiset" shows "M - N \<subseteq># M"
bi_uniqueDl: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e"   and z :: "'d" assumes "bi_unique A"   and "A x y"   and "A z y" shows "x = z"
lemma4_aux: fixes rs :: "dB list"   and rs' :: "dB list"   and ss :: "dB list" assumes "listrelp (\<lambda>t u. t \<rightarrow>\<^sub>s u \<and> (\<forall>r. u \<rightarrow>\<^sub>\<beta> r \<longrightarrow> t \<rightarrow>\<^sub>s r)) rs rs'"   and "rs' => ss" shows "rs [\<rightarrow>\<^sub>s] ss"
map_poly_mult_3: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and p :: "'b poly" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "map_mat (\<lambda>a. [:a:] * p) (A * B) = map_mat (\<lambda>a. [:a:]) A * map_mat (\<lambda>a. [:a:] * p) B"
image_inverse_permutations: fixes S :: "'b set" shows "{inv p |p. p permutes S} = {p. p permutes S}"
not_less: shows "\<not> scf (lift R) \<prec>[set_ext (lift R i)] scf (lift (R(i := Ri')))"
path_connected_path_component: fixes s :: "'b set"   and x :: "'b" shows "path_connected (path_component_set s x)"
restricted_subformulas_superset: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> restricted_subformulas \<phi>" shows "subformulas\<^sub>\<nu> \<psi> \<union> subformulas\<^sub>\<mu> \<psi> \<subseteq> restricted_subformulas \<phi>"
residue_of_zero_1: fixes k :: "nat" shows "\<zero> k = \<zero>\<^bsub>residue_ring (p ^ k)\<^esub>"
that: fixes Sl :: "'node set" assumes "Sl \<in> slices p"   and "Sl \<subseteq> C" shows "thesis"
Re_mono: fixes x :: "complex"   and y :: "complex" assumes "x \<le> y" shows "Re x \<le> Re y"
simps_27: fixes uu :: "cospan_shape.Arr" shows "cospan_shape.comp uu cospan_shape.Null = cospan_shape.Null"
valid_index_nths: fixes "is" :: "nat list"   and ds :: "nat list"   and A :: "nat set" assumes "is \<lhd> ds" shows "nths is A \<lhd> nths ds A"
schroeder_4: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> z) = (- z * y\<^sup>T \<le> - x)"
elims_2: fixes x :: "nat"   and xa :: "(nat \<times> nat) set"   and xb :: "nat list" assumes "is_path x xa xb" obtains n :: "nat" and S :: "(nat \<times> nat) set" and xc :: "nat" and xs :: "nat list" where "x = n" and "xa = S" and "xb = xc # xs" and "if xs = [] then xc < n else ((xc, hd xs) \<in> S \<or> (hd xs, xc) \<in> S) \<and> is_path n S xs"
intro: fixes R :: "('a, 'b) ring_scheme" assumes "cring R" shows "UP_cring R"
uniformity_bcontfun_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
A1: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> RelA"
emeasure_pair_measure_Times: fixes A :: "'b set"   and N :: "'b measure"   and B :: "'a set"   and i :: "'i" assumes "A \<in> sets N"   and "B \<in> sets (M i)" shows "emeasure (N \<Otimes>\<^sub>M M i) (A \<times> B) = emeasure N A * emeasure (M i) B"
less_eq_linord_helper_def: fixes a :: "('a, 'b) linord_helper"   and b :: "('a, 'b) linord_helper" shows "(a \<le> b) = linord_helper_less_eq1 a b"
sum_prog_def: fixes \<Phi> :: "nat \<Rightarrow> bool"   and l :: "nat"   and u :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "sum_prog \<Phi> l u f \<equiv> nfoldli [l..<u] (\<lambda>_. True) (\<lambda>i s. ASSERT (\<Phi> i) \<bind> (\<lambda>_. RETURN (s + f i))) (0::'a)"
L1: fixes Q' :: "pi" assumes "Q \<longmapsto> \<tau> \<prec> Q'" shows "(Q', Q) \<notin> Rel"
no_nonpos_Real_in_ball: fixes z :: "complex"   and t :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0"   and "t \<in> ball z (if Im z = 0 then Re z / 2 else \<bar>Im z\<bar> / 2)" shows "t \<notin> \<real>\<^sub>\<le>\<^sub>0"
