hcju/isabelleps huggingface dataset:
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "comm_ring_hom hom"   and "zero_hom_0 hom" shows "inj_comm_ring_hom hom"
closed: fixes x :: "'a set" assumes "x \<in> S" shows "closed_csubspace x"
eval_G2: fixes s :: "complex" shows "eval_fps G2 (s - 1) = (s - 1) * pre_zeta 1 s + 1"
add_right_mono: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<preceq> b" shows "dbm_add a c \<preceq> dbm_add b c"
append_def: shows "(@) \<equiv> rec_list (\<lambda>ys. ys) (\<lambda>x xs xsa ys. x # xsa ys)"
dom_map_of_conv_image_fst: fixes xys :: "('d \<times> 'e) list" shows "dom (map_of xys) = fst ` set xys"
dom_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "local.dom (trg \<mu>) = trg \<mu>"
intros_2: fixes xs :: "color list" assumes "valid xs" shows "valid (B # xs)"
fds_one_converges: fixes s :: "'a" shows "fds_converges 1 s"
NO_CYC: shows "\<nexists>x. (Pos x, Neg x) \<in> G\<^sup>* \<and> (Neg x, Pos x) \<in> G\<^sup>*"
exhaust: fixes y :: "('a, 'b, 'c) cntxt" obtains "y = \<box>"   | x21 :: "('a, 'b, 'c) cntxt" and x22 :: "('a, 'b, 'c) expr" where "y = ApplyL\<^sub>\<E> x21 x22"   | x31 :: "('a, 'b, 'c) val" and x32 :: "('a, 'b, 'c) cntxt" where "y = ApplyR\<^sub>\<E> x31 x32"   | x41 :: "('a, 'b, 'c) cntxt" and x42 :: "('a, 'b, 'c) expr" and x43 :: "('a, 'b, 'c) expr" where "y = Ite\<^sub>\<E> x41 x42 x43"   | x5 :: "('a, 'b, 'c) cntxt" where "y = Ref\<^sub>\<E> x5"   | x6 :: "('a, 'b, 'c) cntxt" where "y = Read\<^sub>\<E> x6"   | x71 :: "('a, 'b, 'c) cntxt" and x72 :: "('a, 'b, 'c) expr" where "y = AssignL\<^sub>\<E> x71 x72"   | x81 :: "'b" and x82 :: "('a, 'b, 'c) cntxt" where "y = AssignR\<^sub>\<E> x81 x82"   | x9 :: "('a, 'b, 'c) cntxt" where "y = Rjoin\<^sub>\<E> x9"
c: shows "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(R s t) c s t Q,A"
showsp_float_def: fixes p :: "nat"   and x :: "float" shows "showsp_float p x = (let m = mantissa x; e = exponent x in if e = 0 then showsp_int p m else showsp_int p m \<circ> shows_string ''*2^'' \<circ> showsp_int p e)"
strict_implies_order: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "ordering less_eq less"   and "less a b" shows "less_eq a b"
bind_split: fixes P :: "'b option \<Rightarrow> bool"   and m :: "'c option"   and f :: "'c \<Rightarrow> 'b option" shows "P (Option.bind m f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
runit_char_1: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<r>[f] : R f \<Rightarrow> f\<guillemotright>"
subid_aux2: fixes R :: "('a \<times> 'a set) set"   and a :: "'a"   and A :: "'a set" assumes "R \<subseteq> 1\<^sub>\<sigma>"   and "(a, A) \<in> R" shows "A = {a}"
considersD: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula"   and n :: "nat" assumes "considers f iA n" shows "\<exists>xs. nforms (f n) = iA # xs"
lhopital_left_at_top: fixes g :: "real \<Rightarrow> real"   and x :: "real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and y :: "real" assumes "filterlim g at_top (at_left x)"   and "\<forall>\<^sub>F x in at_left x. g' x \<noteq> 0"   and "\<forall>\<^sub>F x in at_left x. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_left x. (g has_real_derivative g' x) (at x)"   and "((\<lambda>x. f' x / g' x) \<longlongrightarrow> y) (at_left x)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> y) (at_left x)"
