hcju/isabelleps huggingface dataset:
induct: fixes P :: "nat list \<Rightarrow> (('a, 'b) vec, 'c) vec \<Rightarrow> nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "(('a, 'b) vec, 'c) vec"   and a2 :: "nat"   and a3 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "\<And>A i bezout. P [] A i bezout"   and "\<And>j xs A i bezout. \<lbrakk>A $ from_nat i $ from_nat i dvd A $ from_nat j $ from_nat j \<Longrightarrow> P xs A i bezout; \<And>x xa y xb ya xc yb xd yc xe. \<lbrakk>\<not> A $ from_nat i $ from_nat i dvd A $ from_nat j $ from_nat j; x = bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j); (xa, y) = x; (xb, ya) = y; (xc, yb) = ya; (xd, yc) = yb; xe = diagonal_step A i j yc xd\<rbrakk> \<Longrightarrow> P xs xe i bezout\<rbrakk> \<Longrightarrow> P (j # xs) A i bezout" shows "P a0 a1 a2 a3"
case_2: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Col of Row \<Rightarrow> f1 | Col \<Rightarrow> f2 | Num \<Rightarrow> f3) = f2"
dim_subset: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set"   and S :: "'e set"   and T :: "'e set" assumes "finite_dimensional_vector_space scale Basis"   and "S \<subseteq> T" shows "vector_space.dim scale S \<le> vector_space.dim scale T"
i_f_join_singleton: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "f \<Join>\<^sub> {n} = [f n]"
restrict_restrict: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" shows "restrict (restrict f A) B = restrict f (A \<inter> B)"
minimal_ttree: fixes S :: "'b ttree" shows "TTree.empty \<sqsubseteq> S"
source_out_not_cycle: fixes \<Delta> :: "('a, 'b) network_scheme"   and p :: "'a list"   and x :: "'a" assumes "flow_attainability \<Delta>"   and "cycle \<Delta> p" shows "(source \<Delta>, x) \<notin> set (cycle_edges p)"
maxvar: fixes vs :: "nat list" shows "\<forall>v\<in>set vs. v \<le> maxvar vs"
pol_expr_deg: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "\<exists>c. pol_coeff S c \<and> deg_n R S X p \<le> fst c \<and> p = polyn_expr R X (deg_n R S X p) c \<and> snd c (deg_n R S X p) \<noteq> \<zero>\<^bsub>S\<^esub>"
fact_group_trivial_iff: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<lhd> G"   and "finite (carrier G)" shows "(carrier (G Mod H) = {\<one>\<^bsub>G Mod H\<^esub>}) = (H = carrier G)"
prj0_simps_3: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "cod \<p>\<^sub>0[f, g] = local.dom g"
abs_mult_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x" shows "times (abs y) x = abs (times y x)"
assms_2: fixes n :: "nat" shows "0 \<le> e n"
D: shows "real (D fs') < real_of_rat reduction * real (D fs)"
simps_1: fixes x :: "'a" shows "cast_Ubx1 (OpUbx1 x) = Some x"
exists_split_characterisations_1: shows "(\<exists>x. ((top \<le> x \<and> x \<sqinter> x \<le> x) \<and> x \<sqinter> id x \<le> top) \<and> x \<squnion> id x = top) = (\<exists>x. (x \<le> - top \<and> x \<sqinter> x \<le> x) \<and> x \<squnion> id x = - top)"
dom_char: fixes f :: "'a" shows "Left.dom f = (if Left.arr f then local.dom f else null)"
indets_plus_subset: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "indets (p + q) \<subseteq> indets p \<union> indets q"
elims_2: fixes x :: "('fsym, 'psym) lit" assumes "Ik.wtL x" obtains a :: "('fsym, 'psym) atm" where "x = Pos a__" and "Ik.wtA a__"   | a :: "('fsym, 'psym) atm" where "x = Neg a" and "Ik.wtA a"
rderivs_lderivs: fixes w :: "'a list"   and r :: "'b rexp" shows "rderivs w r = REV (lderivs w (REV r))"
steps: shows "\<forall>i<k. \<Gamma>\<turnstile> p i \<rightarrow> p (Suc i)"
add_bot: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.aggregation_order plus bot less_eq less" shows "plus x y = bot \<longrightarrow> x = bot"
isPrefixPrefixToElement: fixes e :: "'b"   and t :: "('b \<times> bool) list" shows "isPrefix (prefixToElement e t) t"
iLastWeak_def: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLastWeak t0 I P \<equiv> \<box> t {iprev t0 I} \<down>< t0. P t"
