hcju/isabelleps huggingface dataset:
eq_smooth: fixes d :: "real"   and s :: "real set"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a" assumes "0 < d"   and "\<forall>x\<in>s. \<forall>y. dist x y < d \<longrightarrow> f y = g y"   and "f C1_differentiable_on s" shows "g C1_differentiable_on s"
heap_sort_second_half_sorted_step: fixes l\<^sub>0 :: "int"   and m :: "int"   and r\<^sub>0 :: "int"   and arr :: "int \<Rightarrow> int"   and arr' :: "int \<Rightarrow> int"   and i :: "int"   and j :: "int"   and mod_arr :: "int \<Rightarrow> int" assumes "l\<^sub>0 < m"   and "m \<le> r\<^sub>0"   and "arr = arr' on - {l\<^sub>0..<m}"   and "\<forall>i j. bounded m r\<^sub>0 i \<longrightarrow> bounded m r\<^sub>0 j \<longrightarrow> i < j \<longrightarrow> arr j \<le> arr i"   and "\<forall>x\<in>#arr_mset arr l\<^sub>0 m. \<forall>y\<in>#arr_mset arr m r\<^sub>0. \<not> x < y"   and "bounded (m - 1) r\<^sub>0 i"   and "bounded (m - 1) r\<^sub>0 j"   and "i < j"   and "mod_arr = arr'(m - 1 := Min_mset (arr_mset arr l\<^sub>0 m))" shows "mod_arr j \<le> mod_arr i"
insert_UNIV: fixes x :: "'b" shows "insert x UNIV = UNIV"
simps: fixes u :: "'b"   and d :: "'b"   and v :: "'b"   and l :: "bool list" shows "lgeom_proc u d v l = lrev_geom u d v (rev l)"
accept\<^sub>G\<^sub>R_I: fixes \<phi> :: "'a ltl" assumes "accept\<^sub>G\<^sub>R (\<A> \<Sigma> \<phi>) w" obtains \<pi> :: "'a ltl \<Rightarrow> nat option" where "dom \<pi> \<subseteq> \<^bold>G \<phi>" and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> the (\<pi> \<chi>) < max_rank_of \<Sigma> \<chi>" and "accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (M_fin \<pi>, UNIV) w" and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (Acc\<^sub>\<A> \<Sigma> \<pi> \<chi>) w"
characteriseAx: fixes r :: "'a list \<times> ('b, 'c) sequent" assumes "r \<in> Ax" shows "r = ([],  \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) \<or> (\<exists>i. r = ([],  \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>))"
HashMap_impl_of: fixes t :: "('a, 'b) hashmap" shows "HashMap (impl_of t) = t"
iLastWeak_iff: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLastWeak t0 I P = (iLast t0 I P \<or> I \<down>< t0 = {} \<or> t0 \<notin> I)"
simps_1: fixes S :: "'av Abs_State.st option"   and P :: "'av Abs_State.st option" shows "step' S (SKIP {P}) = SKIP {S}"
converse_inward_2: fixes r :: "('b \<times> 'a) set"   and s :: "('b \<times> 'a) set" shows "(r \<union> s)\<inverse> = r\<inverse> \<union> s\<inverse>"
prSubst_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and SEM :: "('c, 'd, 'e, 'sort, 'f, 'b, 'g) semDom_scheme" shows "prSubst g SEM \<equiv> \<forall>ys Y y s X val. wls (asSort ys) Y \<and> wls s X \<and> sWlsVal SEM val \<longrightarrow> g (X #[Y / y]_ys) val = g X (val (y := g Y val)_ys)"
split_sels_1: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (nat = Suc (nat.pred nat) \<longrightarrow> P (f2 (nat.pred nat))))"
matrix_vector_mul_assoc: fixes A :: "(('d, 'f) vec, 'e) vec"   and B :: "(('d, 'g) vec, 'f) vec"   and x :: "('d, 'g) vec" shows "A *v (B *v x) = A ** B *v x"
proj2_Col_coeff_def: fixes a :: "proj2"   and r :: "proj2"   and t :: "proj2" shows "proj2_Col_coeff a r t \<equiv> SOME i. t = proj2_abs (i *\<^sub>R proj2_rep a + proj2_rep r)"
intro: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transitive A equiv" shows "partial_equivalence_axioms A equiv"
f3: shows "finite {c. [(a, c)] \<bullet> x \<noteq> x \<or> [(b, c)] \<bullet> x \<noteq> x}"