diagonal_step_PQ_invertible_P: fixes P :: "(('a, 'b) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and i :: "nat"   and k :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and p :: "'a"   and q :: "'a"   and u :: "'a"   and v :: "'a"   and d :: "'a" assumes "(P, Q) = diagonal_step_PQ A i k bezout"   and "(p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)"   and "i \<noteq> k"   and "i < min (nrows A) (ncols A)"   and "k < min (nrows A) (ncols A)" shows "invertible P"
pth_4_2: fixes c :: "real" shows "c *\<^sub>R (0::'a) = (0::'a)"
intro: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and cnt :: "'a \<Rightarrow> bool"   and nxt :: "'a \<Rightarrow> 'a"   and sizef :: "'a \<Rightarrow> nat" assumes "dp_consistency lookup update P"   and "iterator cnt nxt sizef" shows "dp_consistency_iterator P lookup update cnt nxt sizef"
inorder_update: fixes t :: "'t"   and a :: "'a"   and b :: "'b" assumes "inv t \<and> sorted1 (inorder t)" shows "inorder (update a b t) = upd_list a b (inorder t)"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'b"   and f2 :: "'b"   and const :: "const" shows "P (case const of Const x \<Rightarrow> f1 x | Any \<Rightarrow> f2) = ((\<forall>x1. const = Const x1 \<longrightarrow> P (f1 x1)) \<and> (const = Any \<longrightarrow> P f2))"
UniqueSucStates_def: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and C :: "'d set" shows "UniqueSucStates F G C \<equiv> \<forall>S\<in>\<Union> (States ` F). \<forall>A\<in>the (G S). if S \<in> C then \<exists>!S'. S' \<in> States A \<and> S' \<in> C else \<forall>S\<in>States A. S \<notin> C"
contlub_Mndet: fixes A :: "'a set" shows "contlub (mndet A)"
n_kat_1_opp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<cdot> y \<cdot> !z = y \<cdot> !z) = (t x \<cdot> y \<cdot> !z = (0::'a))"
uminus_def_2: fixes x :: "'a" shows "- x = (if x = bot then top else bot)"
2: fixes u :: "real" shows "a + u *\<^sub>R (b - a) = (1 - u) *\<^sub>R a + u *\<^sub>R b"
se_star_Cons: fixes c1 :: "('d, 'e) conf"   and l :: "('d, 'e) label"   and ls :: "('d, 'e) label list"   and c2 :: "('d, 'e) conf" shows "se_star c1 (l # ls) c2 = (\<exists>c. se c1 l c \<and> se_star c ls c2)"
TU: shows "topspace X \<inter> (S - U) \<inter> (T - U) \<subseteq> T"
permutes_inv_o_1: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "p \<circ> inv p = id"
assms: shows "bdd_all (nfa_is_node N) bdd"
intro: fixes R :: "'c set"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>a b. \<lbrakk>a \<in> R; b \<in> R\<rbrakk> \<Longrightarrow> multiplication a b = multiplication b a" shows "comm_ring_axioms R multiplication"
in_range_subset: fixes as :: "nat set"   and as' :: "nat set"   and h :: "heap" assumes "as \<subseteq> as'"   and "in_range (h, as')" shows "in_range (h, as)"
RMWWrite: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and \<O>s :: "nat set list"   and i :: "nat"   and A :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "bool" assumes "cond (\<theta>(t \<mapsto> m a))"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> \<O>s ! j"   and "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = _emptyset"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> \<O>s ! j = _emptyset"   and "a \<notin> read_only \<S>" shows "\<O>s,i\<turnstile> (RMW a t (D, f) cond ret A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
less_diff_conv2: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "(j - k < i) = (j < i + k)"