distinct_11: fixes x21 :: "(char list \<times> tid_or_exn) option"   and x22 :: "v list"   and x31 :: "v sem_env"   and x32 :: "char list"   and x33 :: "exp0" shows "Conv x21 x22 \<noteq> Closure x31 x32 x33"
LLq_pls_zz: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> atrm"   and "t2 \<in> atrm"   and "zz \<notin> FvarsT t1"   and "zz \<notin> FvarsT t2" shows "LLq t1 t2 = exi zz (eql t2 (pls (Var zz) t1))"
WP1ConcEnd: fixes xs :: "('a, 'b) Combinators list"   and xa :: "('a, 'b) Combinators" assumes "wellformed_policy1 (xs @ [xa])" shows "wellformed_policy1 xs"
abGrp: shows "\<forall>U\<in>set Us. AbGroup U"
IH_2: fixes as :: "'a list"   and bs :: "'a list" assumes "ys = map (assocs G) as"   and "zs = map (assocs G) bs" shows "\<exists>as'. as <~~> as' \<and> map (assocs G) as' = zs"
composite_functor_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'e" shows "composite_functor A B C F G \<equiv> functor A B F \<and> functor B C G"
h1: shows "(\<Sum>i = 0..<n + 1. f i / B ^ i) * B ^ n = (\<Sum>i = 0..<n + 1. f i / B ^ i * B ^ n)"
binary_union: fixes x :: "hf"   and y :: "hf" shows "\<exists>z. \<forall>u. (u \<^bold>\<in> z) = (u \<^bold>\<in> x \<or> u \<^bold>\<in> y)"
homotopically_trivial_retraction_null_gen: fixes s :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and t :: "'b set"   and k :: "'b \<Rightarrow> 'a"   and U :: "'c set"   and Q :: "('c \<Rightarrow> 'b) \<Rightarrow> bool"   and P :: "('c \<Rightarrow> 'a) \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b" assumes "Retracts s h t k"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P (k \<circ> f)"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q (h \<circ> f)"   and "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> s; P f\<rbrakk> \<Longrightarrow> \<exists>c. homotopic_with_canon P U s f (\<lambda>x. c)"   and "continuous_on U f"   and "f ` U \<subseteq> t"   and "Q f" obtains c :: "'b" where "homotopic_with_canon Q U t f (\<lambda>x. c)"
less_fun_trans: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "less_fun f g"   and "less_fun g h" shows "less_fun f h"
eval_sub_consts: fixes p :: "('e, 'f) form"   and q :: "('e, 'f) form"   and e :: "nat \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'd list \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'd list \<Rightarrow> bool"   and cs :: "'e list" assumes "\<forall>e f g. eval e f g p \<longrightarrow> eval e f g q"   and "eval e f g (sub_consts cs p)" shows "eval e f g (sub_consts cs q)"
R_def: shows "R = 1 + \<bar>B\<bar> + cmod z"
A: fixes fa :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and ha :: "'b \<Rightarrow> 'c"   and ga :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and ia :: "'b \<Rightarrow> 'c" assumes "uniform_limit X fa ha F"   and "\<forall>\<^sub>F y in F. \<forall>x\<in>X. fa y x = ga y x"   and "\<And>x. x \<in> X \<Longrightarrow> ha x = ia x" shows "uniform_limit X ga ia F"
simps_1: fixes uu :: "g_state" shows "g_out uu Any = None"
F1set2_image_Lev1: fixes xe :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xc :: "'d"   and xd :: "bd_type_F"   and xa :: "'d" assumes "xe \<in> Lev1 s1 s2 n xc"   and "xe @ [Inl xd] \<in> Lev1 s1 s2 (Suc n) xc"   and "rv1 s1 s2 xe xc = Inl xa" shows "xd \<in> tobd_F12 s1 xa ` F1set2 (s1 xa)"
iwlsFSbSwTR_qFreshAll_igFreshAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "iwlsFSbSwTR MOD" shows "(qFresh ys y qX \<longrightarrow> igFresh MOD ys y (qInit MOD qX)) \<and> (qFreshAbs ys y qA \<longrightarrow> igFreshAbs MOD ys y (qInitAbs MOD qA))"
closed_segment_Un: fixes b :: "real"   and a :: "real"   and c :: "real" assumes "b \<in> {a--c}" shows "{a--b} \<union> {b--c} = {a--c}"
Ln'_cis: fixes t :: "real" assumes "t \<in> {- pi / 2<..3 / 2 * pi}" shows "Ln' (cis t) = \<i> * complex_of_real t"
shift_ms_aux: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and e :: "real" assumes "is_expansion_aux xs f (b # basis)"   and "basis_wf (b # basis)" shows "is_expansion_aux (shift_ms_aux e xs) (\<lambda>x. b x powr e * f x) (b # basis)"