1: shows "[k] \<langle>(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms)\<rangle> [m] = [k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m]"
is_imag_arg1: fixes z :: "complex" assumes "arg z = pi / 2 \<or> arg z = - pi / 2" shows "is_imag z"
cod_simp: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.VV.cod f = C.VxV.cod f"
update_simps_14: fixes V :: "(nat, 'b) mapping"   and s :: "('a, 'b) state" shows "\<B>\<^sub>i\<^sub>l (\<V>_update V s) = \<B>\<^sub>i\<^sub>l s"
assms_1: shows "(f \<longlongrightarrow> l) (at a within T)"
inj_on_cong: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "inj_on f A = inj_on g A"
map_key_single: fixes f :: "'a \<Rightarrow> 'b"   and k :: "'a"   and v :: "'c" assumes "inj f" shows "Poly_Mapping.map_key f (monomial v (f k)) = monomial v k"
em_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "em P = (\<lambda>(ps, n). \<forall>ex. P (Partial_Evaluation.emb ps ex) \<le> enat n)"
locally_finite_on_def: fixes X :: "'c set"   and I :: "'d set"   and U :: "'d \<Rightarrow> 'c set" shows "locally_finite_on X I U = (\<forall>p\<in>X. \<exists>N. p \<in> N \<and> open N \<and> finite {i \<in> I. U i \<inter> N \<noteq> {}})"
mem_equal_mod_trap: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and new_traps1 :: "Trap set"   and traps2 :: "Trap set" assumes "mem_equal s1 s2 pa" shows "mem_equal (s1\<lparr>traps := new_traps1\<rparr>) (s2\<lparr>traps := traps2\<rparr>) pa"
of_int_0_le_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (ring_1.of_int one plus zero minus z) = (0 \<le> z)"
pdivides_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and q :: "'c list" shows "p pdivides\<^bsub>R\<^esub> q = p divides\<^bsub>poly_ring R\<^esub> q"
locally_connected_open_component: fixes S :: "'b set" shows "locally connected S = (\<forall>t c. openin (top_of_set S) t \<and> c \<in> components t \<longrightarrow> openin (top_of_set S) c)"
c_prod_assoc: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set"   and z :: "('a \<times> 'a set) set" shows "x \<parallel> y \<parallel> z = x \<parallel> (y \<parallel> z)"
is_equivalence_pseudofunctor: shows "equivalence_pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D I\<^sub>D.map I\<^sub>D.cmp"
nat_pow_pow: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier \<G>" shows "(x [^] n) [^] m = x [^] (n * m)"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "P T"   and "P F"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
Delta_AudioPlayer_CTRL: shows "Delta AudioPlayer_CTRL = AudioPlayer_CTRL_Delta"
Inf_V_def: fixes X :: "V set" shows "\<Sqinter> X \<equiv> if X = {} then 0 else set (\<Inter> (elts ` X))"
intro: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme" assumes "g_set_xy_defs_loc ops1 ops2"   and "StdSet ops1"   and "StdSet ops2" shows "g_set_xy_loc ops1 ops2"
llength_ltakeWhile_le: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "llength (ltakeWhile P xs) \<le> llength xs"
ntrs_stack_top_decomp_s: fixes u :: "'a"   and r :: "'a list"   and c :: "'a list multiset"   and ee :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((u # r, c), ee, s', c') \<in> ntrs fg" obtains v :: "'a" and u' :: "'a" and p :: "'b" where "s' = v # u' # r" and "(([u], c), ee, [v, u'], c') \<in> ntrs fg" and "(u, Call p, u') \<in> edges fg"
assms_4: shows "\<forall>x\<in>S. (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
space_lim: shows "space PF.lim = space (Pi\<^sub>M UNIV (\<lambda>_. M))"
