hcju/isabelleps huggingface dataset:
preserves_reflects_arr: fixes f :: "'a \<times> 'a" shows "hseq (fst f) (snd f) = VV.arr f"
SA_trans_TVar: fixes \<Gamma> :: "binding list"   and i :: "nat"   and U :: "POPLmarkRecord.type"   and T :: "POPLmarkRecord.type" assumes "\<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB U\<rfloor>"   and "\<Gamma> \<turnstile> \<up>\<^sub>\<tau> (Suc i) 0 U <: T" shows "\<Gamma> \<turnstile> TVar i <: T"
closed_Union_compact_subsets: fixes S :: "'c set" assumes "closed S" obtains F :: "nat \<Rightarrow> 'c set" where "\<And>n. compact (F n)" and "\<And>n. F n \<subseteq> S" and "\<And>n. F n \<subseteq> F (Suc n)" and "\<Union> (range F) = S" and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> F n"
msg_def: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'd list" shows "msg n s \<equiv> \<forall>t. length (s t) \<le> n"
K_cfg_valid_cfgD_2: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and cfg' :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "state cfg' \<in> set_pmf (action cfg)"
UN_Un_distrib: fixes A :: "'e \<Rightarrow> 'd set"   and B :: "'e \<Rightarrow> 'd set"   and I :: "'e set" shows "(\<Union>i\<in>I. A i \<union> B i) = \<Union> (A ` I) \<union> \<Union> (B ` I)"
simps_4: fixes v :: "'a formula" shows "is_mini_formula (\<^bold>\<not> v) = False"
well_related_set_axioms: shows "well_related_set A (\<sqsubseteq>)"
case_cong: fixes nat :: "nat"   and nat' :: "nat"   and f1 :: "'b"   and g1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and g2 :: "nat \<Rightarrow> 'b" assumes "nat = nat'"   and "nat' = 0 \<Longrightarrow> f1 = g1"   and "\<And>x2. nat' = Suc x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case nat of 0 \<Rightarrow> f1 | Suc x2 \<Rightarrow> f2 x2) = (case nat' of 0 \<Rightarrow> g1 | Suc x2 \<Rightarrow> g2 x2)"
accept_def: shows "\<MM>.accept\<^sub>R' = Alm_all \<MM>.token_succeeds"
hequiv_sym: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set"   and i :: "'c"   and j :: "'c" assumes "Hintikka A H"   and "i \<in> A"   and "j \<in> A" shows "hequiv H i j = hequiv H j i"
Cons_acc_listrel1I: fixes x :: "'g"   and r :: "('g \<times> 'g) set"   and xs :: "'g list" assumes "x \<in> Wellfounded.acc r"   and "xs \<in> Wellfounded.acc (listrel1 r)" shows "x # xs \<in> Wellfounded.acc (listrel1 r)"
WTrt_elim_cases_2: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and v :: "String.literal"   and e :: "'c expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (v:=e) T" obtains T :: "ty" and T' :: "ty" where "Ta__ = Void" and "E v = \<lfloor>T\<rfloor>" and "J_heap_base.WTrt typeof_addr P h E e T'" and "P \<turnstile> T' \<le> T"
sep_add_assoc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y"   and "sep_disj y z"   and "sep_disj x z" shows "plus (plus x y) z = plus x (plus y z)"
wp1n_RS2: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (removeShadowRules2 p)"
is_lub_bool: fixes S :: "bool set" shows "S <<| (True \<in> S)"
Cond: fixes p :: "'a set"   and b :: "'a set"   and c1 :: "'a com"   and q :: "'a set"   and c2 :: "'a com" assumes "\<parallel>- (p \<inter> b) c1 q"   and "\<parallel>- (p \<inter> - b) c2 q" shows "\<parallel>- p Cond b c1 c2 q"
ground_args: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "t \<in> set (args s)"   and "ground s" shows "ground t"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and Vertex :: "Vertex" shows "P (case Vertex of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = ((Vertex = A \<longrightarrow> P f1) \<and> (Vertex = B \<longrightarrow> P f2) \<and> (Vertex = C \<longrightarrow> P f3) \<and> (Vertex = D \<longrightarrow> P f4))"