comp_assoc: fixes h :: "'c"   and g :: "'c"   and f :: "'c" shows "D (D h g) f = D h (D g f)"
i_shrink_last_nth: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "0 < k" shows "(f \<div>\<^sub>l k) n = f (n * k + k - Suc 0)"
shadow_root_h3: shows "h' \<turnstile> get_shadow_root element_ptr \<rightarrow>\<^sub>r Some new_shadow_root_ptr"
is_ring_iso_def: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme" shows "R \<simeq> S = (ring_iso R S \<noteq> {})"
cut_def: fixes f :: "'c \<Rightarrow> 'd"   and R :: "('c \<times> 'c) set"   and x :: "'c" shows "cut f R x = (\<lambda>y. if (y, x) \<in> R then f y else undefined)"
R_Inf: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> action.relation_of ` A} is R healthy"
char_poly_similar: fixes A :: "'c mat"   and B :: "'c mat" assumes "similar_mat A B" shows "char_poly A = char_poly B"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "delete_index 0 (x # xs) = xs"
override_on_apply_notin: fixes a :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "a \<notin> A" shows "override_on f g A a = f a"
diamond_rtrancl: fixes r :: "comb \<Rightarrow> comb \<Rightarrow> bool" assumes "diamond r" shows "diamond r\<^sup>*\<^sup>*"
irreducible_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.irreducible times one zero p = (p \<noteq> zero \<and> \<not> dvd.dvd times p one \<and> (\<forall>a b. p = times a b \<longrightarrow> dvd.dvd times a one \<or> dvd.dvd times b one))"
delay_bisimulation_diverge_axioms: shows "delay_bisimulation_diverge trsys2 trsys3 bisim23 tlsim23 \<tau>move2 \<tau>move3"
simps_9: fixes x :: "name"   and P :: "pi" shows "depth (<\<nu>x>P) = depth P"
Ana_f: fixes s :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list" assumes "Ana s = (K, T)"   and "f \<in> \<Union> (funs_term ` set K)" shows "f \<in> funs_term s"
distinct_distinct_of_odd: fixes p :: "nat \<Rightarrow> nat"   and i :: "nat" shows "distinct_of_odd p i \<le> 1"
mult_le_cancel_left_pos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less zero c" shows "less_eq (times c a) (times c b) = less_eq a b"
cong_3_sym: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C Cong3 A' B' C'" shows "A' B' C' Cong3 A B C"
K: shows "\<forall>b. b \<in># K' \<longrightarrow> (b, a) \<in> r"
nn_integral_cmult_indicator: fixes A :: "'a set"   and M :: "'a measure"   and c :: "ennreal" assumes "A \<in> sets M" shows "\<integral>\<^sup>+x\<in>A. c\<partial>M = c * emeasure M A"
eventually_at_left_to_right: fixes P :: "real \<Rightarrow> bool"   and a :: "real" shows "eventually P (at_left a) = (\<forall>\<^sub>F x in at_right (- a). P (- x))"
simps_3: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and Cs :: "char list list" shows "(P,h,sh \<turnstile>\<^sub>i (C, M, pc, Called Cs)) = (let ins = instrs_of P C M in \<exists>C1 sobj. Called_context P C1 (ins ! pc) \<and> sh C1 = \<lfloor>sobj\<rfloor>)"
A: shows "(\<tau> \<Turnstile> \<delta> (\<lambda>_. self \<tau>)) = (\<tau> \<Turnstile> \<delta> self)"
left_adjoint_determines_right_up_to_iso: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and g' :: "('a S.E.term, 'a) S.arr" assumes "S.adjoint_pair f g"   and "S.adjoint_pair f g'" shows "S.isomorphic g g'"
exp_two_pi_i: shows "exp (2 * complex_of_real pi * \<i>) = 1"
echelon_form_Hermite_of_row: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c" assumes "ass_function ass"   and "res_function res"   and "echelon_form A" shows "echelon_form (Hermite_of_row_i ass res A i)"
sel_4: fixes x1 :: "((('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) list"   and x2 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x3 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x4 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x5 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list"   and x6 :: "('e strand_label \<times> (('f, 'g, 'h) prot_fun, (('f, 'g, 'h) prot_fun, 'g prot_atom) Term.term \<times> nat) stateful_strand_step) list" shows "transaction_checks (Transaction x1 x2 x3 x4 x5 x6) = x4"