l11_62_unicity_bis: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and Y :: "'p" assumes "X OrthAt A B C X U"   and "Y OrthAt A B C Y U" shows "X = Y"
ineq_model_subst': fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a list"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> set X = {}"   and "ineq_model \<theta> X (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)" shows "ineq_model (\<delta> \<circ>\<^sub>s \<theta>) X F"
asymp_equiv: fixes c :: "real" assumes "c \<noteq> - 2 / 3 * e" shows "(\<lambda>x. (f x + c * (x - a)) * exp (f x + c * (x - a)) - x) \<sim>[at_right a] (\<lambda>x. C c * (x - a) powr (3 / 2))"
Infinitesimal_epsilon: shows "\<epsilon> \<in> Infinitesimal"
that: assumes "arc g1"   and "arc g2"   and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g1, pathstart g2}" shows "thesis"
SUP_emeasure_incseq: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union> (range A))"
swapAbs_simp: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "Abs xs x X $[y1 \<and> y2]_ys = Abs xs (x @xs[y1 \<and> y2]_ys) (X #[y1 \<and> y2]_ys)"
simulation2: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and tl2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s2' :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2"   and "t \<turnstile> s2 -2-tl2\<rightarrow> s2'" shows "\<exists>s1' tl1. t \<turnstile> s1 -1-tl1\<rightarrow> s1' \<and> t \<turnstile> s1' \<approx> s2' \<and> tl1 \<sim>m tl2"
columns_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "columns A = {column i A |i. i \<in> UNIV}"
bisim1FAssThrow2: fixes P :: "'b prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and v :: "'addr val" assumes "P,e2,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>ad\<rfloor>)" shows "P,e\<bullet>F{D} := e2,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 e) + pc, \<lfloor>ad\<rfloor>)"
invert_two_sides: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet A B P Q" shows "Tarski_neutral_dimensionless.TS Bet B A P Q"
vifintersectionI: fixes I :: "V"   and a :: "V"   and f :: "V \<Rightarrow> V" assumes "I \<noteq> 0"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> a \<in>\<^sub>\<circ> f i" shows "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"
insertion_var_zero: fixes x :: "'a"   and xs :: "'a list" shows "insertion (nth_default (0::'a) (x # xs)) (Var 0) = x"
ints_1: shows "integrable M (\<lambda>w. max 0 (X w))"
h: fixes x :: "complex" assumes "x \<in> ball \<xi> r" shows "(h has_field_derivative deriv g x / g x) (at x)"
add_cancel_right_left: fixes a :: "'a"   and b :: "'a" shows "(a = b + a) = (b = (0::'a))"
finsum_insert: fixes F :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier G"   and "f a \<in> carrier G" shows "finsum G f (insert a F) = f a \<oplus> finsum G f F"
InfCard_csquare_eq: fixes \<kappa> :: "V" assumes "InfCard \<kappa>" shows "\<kappa> \<otimes> \<kappa> = \<kappa>"
changes_rev: fixes xs :: "'b list" shows "changes (rev xs) = changes xs"
poly_shift_monom: fixes a :: "'a"   and k :: "nat" assumes "a \<in> carrier R" shows "poly_shift (monom P a (Suc k)) = monom P a k"
squarefree_part_dvd': fixes p :: "nat"   and n :: "nat" assumes "p \<in> squarefree_part n" shows "p dvd n"
thmD4: fixes P :: "(symbol \<times> character list) list set"   and k :: "nat" assumes "P \<subseteq> \<PP>" shows "\<W> P k = TokensAt k (Gen P)"
span_add_eq: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> More_Modules.ideal S" shows "(x + y \<in> More_Modules.ideal S) = (y \<in> More_Modules.ideal S)"