sgn_uexpr_def: fixes u :: "('e, 'f) uexpr" shows "sgn u = uop sgn u"
elims_3: fixes x :: "'b aexp" assumes "\<not> is_lit x" obtains v :: "'b" where "x = V v__"   | v :: "'b aexp" and va :: "'b aexp" where "x = Plus va__ vaa__"   | v :: "'b aexp" and va :: "'b aexp" where "x = Minus vb__ vab__"   | v :: "'b aexp" and va :: "'b aexp" where "x = Times v va"
assms_2: shows "STRIPS_Semantics.is_serial_solution_for_problem (\<phi> \<Psi> ) \<pi>"
wf_since_aux_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set"   and args :: "Monitor.args"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and aux :: "'msaux"   and ne :: "nat" shows "wf_since_aux \<sigma> V R args \<phi> \<psi> aux ne = (fv \<phi> \<subseteq> fv \<psi> \<and> (\<exists>cur auxlist. valid_msaux args cur aux auxlist \<and> cur = (if ne = 0 then 0 else \<tau> \<sigma> (ne - 1)) \<and> sorted_wrt (\<lambda>x y. fst y < fst x) auxlist \<and> (\<forall>t X. (t, X) \<in> set auxlist \<longrightarrow> ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right (args_ivl args) \<and> (\<exists>i. \<tau> \<sigma> i = t) \<and> qtable (args_n args) (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne - 1) (Sincep (args_pos args) \<phi> (point (\<tau> \<sigma> (ne - 1) - t)) \<psi>)) X) \<and> (\<forall>t. ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right (args_ivl args) \<and> (\<exists>i. \<tau> \<sigma> i = t) \<longrightarrow> (\<exists>X. (t, X) \<in> set auxlist))))"
div_mult_unit2: 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 c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c one"   and "dvd.dvd times b a" shows "divide a (times b c) = divide (divide a b) c"
h_1: shows "steps (s, l, r) (A, off) (Suc stp) = (s', l', r')"
substitutes_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "substitutes_on A f = (\<not> (\<exists>B\<subseteq>A. \<exists>a b. {a, b} \<subseteq> A - B \<and> b \<notin> f (B \<union> {b}) \<and> b \<in> f (B \<union> {a, b})))"
compare_lists_Cons_2: fixes b :: "real"   and a :: "real"   and as :: "real list"   and bs :: "real list" assumes "b < a" shows "compare_lists (a # as) (b # bs) = GT"
language_coinduct_upto_regular: fixes R :: "'b language \<Rightarrow> 'b language \<Rightarrow> bool"   and L :: "'b language"   and K :: "'b language" assumes "R L K"   and "\<And>L K. R L K \<Longrightarrow> \<oo> L = \<oo> K \<and> (\<forall>x. regular_cong R (\<dd> L x) (\<dd> K x))" shows "L = K"
lconf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and E :: "char list \<Rightarrow> ty option" shows "P,h \<turnstile> l (:\<le>) E \<equiv> \<forall>V v. l V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. E V = \<lfloor>T\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
rep_eq: fixes x :: "'c \<Rightarrow> 'a"   and xa :: "'c set['b]" shows "set_bset (map_bset x xa) = x ` set_bset xa"
nprv_cut2: fixes F :: "'fmla set"   and \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F \<phi>1"   and "nprv F \<phi>2"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla"   and "nprv (insert \<phi>2 (insert \<phi>1 F)) \<psi>" shows "nprv F \<psi>"
that: fixes uty :: "typ" assumes "typ_of1 Ts t = Some (uty \<rightarrow> ty)"   and "typ_of1 Ts u = Some uty" shows "thesis"
exhaust_sel: fixes tm :: "('a, 'b) tm" obtains "tm = Hd (head tm)"   | "tm = App (fun tm) (arg tm)"