open_segment_bound1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> open_segment a b" shows "norm (x - a) < norm (b - a)"
in_rel: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R3 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "('c, 'e, 'g) F2"   and b :: "('d, 'f, 'h) F2" shows "F2rel R1 R2 R3 a b = (\<exists>z. z \<in> F2in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F2map fst fst fst z = a \<and> F2map snd snd snd z = b)"
incr_approx: fixes j :: "nat"   and h :: "real"   and x :: "'a" assumes "t j \<le> t1" shows "dist (incr h (t j) x) (incr' h (t j) x) \<le> B * stepsize j ^ p"
wqo_onI: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "transp_on P A"   and "almost_full_on P A" shows "wqo_on P A"
untilEA: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and \<gamma>' :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "sb.eval c t t' n (\<gamma>' \<UU>\<^sub>b \<gamma>)" shows "\<exists>n'\<ge>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> (\<forall>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub>. n'' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n'\<^esub> \<longrightarrow> sb.eval c t t' n'' \<gamma>) \<and> (\<forall>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub>. n'' < \<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<longrightarrow> sb.eval c t t' n'' \<gamma>') \<or> \<not> (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> sb.eval c t t' n' \<gamma> \<and> (\<forall>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub>. n'' < n' \<longrightarrow> sb.eval c t t' n'' \<gamma>')"
step: fixes us :: "[:'a:]" assumes "\<And>us'. lt\<cdot>(slength\<cdot>us')\<cdot>(slength\<cdot>us) = TT \<Longrightarrow> P us'" shows "P us"
continuous_on_closed_vimageI: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "closed s"   and "continuous_on s f"   and "closed B" shows "closed (f -` B \<inter> s)"
simps_1: fixes c :: "poly"   and p :: "poly" shows "coefficients (CN c 0 p) = c # coefficients p"
map_upds_twist: fixes a :: "'a"   and as :: "'a list"   and m :: "'a \<Rightarrow> 'b option"   and b :: "'b"   and bs :: "'b list" assumes "a \<notin> set as" shows "m(a \<mapsto> b, as [\<mapsto>] bs) = m(as [\<mapsto>] bs, a \<mapsto> b)"
igSubstAbsIPresIGWlsAbsSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstAbsIPresIGWlsAbsSTR MOD \<equiv> \<forall>ys Y y us s A. igWlsAbs MOD (us, s) (igSubstAbs MOD ys Y y A) = (igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A)"
accp_wfPI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>x. Wellfounded.accp r x" shows "wfP r"
plus_except: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c" shows "p = monomial (lookup p t) t + except p {t}"
SemCall: fixes s :: "Var \<Rightarrow> Val"   and n :: "nat"   and t :: "Var \<Rightarrow> Val" assumes " s , body \<rightarrow>\<^sub>n  t " shows " s , Call \<rightarrow>\<^sub>n + 1  t "
extremum_strict: fixes a :: "nat" shows "\<not> (0 dvd a \<and> 0 \<noteq> a)"
q': shows "q' = \<lparr>sprFst = uq, sprLst = vq, sprCRel = tObsC_abs p\<rparr>"
init: assumes "CHOinitConfig A (rho 0) (coords 0)" shows "P 0"
multiplicity_prime_elem_times_other: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "\<not> dvd.dvd times p q" shows "factorial_semiring.multiplicity times one p (times q x) = factorial_semiring.multiplicity times one p x"
tautologies_are_valid: fixes C :: "'at Literal set"   and I :: "'at set" assumes "tautology C" shows "I \<Turnstile> C"
greaterThanAtMost_empty_iff2: fixes k :: "'a multiset"   and l :: "'a multiset" shows "({} = subset_mset.greaterThanAtMost k l) = (\<not> k \<subset># l)"