map: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "map_discr f (Discr x) = Discr (f x)"
simps_1: fixes p :: "fm"   and q :: "fm" shows "\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)"
wf_bs_polysub: fixes bs :: "'a list"   and p :: "poly"   and q :: "poly" assumes "wf_bs bs p"   and "wf_bs bs q" shows "wf_bs bs (p -\<^sub>p q)"
set_rel_def: fixes R :: "('b \<times> 'c) set" shows "\<langle>R\<rangle>set_rel \<equiv> {(A, B). (\<forall>x\<in>A. \<exists>y\<in>B. (x, y) \<in> R) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. (x, y) \<in> R)}"
cfs_monom_mult_l: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat"   and k :: "nat" assumes "p \<in> carrier P"   and "a \<in> carrier R" shows "(p \<otimes>\<^bsub>P\<^esub> monom P a n) (k + n) = a \<otimes> p k"
are_operator_effects_consistent_set: fixes op\<^sub>1 :: "('a, 'b) strips_operator_scheme"   and ops :: "('a, 'b) strips_operator_scheme list"   and op\<^sub>2 :: "('a, 'b) strips_operator_scheme" assumes "op\<^sub>1 \<in> set ops"   and "op\<^sub>2 \<in> set ops" shows "are_operator_effects_consistent op\<^sub>1 op\<^sub>2 = (set (add_effects_of op\<^sub>1) \<inter> set (delete_effects_of op\<^sub>2) = {} \<and> set (delete_effects_of op\<^sub>1) \<inter> set (add_effects_of op\<^sub>2) = {})"
birkhoff_theorem_AE_nonergodic: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "AE x in M. (\<lambda>n. birkhoff_sum f n x / real n) \<longlonglongrightarrow> real_cond_exp M Invariants f x"
assms_4: shows "(H \<longlongrightarrow> 0) (at x within s)"
1: shows "\<guillemotleft>h : y \<rightarrow>\<^sub>D G x\<guillemotright>"
eq_foldr: fixes xs :: "'b list" shows "sum_list xs = foldr (+) xs (0::'b)"
get_shadow_root_locs_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "get_shadow_root_locs = (\<lambda>element_ptr. {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt)})"
assms: fixes i :: "'a" assumes "i \<in> CBasis" shows "continuous F (\<lambda>x. cblinfun_apply (f x) i)"
sp_fundef_def: fixes F :: "'e \<Rightarrow> 'f fundef option"   and fd :: "'g fundef"   and xs :: "('h, 'i, 'e, 'op, 'opinl, 'opubx, 'j, 'k) Ubx.instr list" shows "sp_fundef F fd xs \<equiv> Subx.sp F xs (replicate (arity fd) None)"
discr0_MtransC: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and s :: "'a"   and c' :: "('b, 'c) com"   and s' :: "'a" assumes "PL_Indis indis"   and "PL_Indis.discr0 tval aval indis c"   and "PL.mustT tval aval c s"   and "PL.MtransC tval aval (c, s) (c', s')" shows "PL_Indis.discr0 tval aval indis c'"
eval_wp_Skip: fixes P :: "'c \<Rightarrow> real" shows "wp Skip P = P"
eq: shows "map Val (v' # vs') = map Val ws @ throw ex # es"
to_rat_of_rat: fixes x :: "rat" shows "to_rat (of_rat x) = x"
msubst_disj: fixes P :: "'b \<Rightarrow> 'a upred"   and Q :: "'b \<Rightarrow> 'a upred"   and v :: "('b, 'a) uexpr" shows "(P x \<or> Q x)\<lbrakk>x\<rightarrow>v\<rbrakk> = (msubst P v \<or> msubst Q v)"
pw_leof_iff: fixes m :: "'b nres"   and m' :: "'b nres" shows "(m \<le>\<^sub>n m') = (nofail m \<longrightarrow> (\<forall>x. inres m x \<longrightarrow> inres m' x))"
set_sel_2: fixes a :: "('a, 'b) hd" assumes "is_Var a" shows "var a \<in> vars_hd a"
trans: fixes r :: "'d"   and s :: "'d"   and t :: "'d" assumes "r = s"   and "s = t" shows "r = t"
collision: shows "hash (Inj1 Unit) = hash (Inj2 Unit)"