is_empty_set: fixes xs :: "'k list" shows "Set.is_empty (set xs) = List.null xs"
lossless_inline: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and \<I>' :: "('e, 'f) \<I>"   and callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'd, 'e, 'f) gpv"   and s :: "'d" assumes "lossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> lossless_gpv \<I>' (callee s x)"   and "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV" shows "lossless_gpv \<I>' (inline callee gpv s)"
compl_struct_gb_red: shows "punit.compl_struct punit.gb_red"
eq_neutr_iff: fixes a :: "'d"   and b :: "'d" shows "(sup a b = bot) = (a = bot \<and> b = bot)"
b_c: shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) v = type0 init y x"
distinct_1: shows "cospan_shape.Null \<noteq> cospan_shape.AA"
linear_comb_eqTr: fixes H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M" shows "s \<in> {j. j \<le> n} \<rightarrow> carrier R \<and> f \<in> {j. j \<le> n} \<rightarrow> H \<and> g \<in> {j. j \<le> n} \<rightarrow> H \<and> (\<forall>j\<in>{j. j \<le> n}. f j = g j) \<longrightarrow> l_comb R M n s f = l_comb R M n s g"
converse_trancl_induct: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>+"   and "\<And>y. (y, b) \<in> r \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>+; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
stut_and: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "STUTINV F"   and "STUTINV G" shows "STUTINV (F \<and> G)"
qfree_subst_peps: fixes \<phi> :: "atom fm"   and k :: "nat" assumes "nqfree \<phi>" shows "qfree (subst\<^sub>+ \<phi> k)"
linorder: fixes comp :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "comparator comp" shows "class.linorder (le_of_comp comp) (lt_of_comp comp)"
ltln_to_ltl_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile> ltln_to_ltl \<phi> = w \<Turnstile>\<^sub>n \<phi>"
rep_eq: fixes x :: "float" shows "real_of_float (normfloat x) = real_of_float x"
chainmap_preserves_length: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and \<Pi> :: "('e, 'f) chain" shows "chainlen (chainmap f g \<Pi>) = chainlen \<Pi>"
retraction_is_epi: fixes g :: "('a, 'a set) Maps.arr" assumes "Maps.retraction g" shows "Maps.epi g"
diff_conv_add_uminus: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus a b = plus a (uminus b)"
length_pos_if_in_set: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> ran\<^sub>l xs" shows "0 < length xs"
skip\<^sub>S\<^sub>E_def: shows "skip\<^sub>S\<^sub>E = unit\<^sub>S\<^sub>E ()"
min_def: fixes q :: "('c, 'd) pq" shows "PQ.min q = fst (hd (pq.alist_of q))"
Pow_mono: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A" shows "order.greater_eq (Pow B) (Pow A)"
Units_r_inv: fixes x :: "'a" assumes "x \<in> Units R" shows "x \<otimes> inv x = \<one>"
wls_swapAbs_simp: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and xs :: "'varSort"   and x :: "'var" assumes "wls s X" shows "Abs xs x X $[y1 \<and> y2]_ys = Abs xs (x @xs[y1 \<and> y2]_ys) (X #[y1 \<and> y2]_ys)"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Cod \<nu>)"
invertible_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "invertible A = (\<exists>A'. A ** A' = mat (1::'a) \<and> A' ** A = mat (1::'a))"
isLubD3: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isLub R S x" shows "leastP (isUb R S) x"