zFresh: shows "qAFresh xs z X \<and> qAFresh xs z X'"
pdigraph_base_axioms: shows "pdigraph_base \<alpha> K \<AA>"
tr_NegChecks_constr_iff_1: fixes L :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and M :: "('fun, 'var) Term.term set" shows "(\<forall>G\<in>set L. ineq_model \<I> X (F @ G)) = \<lbrakk>M; map (\<lambda>G. \<forall>X\<langle>\<or>\<noteq>: (F @ G)\<rangle>\<^sub>s\<^sub>t) L\<rbrakk>\<^sub>d \<I>"
f'_def: shows "f' \<equiv> \<lambda>z u. Ln (- Ln (complex_of_real u)) * f z u"
sup_bot_right: fixes x :: "'c \<times> 'd" shows "sup_prod x bot_prod = x"
statEqvt': fixes p :: "(name \<times> name) list"   and \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "p \<bullet> \<Psi> \<otimes> \<Psi>' = (p \<bullet> \<Psi>) \<otimes> (p \<bullet> \<Psi>')"
small_big_asymmetric: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)"   and "g \<in> O[F](f)" shows "\<forall>\<^sub>F x in F. f x = (0::'c)"
select_convs_4: fixes s_inst :: "t_context list"   and s_funcs :: "tf list"   and s_tab :: "nat list"   and s_mem :: "nat list"   and s_globs :: "tg list"   and more :: "'a" shows "s_mem \<lparr>s_inst = s_inst, s_funcs = s_funcs, s_tab = s_tab, s_mem = s_mem, s_globs = s_globs, \<dots> = more\<rparr> = s_mem"
subst_not_fin1: fixes M :: "trm"   and x :: "name"   and c :: "coname"   and P :: "trm" shows "\<not> fin (M{x:=<c>.P}) x"
create_initial_object_simps: fixes allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set"   and h :: "'a"   and ads :: "'b list"   and b :: "bool"   and C :: "String.literal" shows "heap_base.create_initial_object allocate (h, ads, b) C = (if b then let HA = allocate h (Class_type C) in if HA = {} then (h, ads, False) else let (h', a'') = SOME ha. ha \<in> HA in (h', ads @ [a''], True) else (h, ads, False))"
assms_3: fixes x :: "'a" assumes "x \<in> space M \<inter> A" shows "f x = g x"
matachAny_matchNone: fixes m :: "'c match_expr" assumes "\<not> has_primitive m" shows "matcheq_matchAny m = (\<not> matcheq_matchNone m)"
row_echelon_carrier: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "row_echelon A \<in> carrier_mat nr nc"
ball_divide_subset: fixes d :: "real"   and x :: "'b"   and e :: "real" assumes "1 \<le> d" shows "ball x (e / d) \<subseteq> ball x e"
lookup_proj_poly: fixes k :: "'k"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" shows "lookup (proj_poly k p) t = lookup p (term_of_pair (t, k))"
state_transition_in_refl_def: fixes s :: "infrastructure"   and s' :: "infrastructure" shows "s \<rightarrow>\<^sub>n* s' \<equiv> (s, s') \<in> {(x, y). x \<rightarrow>\<^sub>n y}\<^sup>*"
prems_1: fixes Y1 :: "'a multiset"   and Y2 :: "'a multiset" assumes "Y = Y1 + Y2"   and "(X1, Y1) \<in> multpw ns"   and "(add_mset x2 X2, Y2) \<in> multpw ns" shows "thesis"
pcr_msg_def: shows "pcr_msg \<equiv> (=) OO cr_msg"
adjoin_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" shows "adjoin G H a = {x \<otimes>\<^bsub>G\<^esub> a [^]\<^bsub>G\<^esub> k |x k. x \<in> H \<and> k < subgroup_indicator G H a}"
akra_bazzi_params_nonzero_axioms: shows "akra_bazzi_params_nonzero k as bs"
abs_eq: fixes x :: "('a, 'b) pfun" assumes "eq_onp (\<lambda>f. finite (pdom f)) x x" shows "fdom (ffun.Abs_pfun x) = pdom x"
lnths_LCons: fixes x :: "'c"   and xs :: "'c llist"   and A :: "nat set" shows "lnths (LCons x xs) A = (if 0 \<in> A then LCons x (lnths xs {n. Suc n \<in> A}) else lnths xs {n. Suc n \<in> A})"
ifex_ordered_not_part: fixes b :: "'a ifex"   and v :: "'a"   and b1 :: "'a ifex"   and b2 :: "'a ifex"   and w :: "'a" assumes "ifex_ordered b"   and "b = IF v b1 b2"   and "w < v" shows "w \<notin> ifex_var_set b"