tuple_prj: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "(\<r>[a] \<cdot> (a \<otimes> \<t>[b]) \<otimes> \<l>[b] \<cdot> (\<t>[a] \<otimes> b)) \<cdot> \<d>[a \<otimes> b] = a \<otimes> b"
apply_outputs_nth: fixes i :: "nat"   and p :: "'a aexp list"   and s :: "'a \<Rightarrow> value option" assumes "i < length p" shows "apply_outputs p s ! i = aval (p ! i) s"
length_lconjseq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and xs :: "'b list" assumes "class.group_add minus plus zero uminus" shows "length (group_add.lconjseq minus plus xs) = length xs"
inf_bot_right: fixes x :: "'a" shows "x \<sqinter> bot = bot"
4: fixes x :: "brick list"   and xs :: "wall"   and y :: "brick list"   and ys :: "wall" shows "x * xs \<otimes> y * ys = (x \<otimes> y) * (xs \<otimes> ys)"
times_complex_interval: fixes x :: "complex"   and X :: "complex_interval"   and y :: "complex"   and Y :: "complex_interval" assumes "x \<in>\<^sub>c X"   and "y \<in>\<^sub>c Y" shows "x * y \<in>\<^sub>c X * Y"
usolves_ode_from_def: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and t0 :: "real"   and T :: "real set"   and X :: "'b set" shows "(y usolves_ode f from t0) T X = ((y solves_ode f) T X \<and> t0 \<in> T \<and> is_interval T \<and> (\<forall>z T'. t0 \<in> T' \<and> is_interval T' \<and> T' \<subseteq> T \<and> (z solves_ode f) T' X \<longrightarrow> z t0 = y t0 \<longrightarrow> (\<forall>t\<in>T'. z t = y t)))"
conv_concat_helper: fixes s :: "'a"   and ls :: "(bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss2 :: "'a \<times> (bool \<times> 'b \<times> 'a) list" assumes "(s, ls) \<in> conv ars"   and "ss2 \<in> conv ars"   and "lst_conv (s, ls) = fst ss2" shows "(s, ls @ snd ss2) \<in> conv ars \<and> lst_conv (s, ls @ snd ss2) = lst_conv ss2"
assms_5: shows "distributed M T Y (\<lambda>x. ennreal (Py x))"
RothsTheorem: shows "\<forall>\<xi> \<kappa>. algebraic \<xi> \<and> \<xi> \<notin> \<rat> \<and> infinite {(p, q). 0 < q \<and> coprime p q \<and> \<bar>\<xi> - real_of_int p / real_of_int q\<bar> < 1 / real_of_int q powr \<kappa>} \<longrightarrow> \<kappa> \<le> 2"
Inf_less_eq: fixes Inf :: "'f set \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f"   and inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and bot :: "'f"   and top :: "'f"   and A :: "'f set"   and u :: "'f" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v \<in> A \<Longrightarrow> less_eq v u"   and "A \<noteq> {}" shows "less_eq (Inf A) u"
subst_typ_comp_single_left: fixes single :: "(variable \<times> String.literal set) \<times> typ"   and insts :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" shows "subst_typ [single] (subst_typ insts T) = subst_typ (map (apsnd (subst_typ [single])) insts @ [single]) T"
nodes_connected_ind_sym: fixes F :: "(nat \<times> int \<times> nat) set" assumes "F \<subseteq> E" shows "sym {(x, y) |x y. nodes_connected \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = F\<rparr> x y}"
le_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 "less_eq zero zero"
simps_1: fixes uu :: "'c DBMEntry" shows "dbm_add \<infinity> uu = \<infinity>"
secure_unique_session_keys: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and evs :: "event list"   and A' :: "agent"   and B' :: "agent"   and na' :: "msg" assumes "Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace> \<in> analz (knows Spy evs)"   and "Crypt (shrK A') \<lbrace>Agent B', Key K, na'\<rbrace> \<in> analz (knows Spy evs)"   and "Key K \<notin> analz (knows Spy evs)"   and "evs \<in> yahalom" shows "A = A' \<and> B = B'"