consistent_init: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option"   and i :: "nat"   and n :: "nat" assumes "\<psi> \<in> \<R>\<^sup>2"   and "f \<in> \<R>" shows "consistent \<psi> i (f \<triangleright> n) = (if \<psi> i \<triangleright> n = f \<triangleright> n then Some 0 else Some 1)"
almost_full_on_pp_iff: fixes A :: "('a, 'b) pp set" shows "almost_full_on (adds) A = almost_full_on (adds) (pp.mapping_of ` A)"
pref_cancel_conv: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs \<cdot> ys \<le>p xs \<cdot> zs) = (ys \<le>p zs)"
upper_bound_CD: shows "\<exists>C D. (\<forall>n. B (Suc n) \<le> D \<or> B (Suc n) \<le> C * B n) \<and> 1 \<le> C"
orderE: fixes b :: "'a word"   and a :: "'a word" assumes "b \<le>s a" obtains "a = signed.max a b"
Gamma_integral_real: fixes x :: "real" assumes "0 < x" shows "((\<lambda>t. t powr (x - 1) / exp t) has_integral Gamma x) {0..}"
B: shows "\<forall>X\<in>set PS. weakly_sequential X"
add2_assoc: fixes a :: "'b option"   and b :: "'b option"   and c :: "'b option" shows "total_partial.add2 a (total_partial.add2 b c) = total_partial.add2 (total_partial.add2 a b) c"
gd_powerprod_ord_pp_punit: fixes cmp_term :: "'a nat_term_order" shows "gd_powerprod (ord_pp_punit cmp_term) (ord_pp_strict_punit cmp_term)"
index_distinct: fixes G1 :: "('a \<times> 'b) set"   and G2 :: "('a \<times> 'b) set"   and x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" assumes "G1 \<in> {{x} \<times> I |x. x \<in> bl}"   and "G2 \<in> {{x} \<times> I |x. x \<in> bl}"   and "G1 \<noteq> G2"   and "x \<in> G1"   and "y \<in> G2" shows "int (f bl index {x, y}) = 1"
hd_conv_nth: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "hd xs = xs ! 0"
vlrestriction_atE2: fixes r :: "V"   and x :: "V"   and A :: "V" assumes "vsv r"   and "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> A" and "r\<lparr>a\<rparr> = b"
absorb_iff1: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (min a b = a)"
ide_cod: fixes f :: "'t" assumes "S'.arr f" shows "S'.ide (S'.cod f)"
assms_5: shows "(\<Sum>x\<in>X. \<beta> x * exp x) = 0"
simps_6: fixes a :: "MIR.num" shows "bound0 (Ge a) = numbound0 a"
5: shows "[a] @ w =\<^sub>F u @ v\<^sub>1 @ [a] @ v\<^sub>2"
lappend_inf: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lfinite xs" shows "lappend xs ys = xs"
order_refl: fixes x :: "'a multiset" shows "x \<subseteq># x"
strict_trans1: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry"   and c :: "'d DBMEntry" assumes "b \<preceq> a"   and "c \<prec> b" shows "c \<prec> a"
subsumes_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" shows "substitution_ops.subsumes subst_atm C D = (\<exists>\<sigma>. substitution_ops.subst_cls subst_atm C \<sigma> \<subseteq># D)"
left_ac_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * c * b"
equiv_out_trans: fixes a :: "'prog_out"   and b :: "'prog_out"   and c :: "'prog_out" assumes "equiv_out a b"   and "equiv_out b c" shows "equiv_out a c"
surj_lift_map: fixes i :: "nat" shows "surj (lift_map i)"
face_cycle_set_parts: fixes a :: "'b"   and b :: "'b" shows "face_cycle_set a = face_cycle_set b \<or> face_cycle_set a \<inter> face_cycle_set b = {}"
ID_ccompare_neq_None: shows "ID ccompare \<noteq> None"
the_Some_image: fixes A :: "'c set" shows "the ` Some ` A = A"
ty_term_mono: fixes varT :: "variable \<Rightarrow> 'a option"   and varT' :: "variable \<Rightarrow> 'a option"   and objT :: "object \<Rightarrow> 'a option"   and objT' :: "object \<Rightarrow> 'a option" assumes "varT \<subseteq>\<^sub>m varT'"   and "objT \<subseteq>\<^sub>m objT'" shows "ty_term varT objT \<subseteq>\<^sub>m ty_term varT' objT'"