Cd_inj_on_Xd: fixes d :: "'d"   and X :: "'x set" shows "inj_on Xd (Cd d X)"
auto_local_lipschitz: shows "local_lipschitz UNIV X (\<lambda>_. f)"
monoid_axioms: shows "Group.monoid (ring_mat TYPE('a) n b)"
polymul_normh: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "isnpolyh (p *\<^sub>p q) (min n0 n1)"
ppos: fixes arity :: "'a \<Rightarrow> nat"   and exec :: "'a \<Rightarrow> 'b list \<Rightarrow> 'b"   and inv :: "'b \<Rightarrow> bool"   and cost :: "'a \<Rightarrow> 'b list \<Rightarrow> nat"   and \<Phi> :: "'b \<Rightarrow> real"   and U :: "'a \<Rightarrow> 'b list \<Rightarrow> real"   and s :: "'b" assumes "Amortized arity exec inv cost \<Phi> U"   and "inv s" shows "0 \<le> \<Phi> s"
VectorSpaceHomSetI: fixes fsmult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and fsmult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and W :: "'c set" assumes "VectorSpaceHom fsmult V fsmult' T"   and "T ` V \<subseteq> W" shows "T \<in> VectorSpaceHomSet fsmult V fsmult' W"
Pi_I': fixes A :: "'ag set"   and f :: "'ag \<Rightarrow> 'ah"   and B :: "'ag \<Rightarrow> 'ah set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
bind_def2: fixes m :: "'e\<cdot>'d"   and k :: "'e \<rightarrow> 'c\<cdot>'d" shows "m \<bind> k = join\<cdot>(fmap\<cdot>k\<cdot>m)"
falling_fact_pochhammer: fixes a :: "int"   and k :: "nat" shows "(\<Prod>i = 0..<k. a - int i) = (- 1) ^ k * pochhammer (- a) k"
euler_incr_slp_impl_def: fixes ode_ops :: "'c \<times> ('d \<times> 'a \<times> 'e \<times> 'f \<times> 'g) \<times> 'h \<times> 'i" shows "euler_incr_slp_impl ode_ops = (case ode_ops of (xa, (xb, x, xc, xd, xe), xf, xg) \<Rightarrow> x)"
rank_def: fixes SkewBinomialTree :: "('c, 'd) SkewBinomialTree" shows "rank SkewBinomialTree = (case SkewBinomialTree of Node x1 x2 x3 x4 \<Rightarrow> x3)"
has_absolute_integral_change_of_variables_invertible: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and h :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and b :: "(real, 'b) vec" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "\<And>x. x \<in> S \<Longrightarrow> h (g x) = x"   and "continuous_on (g ` S) h" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \<and> integral (g ` S) f = b)"
subgroup_imp_group: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H\<rparr>)" shows "Group.group (G\<lparr>carrier := H, carrier := H\<rparr>)"
P7: shows "Bet C X A \<and> Bet B D X \<longrightarrow> A B OS C D"
strict_trans: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term"   and c :: "('a, 'b) Term.term" assumes "a \<sqsubset> b"   and "b \<sqsubset> c" shows "a \<sqsubset> c"
no_two_\<omega>_limit_points: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and u :: "'a"   and v :: "'a" assumes "transversal_segment a b"   and "{0..} \<subseteq> existence_ivl0 x"   and "\<omega>_limit_point x u"   and "u \<in> {a<--<b}"   and "\<omega>_limit_point x v"   and "v \<in> {a<--<b}"   and "v \<in> {u<--<b}" shows "False"
in_trancl_imp: fixes v :: "'a \<times> nat"   and v' :: "'a \<times> nat" assumes "(v, v') \<in> subs\<^sup>+" shows "(v, v') \<in> subs"
acute_not_per: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Acute A B C" shows "\<not> Per A B C"
meet_le1: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and meet :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.L_top_bot le join Top meet bot" shows "le (meet x y) x"