Conj_rep_eq: fixes xset :: "('a, 'b, 'c) formula set['a]" assumes "finite (supp xset)" shows "Rep_formula (Conj xset) = Conj\<^sub>\<alpha> (map_bset Rep_formula xset)"
h1: shows "emptyseq (s |\<^sub>s Suc n)"
rw: shows "A * (Q * make_mat (dim_col A) (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then S $$ (0, 0) else S $$ (0, Suc i)))))) = A * Q * make_mat (dim_col A) (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then S $$ (0, 0) else S $$ (0, Suc i)))))"
ileq_interleave: fixes w :: "'d list"   and w1 :: "'d list"   and w2 :: "'d list" assumes "w \<in> w1 \<otimes> w2" shows "w1 \<preceq> w \<and> w2 \<preceq> w"
fd_4: shows "f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B)"
assms: shows "\<C> \<turnstile> map EConst vs : ts' _> ts' @ ts"
hfun_in_hexp: fixes B :: "hf"   and C :: "hf"   and F :: "hf" assumes "hfun B C F" shows "F \<^bold>\<in> hexp B C"
rbt_lookup_fold_rbt_insertwk: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a" assumes "rbt_sorted t1"   and "rbt_sorted t2" shows "rbt_lookup (RBT_Impl.fold (rbt_insert_with_key f) t1 t2) k = (case rbt_lookup t1 k of None \<Rightarrow> rbt_lookup t2 k | Some v \<Rightarrow> case rbt_lookup t2 k of None \<Rightarrow> Some v | Some w \<Rightarrow> Some (f k w v))"
viewR_correct_1: fixes t :: "('a, 'b) FingerTree" assumes "t = FingerTree.empty" shows "FingerTree.viewR t = None"
sum_list_closed: fixes G :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and as :: "'b list" assumes "Group G"   and "set (map f as) \<subseteq> G" shows "sum_list (map f as) \<in> G"
circuit_imp_circuit_in: fixes C :: "'a set" assumes "circuit C" shows "circuit_in carrier C"
igFreshIGAbs2_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igFreshIGAbs2 MOD \<equiv> \<forall>ys y xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD ys y X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD xs x X)"
diamond_bottom_1: fixes a :: "'action"   and b :: "'action"   and p :: "'state" assumes "ind a b"   and "en a p"   and "en b p" shows "en a (ex b p)"
eq2: shows "Y -` (A \<inter> range Y) \<inter> space M = (\<Union>i. prY i \<inter> space M)"
index_unit_vec_2: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "unit_vec n i $ i = (1::'a)"
map_add_apply: fixes m\<^sub>1 :: "'c \<Rightarrow> 'b option"   and m\<^sub>2 :: "'c \<Rightarrow> 'b option"   and k :: "'c" shows "(m\<^sub>1 ++ m\<^sub>2) k = (case m\<^sub>2 k of None \<Rightarrow> m\<^sub>1 k | Some x \<Rightarrow> Some x)"
le: shows "2 ^ (LENGTH('a) - 1) \<le> 2 ^ (LENGTH('b) - 1)"
surjective: fixes r :: "('c, 'd, 'e) automaton_scheme" shows "r = \<lparr>init = init r, trans = automaton.trans r, \<dots> = more r\<rparr>"
ih_4: shows "InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
simps_8: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) form" shows "eval e f g (Forall p) = (\<forall>z. eval (e\<langle>0:z\<rangle>) f g p)"
minimal_ifex_ite_invar: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" assumes "ifex_minimal i"   and "ifex_minimal t"   and "ifex_minimal e" shows "ifex_minimal (ifex_ite i t e)"
lossless_S1: fixes M :: "'grp \<times> 'grp"   and out1 :: "unit" shows "lossless_spmf (S1 M out1)"
no_pending_succ_impl_path_in_tree: fixes v :: "'v"   and w :: "'v" assumes "v \<in> dom (discovered s)"   and "pending s `` {v} = {}"   and "w \<in> E `` {v}"   and "\<delta> s v < \<delta> s w" shows "(v, w) \<in> (tree_edges s)\<^sup>+"