word_less_cases: fixes x :: "'b word"   and y :: "'b word" assumes "x < y" shows "x = y - 1 \<or> x < y - 1"
NMPrm: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and p :: "'a list" assumes "not_MT C p" shows "not_MT C (rm_MT_rules C p)"
reachable_awalkI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "H.awalk u p v" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"
top_apply: fixes x :: "'f" shows "\<top> x = \<top>"
ce_rel_lm_23: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and r :: "(nat \<times> nat) set" assumes "p \<in> PrimRec3"   and "\<forall>x y. ((x, y) \<in> r) = (\<exists>u. p x y u = 0)" shows "r \<in> ce_rels"
induct: fixes P :: "Cooper.num \<Rightarrow> bool"   and a0 :: "Cooper.num" assumes "\<And>j. P (C j)"   and "\<And>n. P (Bound n)"   and "\<And>t. P t \<Longrightarrow> P (Neg t)"   and "\<And>t s. \<lbrakk>P t; P s\<rbrakk> \<Longrightarrow> P (Add t s)"   and "\<And>t s. \<lbrakk>P t; P s\<rbrakk> \<Longrightarrow> P (Sub t s)"   and "\<And>i t. (i \<noteq> 0 \<Longrightarrow> P t) \<Longrightarrow> P (Mul i t)"   and "\<And>v va vb. P (CN v va vb)" shows "P a0"
declrefT_simp: fixes r :: "ref_ty"   and m :: "mhead" shows "declrefT (r, m) = r"
emeasure_count_space: fixes X :: "'b set"   and A :: "'b set" assumes "X \<subseteq> A" shows "emeasure (count_space A) X = (if finite X then of_nat (card X) else \<infinity>)"
B_def: shows "B = (\<Sum>\<^sub>ai\<in>A. norm (x i * x i)) + (\<Sum>\<^sub>ai\<in>A. norm (y i * y i))"
T: fixes e :: "real" shows "0 < e \<longrightarrow> (\<exists>T\<in>fmeasurable M. T \<subseteq> S \<and> m - e < Sigma_Algebra.measure M T)"
R_mult_right_mono: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "R a b"   and "0 \<le> c" shows "R (a * c) (b * c)"
1: shows "! (! x \<sqinter> ! y) = x \<squnion> y"
wpl_opt_bst: fixes i :: "int"   and j :: "int" shows "local.wpl i j (local.opt_bst i j) = local.min_wpl i j"
not_less_Least: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "less k (ord.Least less_eq P)" shows "\<not> P k"
chanEqSym: fixes \<Psi> :: "'b"   and M :: "'a"   and N :: "'a" assumes "\<Psi> \<turnstile> M \<leftrightarrow> N" shows "\<Psi> \<turnstile> N \<leftrightarrow> M"
zunion_def: fixes a :: "'b zet"   and b :: "'b zet" shows "zunion a b \<equiv> Abs_zet (Rep_zet a \<union> Rep_zet b)"
continuous_within_eps_delta: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous (at x within s) f = (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
the_inv_into_map_impl_correct: fixes V :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite V"   and "inj_on f V" shows "the_inv_into_map_impl V f \<le> SPEC (\<lambda>r. r = the_inv_into_map V f)"
eta_expansion_safe: fixes T :: "stack_elem list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "set T \<subseteq> range Arg" shows "(\<Gamma>, eta_expand (length T) e, T @ S) \<Rightarrow>\<^sup>* (\<Gamma>, e, T @ S)"
assms_3: shows "formulaFalse (getF state) (elements (getM state))"
snull_eq_snil: fixes xs :: "[:'r:]" shows "snull\<cdot>xs = eq\<cdot>xs\<cdot>[::]"
linear_sum: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'b"   and S :: "'e set" assumes "\<forall>x. x \<in> S1 \<longrightarrow> f x \<in> S2"   and "\<forall>x. g x \<in> S1"   and "linear_on S1 S2 scale1 scale2 f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
gen_binomial_complex': fixes x :: "real"   and y :: "real"   and a :: "complex" assumes "\<bar>x\<bar> < \<bar>y\<bar>" shows "(\<lambda>n. (a gchoose n) * complex_of_real x ^ n * complex_of_real y powr (a - of_nat n)) sums complex_of_real (x + y) powr a"
assms: shows "S = {} \<or> S retract_of_space X"
currentD_weight_IN: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f" shows "d_IN f x \<le> weight \<Gamma> x"
le_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "le\<cdot>x\<cdot>y = TT"   and "le\<cdot>y\<cdot>z = TT" shows "le\<cdot>x\<cdot>z = TT"