one_F: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "(0::'a) + sum_0 g A = sum_0 g A"
setdist_sing_in_set: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "setdist {x} S = 0"
basis_for_Span_ex: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and vs :: "'b list" assumes "VectorSpace smult V"   and "set vs \<subseteq> V" shows "\<exists>us. set us \<subseteq> R_scalar_mult.RSpan UNIV smult vs \<and> R_scalar_mult.RSpan UNIV smult vs = R_scalar_mult.RSpan UNIV smult us \<and> R_scalar_mult.R_lin_independent UNIV smult us"
H2_DR: shows "H2_DR \<in> carrier_mat (m - 1) (n - 1)"
one_plus_BitM: fixes n :: "num" shows "num.One + Num.BitM n = num.Bit0 n"
indicator_leI: fixes x :: "'a"   and A :: "'a set"   and y :: "'b"   and B :: "'b set" assumes "x \<in> A \<Longrightarrow> y \<in> B" shows "indicator A x \<le> indicator B y"
elims: fixes x :: "'a mat"   and xa :: "rat vec"   and y :: "'a mat" assumes "reduce_mat_cols x xa = y" obtains A :: "'a mat" and lhs_vec :: "rat vec" where "x = A" and "xa = lhs_vec" and "y = take_cols_from_matrix A (find_nonzeros_from_input_vec lhs_vec)"
prv_softSubst_imp_subst: fixes x :: "'var"   and t :: "'trm"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "t \<in> trm"   and "\<phi> \<in> fmla"   and "x \<notin> FvarsT t" shows "prv (imp (softSubst \<phi> t x) (subst \<phi> t x))"
rsp: shows "eq_onp (\<lambda>A. finite A \<and> incomparable A) {} {}"
rcs_Unit2: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and h :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "h \<in> H" shows "H \<bullet>\<^bsub>G\<^esub> h = H"
augment_edge_other: fixes e :: "nat \<times> nat"   and e' :: "nat \<times> nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and \<Delta> :: "'a" assumes "e \<in> E"   and "e' \<noteq> e" shows "augment_edge f e \<Delta> e' = f e'"
atLeastatMost_empty_iff: fixes a :: "'a"   and b :: "'a" shows "({a..b} = {}) = (\<not> a \<le> b)"
arith_special_3: shows "(1::'b) + (1::'b) = (2::'b)"
simps_5: fixes x2 :: "nat"   and x1 :: "nat" shows "Regions.intv.Intv x2 \<noteq> Regions.intv.Const x1"
Bit_in_uints_Suc: fixes w :: "int"   and m :: "nat"   and c :: "bool" assumes "w \<in> uints m" shows "of_bool c + 2 * w \<in> uints (Suc m)"
intro: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and N :: "('b, 'f, 'g) module_scheme"   and f :: "'d \<Rightarrow> 'f" assumes "module R M"   and "module R N"   and "mod_hom_axioms R M N f" shows "mod_hom R M N f"
COMPL: fixes \<sigma> :: "'b" assumes "I S\<^sub>0 {} \<sigma>"   and "c \<sigma>" shows "\<Phi> \<sigma>"
take_bit_int_greater_eq: fixes k :: "int"   and n :: "nat" assumes "k < 0" shows "k + 2 ^ n \<le> take_bit n k"
fimage_eqI: fixes b :: "'x"   and f :: "'y \<Rightarrow> 'x"   and x :: "'y"   and A :: "'y fset" assumes "b = f x"   and "x |\<in>| A" shows "b |\<in>| f |`| A"
contains_order2_nreduced: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and A :: "'b set"   and as :: "'b list"   and bs :: "'b list" assumes "class.monoid_add plus zero"   and "plus a a = zero" shows "\<not> monoid_add.reduced_word_for plus zero A (monoid_add.sum_list plus zero (as @ [a, a] @ bs)) (as @ [a, a] @ bs)"
SK_addS_distrib: fixes x :: "nat discr llist"   and y :: "nat discr llist" shows "SK\<cdot>(addS\<cdot>x\<cdot>y) = addK\<cdot>(SK\<cdot>x)\<cdot>(SK\<cdot>y)"
simps_2: fixes \<sigma> :: "'tp" shows "TE_wtFsym (Tag \<sigma>) = True"
unique_eval: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and w :: "vals"   and s' :: "state"   and w' :: "vals"   and s'' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w, s')"   and "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w', s'')" shows "w' = w \<and> s'' = s'"