rep_eq: fixes x :: "'a scene" shows "idem_scene x = idem_overrider (Rep_scene x)"
assms_2: fixes s :: "'a set" assumes "s \<in> set ss" shows "(0::'a) \<in> s"
prime_power_eq_imp_eq: fixes p :: "'b"   and q :: "'b"   and m :: "nat"   and n :: "nat" assumes "prime p"   and "prime q"   and "0 < m"   and "p ^ m = q ^ n" shows "p = q"
bcontfun_eqI: fixes f :: "'c \<Rightarrow>\<^sub>C 'd"   and g :: "'c \<Rightarrow>\<^sub>C 'd" assumes "\<And>x. apply_bcontfun f x = apply_bcontfun g x" shows "f = g"
components_are_iso: fixes a :: "'b" assumes "D.ide a" shows "D.iso (D.L a)"
real_linearD: fixes f :: "real \<Rightarrow> real" assumes "linear f" obtains c :: "real" where "f = (*) c"
mset_add: fixes a :: "'k"   and A :: "'k multiset" assumes "a \<in># A" obtains B :: "'k multiset" where "A = add_mset a B"
check_poly_weak_mono_def: fixes p :: "('c monom \<times> 'd) list"   and v :: "'c" shows "check_poly_weak_mono p v \<equiv> list_all (\<lambda>(m, c). (0::'d) \<le> c \<or> v \<notin> monom_vars m) p"
items_le_Predict: fixes k :: "nat"   and I :: "item set" shows "items_le k (Predict k I) = Predict k (items_le k I)"
simps_16: fixes v :: "bexp"   and va :: "'b acom"   and vb :: "'b acom"   and vc :: "'b"   and vd :: "'b acom"   and ve :: "'b acom" shows "(IF v THEN va ELSE vb {vc} \<sqsubseteq> vd;; ve) = False"
dim_le_card: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and V :: "'e set"   and W :: "'e set" assumes "vector_space scale"   and "V \<subseteq> module.span scale W"   and "finite W" shows "vector_space.dim scale V \<le> card W"
injective_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "injective x" shows "injective (x \<sqinter> y)"
simps_2: fixes uv :: "'b" shows "is_Inl (Inr uv) = False"
suffix_eq_distinct_list: fixes xs :: "'b list"   and ys :: "'b list"   and suf1 :: "'b list"   and suf2 :: "'b list" assumes "distinct xs"   and "ys @ suf1 = xs"   and "ys @ suf2 = xs" shows "suf1 = suf2"
axioms_2: fixes x :: "'d \<Longrightarrow> 'e" assumes "pbij_lens x" shows "pbij_lens_axioms x"
R1_def: shows "R1 = (\<lambda>x. sum_upto real x - x\<^sup>2 / 2)"
subset_image_Int: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "A \<subseteq> B" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
ex_Zleast_in_mI: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "j \<le> n" shows "\<exists>x\<in>I. (\<nu>\<^bsub>K P j\<^esub>) x = LI K (\<nu>\<^bsub>K P j\<^esub>) I"
terminates_upto_raw: fixes n :: "int" shows "terminates (upto_raw n)"
disc_2: fixes x2 :: "'a" shows "is_neg (Neg x2)"
iso_eval_Can: fixes t :: "'a local.term" assumes "Can t" shows "local.iso \<lbrace>t\<rbrace>"
path_valid_edges: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "\<forall>a\<in>set as. valid_edge a"
notindom: fixes a :: "((int \<times> int) set set, int) Combinators"   and p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "a \<in> set p"   and "x \<notin> dom (C (list2FWpolicy p))" shows "x \<notin> dom (C a)"
split_pairs: fixes A :: "'a"   and B :: "'b"   and X :: "'a \<times> 'b" shows "((A, B) = X) = (fst X = A \<and> snd X = B)"
lift2_def: fixes h :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and inp' :: "'e \<Rightarrow> 'g option"   and inp :: "'e \<Rightarrow> 'h option" shows "lift2 h inp' inp \<equiv> \<lambda>i. case (inp' i, inp i) of (None, b) \<Rightarrow> None | (Some v', None) \<Rightarrow> None | (Some v', Some v) \<Rightarrow> Some (h v' v)"
Infinitesimal_minus_approx: fixes x :: "'a star" assumes "x \<in> Infinitesimal" shows "- x \<approx> x"