edgesT_diff_sng_inv_eq: fixes T :: "'a ugraph"   and x :: "'a"   and y :: "'a" shows "(edges T - {(x, y), (y, x)})\<inverse> = edges T - {(x, y), (y, x)}"
CASE_4: shows "(([entry fg p], c), w, [v], c') \<in> trcl (trss fg)"
poly_eq_iff: fixes p :: "'b poly"   and q :: "'b poly" shows "(p = q) = (\<forall>n. poly.coeff p n = poly.coeff q n)"
subcls1D: fixes C :: "char list"   and D :: "char list"   and P :: "'a prog" assumes "P \<turnstile> C \<prec>\<^sup>1 D" shows "C \<noteq> Object \<and> (\<exists>fs ms. class P C = \<lfloor>(D, fs, ms)\<rfloor>)"
Suc_1: fixes pv :: "'a poly vec" assumes "dim_row A = dim_vec pv" shows "mk_poly2_col A pv j $ i = pv $ i + (\<Sum>j'<j. monom (A $$ (i, dim_col A - Suc j')) j')"
KS_SK_id: fixes xs :: "nat discr llist" shows "KS\<cdot>(SK\<cdot>xs) = xs"
defined7: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and Y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<delta> (X \<triangleq> Y) = true"
lit_ord_dominating_term: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and s1 :: "'b trm"   and s2 :: "'b trm"   and t2 :: "'b trm"   and x1 :: "'b literal"   and t1 :: "'b trm"   and p1 :: "sign"   and x2 :: "'b literal"   and p2 :: "sign" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "(s1, s2) \<in> trm_ord \<or> (s1, t2) \<in> trm_ord"   and "basic_superposition.orient_lit trm_ord x1 s1 t1 p1"   and "basic_superposition.orient_lit trm_ord x2 s2 t2 p2"   and "vars_of_lit x1 = {}"   and "vars_of_lit x2 = {}" shows "(x1, x2) \<in> basic_superposition.lit_ord trm_ord"
stepsI: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "E x (hd xs)"   and "Graph_Defs.steps E xs" shows "Graph_Defs.steps E (x # xs)"
hyps: fixes xs :: "'a llist" assumes "\<And>m. m < n \<Longrightarrow> lnth xs m \<in> A" shows "lset (ltake (enat n) xs) \<subseteq> A"
w_simps_3: shows "src \<mu>.chine = src (tab\<^sub>0 (local.dom \<mu>))"
r_limr_2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and i :: "nat" assumes "learn_lim \<psi> U (\<phi> i)" shows "learn_lim \<psi> U (\<phi> (the (eval r_limr [i])))"
hom_functor_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b" shows "hom_functor C S \<phi> \<equiv> (category C \<and> dual_category C) \<and> product_category (dual_category.comp C) C \<and> replete_set_category S \<and> hom_functor_axioms C S \<phi>"
min_elt_minimal: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set"   and y :: "'a" assumes "minimal_element P A"   and "B \<subseteq> A"   and "B \<noteq> {}"   and "y \<in> A"   and "P y (minimal_element.min_elt P A B)" shows "y \<notin> B"
cd_not_ret: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "\<pi> k \<noteq> local.return"
simps_6: fixes i :: "nat" shows "max_Var_floatarith (Var i) = Suc i"
nprv_exuE_exi: fixes F :: "'fmla set"   and x :: "'var"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F (exu x \<phi>)"   and "nprv (insert \<phi> F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "x \<in> var"   and "\<psi> \<in> fmla"   and "x \<notin> Fvars \<psi>"   and "x \<notin> \<Union> (Fvars ` F)" shows "nprv F \<psi>"
cut_le_singleton: fixes a :: "'a"   and t :: "'a" shows "{a} \<down>\<le> t = (if a \<le> t then {a} else {})"
uminus_add_conv_diff: fixes 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 b :: "'b" assumes "class.ab_group_add plus zero minus uminus" shows "plus (uminus a) b = minus b a"
gcd_proj1_if_dvd: fixes a :: "'a"   and b :: "'a" assumes "a dvd b" shows "gcd a b = normalize a"
set_iterator_genord_foldli_conv: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'d set"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "set_iterator_genord iti S R = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0)"
\<zeta>: fixes p :: "fm" assumes "iszlfm p" shows "0 < \<zeta> p \<and> d_\<beta> p (\<zeta> p)"
q'_5: fixes K1 :: "'n set"   and K2 :: "'n set" assumes "K1 \<in> q"   and "K2 \<in> q"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"