prems_3: shows "p \<in> set (comp_red_basis_aux (a # xs) ys)"
short_exact_sequence_hom_induced_relativization: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and p :: "int" assumes "homotopic_with (\<lambda>x. True) X X id f"   and "f ` topspace X \<subseteq> S" shows "short_exact_sequence (homology_group (p - 1) X) (homology_group (p - 1) (subtopology X S)) (relative_homology_group p X S) (hom_induced (p - 1) (subtopology X S) {} X {} id) (hom_boundary p X S)"
subs: shows "{x \<in> carrier R. x [^]\<^bsub>R\<^esub> n = int (c mod p)} \<subseteq> {a \<in> carrier R. eval R R id a (monom (UP R) \<one>\<^bsub>R\<^esub> n \<ominus>\<^bsub>UP R\<^esub> monom (UP R) (int (c mod p)) 0) = \<zero>\<^bsub>R\<^esub>}"
simps_2: 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 D :: "char list" shows "is_subobj P (C, [D]) = (is_class P C \<and> C = D \<or> (\<exists>X. P \<turnstile> C \<preceq>\<^sup>* X \<and> P \<turnstile> X \<prec>\<^sub>S D))"
eq: fixes s :: "complex" assumes "0 < Re s"   and "s \<noteq> 1" shows "zeta s = zeta' s"
grey_def: fixes r :: "'h" shows "grey r = (\<lambda>s. \<exists>x. r \<in> WL x s)"
assms_1: shows "x \<in> set_spmf sample_primes"
foldli_transform: shows "\<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
to_fun: fixes x :: "('d, 'c) sq_matrix" shows "to_fun x \<in> UNIV"
eq1: shows "post * (post - proj_psi_l R) = post - proj_psi_l R"
h_def: shows "h \<equiv> \<lambda>x. ln (b * x) powr (- e / 2) * g x"
BS2: fixes f :: "'a"   and g :: "'a" assumes "is_left_adjoint f"   and "is_left_adjoint g"   and "src f = src g" shows "\<exists>r \<rho>. tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho> f g"
h3a: assumes "0 < Av * Bv" shows "Av + Bv * sqrt Cv \<noteq> 0"
arrow_of_spans_of_maps_to_tabulation_data_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and w :: "'b" shows "arrow_of_spans_of_maps_to_tabulation_data V H \<a> \<i> src trg r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 w \<equiv> bicategory_of_spans V H \<a> \<i> src trg \<and> arrow_of_spans_of_maps V H \<a> \<i> src trg r\<^sub>0 r\<^sub>1 s\<^sub>0 s\<^sub>1 w \<and> tabulation_data V H \<a> \<i> src trg s \<sigma> s\<^sub>0 s\<^sub>1"
match_complete: fixes t :: "('e, 'd) Term.term"   and p :: "('e, 'd) Term.term" assumes "match t p = None" shows "matchers {(p, t)} = {}"
fps_compose_mult_distrib: fixes c :: "'a fps"   and a :: "'a fps"   and b :: "'a fps" assumes "c $ 0 = (0::'a)" shows "a * b oo c = (a oo c) * (b oo c)"
sel: fixes x :: "('b, 'c, ('b, 'c) sp\<^sub>\<nu>) sp\<^sub>\<mu>" shows "out (In x) = x"
wave_restrict_current: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current \<Gamma> g"   and "wave \<Gamma> g" shows "wave (quotient_web \<Gamma> f) (g \<upharpoonleft> \<Gamma> / f)"
dom_ran_restrict: fixes f :: "'c \<Rightarrow> 'd option"   and B :: "'d set" shows "dom (f\<upharpoonleft>\<^bsub>B\<^esub>) \<subseteq> dom f"
iso_is_arr: fixes f :: "'c" assumes "Cop.iso f" shows "Cop.arr f"
newvar_def: fixes vs :: "nat list" shows "newvar vs = (if vs = [] then 0 else Suc (maxvar vs))"
indicator_times_eq_if_2: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and S :: "'c set" shows "f x * indicator S x = (if x \<in> S then f x else (0::'b))"
eventually_at_top_to_right: fixes P :: "real \<Rightarrow> bool" shows "eventually P at_top = (\<forall>\<^sub>F x in at_right 0. P (inverse x))"