lcoset_refl: fixes A :: "'b set"   and a :: "'b" assumes "(0::'b) \<in> A" shows "a \<in> a +o A"
polyn_deg_add5: fixes p :: "'a"   and q :: "'a"   and a :: "ant" assumes "p \<in> carrier R"   and "q \<in> carrier R"   and "deg R S X p \<le> a"   and "deg R S X q \<le> a" shows "deg R S X (p \<plusminus> q) \<le> a"
sat_Ands: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and l :: "Formula.formula list" shows "Formula.sat \<sigma> V v i (formula.Ands l) = (\<forall>\<phi>\<in>set l. Formula.sat \<sigma> V v i \<phi>)"
cutoff_branch: fixes T :: "tree"   and red :: "bool list \<Rightarrow> bool" assumes "anybranch T red" shows "anybranch (cutoff red [] T) red"
measurable_uniform_real: shows "uniform_real \<in> stock_measure (PRODUCT REAL REAL) \<rightarrow>\<^sub>M subprob_algebra (stock_measure REAL)"
wt_ge_vars: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wt t \<ge>\<^sub>p wt s" shows "vars s \<subseteq> vars t"
distinct_3: fixes name :: "name"   and name' :: "name"   and trm' :: "trm" shows "Var name \<noteq> \<Lambda> name' . trm'"
fun_dual2: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c dual" shows "(\<partial> \<circ> f = g) = (f = \<partial>\<^sup>- \<circ> g)"
constant_deriv_inner: fixes I :: "('e, 'f, 'd) interp"   and id1 :: "'e"   and \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec" assumes "\<forall>x i. (Functions I i has_derivative FunctionFrechet I i x) (at x)" shows "FunctionFrechet I id1 (\<chi>i. sterm_sem I (local.empty i) (fst \<nu>)) (\<chi>i. frechet I (local.empty i) (fst \<nu>) (snd \<nu>)) = 0"
inres_simps_4: shows "inres SUCCEED = (\<lambda>_. False)"
Poly_times_one_minus_x_eq_psums: fixes xs :: "'a list"   and ys :: "'a list" assumes "length xs = length ys"   and "Poly xs = Poly ys * [:1::'a, - (1::'a):]" shows "ys = psums xs"
hm_lookup_impl: shows "imp_map_lookup is_hashmap hm_lookup"
bij_is_inj: fixes f :: "'c \<Rightarrow> 'd" assumes "bij f" shows "inj f"
ctxt_inj: fixes E :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com"   and fctxt :: "('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list"   and x :: "('g, 'h, 'i, 'j) com"   and y :: "('g, 'h, 'i, 'j) com" assumes "(E, fctxt) \<in> ctxt"   and "E x = E y" shows "x = y"
diff_minus_eq_add: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "minus a (uminus b) = plus a b"
card_UN_le: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "finite I" shows "card (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. card (A i))"
rel_prime_pols_def: fixes R :: "('e, 'f) Ring_scheme"   and S :: "('e, 'g) Ring_scheme"   and X :: "'e"   and p :: "'e"   and q :: "'e" shows "rel_prime_pols R S X p q = (1\<^sub>r\<^bsub>R\<^esub> \<in> R \<diamondsuit>\<^sub>p p \<minusplus>\<^bsub>R\<^esub> R \<diamondsuit>\<^sub>p q)"
Keys_def: fixes F :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "Keys F = \<Union> (keys ` F)"
helper_max_prime: fixes a :: "'a" assumes "cring R"   and "a \<in> carrier R" shows "ideal {x \<in> carrier R. a \<otimes> x \<in> I} R"
assms_2: fixes xs :: "'a llist" assumes "\<not> lnull xs" shows "P (lhd xs) xs"
zorder_divide: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and g :: "complex \<Rightarrow> complex" assumes "isolated_singularity_at f z"   and "isolated_singularity_at g z"   and "not_essential f z"   and "not_essential g z"   and "\<exists>\<^sub>F w in at z. f w * g w \<noteq> 0" shows "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