set_intros_1: fixes x21 :: "'f"   and x22 :: "'f list" shows "x21 \<in> set (x21 # x22)"
lookup_insert: fixes k :: "'a"   and v :: "'b"   and t :: "('a, 'b) RBT.rbt" shows "RBT.lookup (RBT.insert k v t) = RBT.lookup t(k \<mapsto> v)"
integrable_imp_measurable: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f integrable_on S" shows "f \<in> borel_measurable (lebesgue_on S)"
map_of_eq_None_iff: fixes xys :: "('b \<times> 'a) list"   and x :: "'b" shows "(map_of xys x = None) = (x \<notin> fst ` set xys)"
p: shows "urel32 (pp - 2) (int (nat (p - 2)))"
inj_on_conv_Ex1: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "inj_on f A = (\<forall>y\<in>f ` A. \<exists>!x. x \<in> A \<and> y = f x)"
ideal_npow_closedTr: fixes I :: "'a set"   and x :: "'a"   and n :: "nat" assumes "ideal R I"   and "x \<in> I" shows "0 < n \<longrightarrow> x^\<^bsup>R n\<^esup> \<in> I"
A: shows "((pp_init_f, pp_init_l), p, f, l) \<in> trcl pr_algo_lts"
cardSuc_ordLeq_ordLess: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Card_order r"   and "Card_order r'" shows "(r' <o cardSuc r) = (r' \<le>o r)"
default_def: fixes df :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd option" shows "Lib.default df f = (\<lambda>i. case f i of None \<Rightarrow> df i | Some s \<Rightarrow> s)"
bind_return': fixes M :: "'d measure" shows "M \<bind> return M = M"
ALG_P'_rest2: fixes n :: "nat"   and qs :: "'c list"   and Strat :: "(nat \<times> nat list) list"   and init :: "'c list"   and x :: "'c"   and y :: "'c"   and r1 :: "(nat \<times> nat list) list"   and r2 :: "'c list" assumes "n < length qs"   and "n < length Strat" shows "ALG_P' Strat qs init n x y = ALG_P' (Strat @ r1) (qs @ r2) init n x y"
stut_or: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "STUTINV F"   and "STUTINV G" shows "STUTINV (F \<or> G)"
summable_diff: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "summable f"   and "summable g" shows "summable (\<lambda>n. f n - g n)"
fplus_def: shows "fplus = COERCE(udom\<cdot>'d \<rightarrow> udom\<cdot>'d \<rightarrow> udom\<cdot>'d, 'c\<cdot>'d \<rightarrow> 'c\<cdot>'d \<rightarrow> 'c\<cdot>'d)\<cdot>plusU"
small_vsubset_set: fixes B :: "V set"   and A :: "V set" assumes "small B"   and "A \<subseteq> B" shows "ZFC_in_HOL.set A \<subseteq>\<^sub>\<circ> ZFC_in_HOL.set B"
inj_injf_max: fixes E :: "'a set" assumes "\<forall>x. \<exists>y\<in>E. x < y" shows "inj (\<lambda>n. injf_max n E)"
sigma_finite_countable: shows "\<exists>A. countable A \<and> A \<subseteq> sets (dens_ctxt_measure \<Y> \<rho>) \<and> \<Union> A = space (dens_ctxt_measure \<Y> \<rho>) \<and> (\<forall>a\<in>A. emeasure (dens_ctxt_measure \<Y> \<rho>) a \<noteq> \<infinity>)"
order: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> '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 n :: "nat"   and p :: "'b list" assumes "class.idom_char_0 one times plus zero minus uminus" shows "(semiring_0.divides plus zero times (semiring_1.pexp one times plus zero [uminus a, one] n) p \<and> \<not> semiring_0.divides plus zero times (semiring_1.pexp one times plus zero [uminus a, one] (Suc n)) p) = (n = ring_1.order one times plus zero uminus a p \<and> semiring_0.poly plus zero times p \<noteq> semiring_0.poly plus zero times [])"