Fst: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> analz H" shows "X \<in> analz H"
x: fixes p :: "nat" assumes "p \<in> P" shows "[x = g p] (mod p ^ k p)"
order_iff: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (a = max a b)"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 (\<nu> \<star> \<pi>) \<rho>"
red2_simps_4: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and u :: "'a local.term"   and v :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Ide V src trg u"   and "bicategorical_language.Nml V src trg u"   and "bicategorical_language.Ide V src trg v"   and "bicategorical_language.Nml V src trg v"   and "bicategorical_language.Src src u = bicategorical_language.Trg trg v" shows "bicategorical_language.Cod V src trg (bicategorical_language.red2 V u v) = bicategorical_language.Nmlize V (u \<^bold>\<star> v)"
ch: shows "Complete_Partial_Order.chain (\<le>) (M ` A)"
adopt_node_get_shadow_root: fixes adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" assumes "l_adopt_node_get_shadow_root adopt_node_locs get_shadow_root_locs" shows "\<forall>w\<in>adopt_node_locs parent owner_document document_ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
Suc: fixes f :: "nat \<Rightarrow> 'a" shows "convergent (\<lambda>n. f (n + m)) = convergent f"
rel_mono: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R1a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R3 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R3a :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a"   and "R3 \<le> R3a" shows "rel_pre_coGcoF\<^sub>M R1 R2 R3 \<le> rel_pre_coGcoF\<^sub>M R1a R2a R3a"
gfp_trans_lemma2: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>t u. \<lbrakk>le_utrans t u; \<And>P. unitary P \<Longrightarrow> unitary (t P); \<And>P. unitary P \<Longrightarrow> unitary (u P)\<rbrakk> \<Longrightarrow> le_utrans (T t) (T u)"   and "\<And>t P. \<lbrakk>\<And>Q. unitary Q \<Longrightarrow> unitary (t Q); unitary P\<rbrakk> \<Longrightarrow> unitary (T t P)" shows "le_utrans (gfp_trans T) (T (gfp_trans T))"
arctan_0_1_bounds_round: fixes x :: "float"   and p1 :: "nat"   and n :: "nat"   and p2 :: "nat"   and p3 :: "nat"   and p4 :: "nat" assumes "0 \<le> real_of_float x"   and "real_of_float x \<le> 1" shows "arctan (real_of_float x) \<in> {real_of_float x * real_of_float (lb_arctan_horner p1 (get_even n) 1 (float_round_up (Suc p2) (x * x)))..real_of_float x * real_of_float (ub_arctan_horner p3 (get_odd n) 1 (float_round_down (Suc p4) (x * x)))}"
set_integral_add_2: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "set_integrable M A f"   and "set_integrable M A g" shows "\<integral>x\<in>A. (f x + g x)\<partial>M = set_lebesgue_integral M A f + set_lebesgue_integral M A g"
filter_leD: fixes F :: "'b filter"   and F' :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "F \<le> F'"   and "eventually P F'" shows "eventually P F"
non_empty_cycle_root_tc_start_points: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "x\<^sup>+ ; r = x ; 1"
path_singleton: fixes s :: "'sum"   and l1 :: "'loc"   and l2 :: "'loc" assumes "s \<in>\<^sub>A summary l1 l2" shows "path l1 l2 [(l1, s, l2)]"
substrT_setD: fixes a :: "char list"   and T :: "POPLmarkRecord.type"   and fs :: "(char list \<times> POPLmarkRecord.type) list"   and k :: "nat"   and U :: "POPLmarkRecord.type" assumes "(a, T) \<in> set (fs[k \<mapsto>\<^sub>\<tau> U]\<^sub>r\<^sub>\<tau>)" shows "\<exists>T'. (a, T') \<in> set fs \<and> T = T'[k \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau>"