le: fixes x :: "'a" assumes "norm x \<le> \<delta>" shows "dist (f' x) id_blinfun \<le> 1 / 2"
cmf: shows "continuous_map (subtopology (Euclidean_space m) S) (subtopology (Euclidean_space n) T) f"
fusion_gfp_eq: fixes H :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and G :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice" assumes "mono (H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "mono (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "dist_over_inf (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"   and "\<And>x::'b::refinement_lattice. ((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)) x = ((H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> F) x" shows "(F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (gfp (G::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)) = gfp (H::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)"
appendSubobj: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and Cs' :: "char list list" assumes "Subobjs P C (Cs @ Cs')"   and "Cs \<noteq> []" shows "Subobjs P C Cs"
NOUA1: shows "OUadmitFO \<sigma> ODE1 U"
part_brn_disj1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and n1 :: "nat"   and n2 :: "nat"   and II1 :: "nat set"   and II2 :: "nat set" assumes "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n)"   and "n1 < length cl"   and "n2 < length cl"   and "II1 \<in> local.shift cl n1 ` P n1"   and "II2 \<in> local.shift cl n2 ` P n2"   and "n1 \<noteq> n2" shows "II1 \<inter> II2 = {}"
simps_1: fixes nil :: "'a"   and lnk :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and lbl :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "pointer nil" shows "pointer.stack nil lnk lbl x [] = (x = nil)"
acquired_share_owns_shared: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and pending_write :: "bool" assumes "sharing_consistent \<S> \<O> sb" shows "acquired pending_write sb \<O> \<union> dom (share sb \<S>) \<subseteq> \<O> \<union> dom \<S>"
konigDtree_ipath: fixes inv :: "'sequent \<times> 'a \<Rightarrow> bool"   and pred :: "'sequent \<times> 'a \<Rightarrow> 'rule \<Rightarrow> 'sequent \<times> 'a \<Rightarrow> bool"   and t :: "('sequent \<times> 'rule) tree"   and a :: "'a" assumes "\<And>r s sl a. \<lbrakk>r \<in> R; eff r s sl; inv (s, a)\<rbrakk> \<Longrightarrow> \<exists>s' a'. s' |\<in>| sl \<and> inv (s', a') \<and> pred (s, a) r (s', a')"   and "local.wf t"   and "inv (fst (tree.root t), a)" shows "ipath t (smap fst (konigDtree inv pred t a))"
cosh_def: fixes x :: "'a" shows "cosh x = (exp x + exp (- x)) /\<^sub>R 2"
simps_1: fixes M :: "'b dfa"   and q :: "hf" assumes "dfa M" shows "dfa.nextl M q [] = q"
paths_cross_at: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set"   and R :: "'b set"   and x :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "Q \<in> \<P>"   and "R \<in> \<P>"   and "Q \<noteq> R"   and "Q \<inter> R \<noteq> {}"   and "x \<in> Q"   and "x \<in> R" shows "Q \<inter> R = {x}"
param_opt_ifex_ite_eq: fixes i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex"   and r :: "'a ifex" assumes "ro_ifex i"   and "ro_ifex t"   and "ro_ifex e"   and "param_opt i t e = Some r" shows "r = ifex_ite i t e"
assms: shows "n -as\<rightarrow>\<^sub>\<iota>* n'"
triangulize_def: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and A :: "'a mat" shows "triangulize sel_fun mf A = sub3 sel_fun mf (dim_row A) (dim_row A) (1::'a, A)"