hcju/isabelleps huggingface dataset:
translation_subtract_Compl: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and t :: "'a set" assumes "class.ab_group_add plus zero minus uminus" shows "(\<lambda>x. minus x a) ` (- t) = - (\<lambda>x. minus x a) ` t"
wp_if_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp \<pi> (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q s = (if bval b s then wp \<pi> c\<^sub>1 Q s else wp \<pi> c\<^sub>2 Q s)"
lookup_single: fixes k :: "'d"   and v :: "'c"   and k' :: "'d" shows "lookup (monomial v k) k' = (v when k = k')"
linears_appI: fixes xs :: "term list"   and ys :: "term list" assumes "linears xs"   and "linears ys"   and "fdisjnt (freess xs) (freess ys)" shows "linears (xs @ ys)"
Unifier_strand_fv_subset: fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list"   and f :: "'c"   and X :: "('c, 'd) Term.term list"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "t \<in> ik\<^sub>s\<^sub>t S"   and "Unifier \<delta> (Fun f X) t"   and "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv (Fun f X \<cdot> \<delta>) \<subseteq> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v (S \<cdot>\<^sub>s\<^sub>t \<delta>)))"
simps: fixes a :: "'form"   and t :: "('form, 'rule, 'subst, 'b) itnode rose_tree"   and "is" :: "nat list" shows "(a \<in> hyps_along t is) = (\<exists>is' i h. a = subst (iSubst (tree_at t is')) (freshen (iAnnot (tree_at t is')) (labelsOut (iNodeOf (tree_at t is')) h)) \<and> prefix (is' @ [i]) is \<and> i < length (inPorts' (iNodeOf (tree_at t is'))) \<and> hyps (iNodeOf (tree_at t is')) h = Some (inPorts' (iNodeOf (tree_at t is')) ! i))"
srelated_subst': fixes penv :: "(name, pterm) fmap"   and senv :: "(name, sterm) fmap"   and t :: "pterm" assumes "srelated'.P_env penv senv"   and "pre_strong_term_class.wellformed t" shows "pterm_to_sterm (subst t penv) = subst (pterm_to_sterm t) senv"
uminus_pred_def: fixes P :: "'a pred" shows "- P = Pred (- eval P)"
trans_IfTrue: fixes tst :: "'test"   and s :: "'state"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "tval tst s" shows "(if tst then c1 else c2, s) \<rightarrow>c (c1, s)"
devExt_nopt_leq: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat" assumes "\<not> Option.is_none (devBC t n nid n')" shows "n' \<le> n"
asubg_pOp_closed: fixes H :: "'a set"   and x :: "'a"   and y :: "'a" assumes "M +> H"   and "x \<in> H"   and "y \<in> H" shows "x \<plusminus> y \<in> H"
af_simps_1: fixes w :: "'a set list" shows "af true\<^sub>n w = true\<^sub>n"
fls_divide_shift_both_nonzero: fixes f :: "'b fls"   and g :: "'b fls"   and n :: "int"   and m :: "int" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "fls_shift n f / fls_shift m g = fls_shift (n - m) (f / g)"
proj_list_5_same: fixes xs :: "'b list"   and ys :: "'c list"   and w :: "'c"   and ws :: "'c list"   and z :: "'c"   and zs :: "'c list"   and ys' :: "'b list"   and w' :: "'b"   and ws' :: "'b list"   and z' :: "'b"   and zs' :: "'b list" assumes "proj_list_5 xs (ys, w, ws, z, zs) = (ys', w', ws', z', zs')"   and "length xs = length ys + 1 + length ws + 1 + length zs" shows "xs = ys' @ w' # ws' @ z' # zs'"
compute_float_sgn: fixes m1 :: "int"   and e1 :: "int" shows "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then - 1 else 0)"
mix_pmf_preferred_independence: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and r :: "'a pmf"   and \<alpha> :: "real"   and p :: "'a pmf"   and q :: "'a pmf" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "r \<in> \<P> outcomes"   and "\<alpha> \<in> {0..1}"   and "p \<succeq>[\<R>] q" shows "mix_pmf \<alpha> p r \<succeq>[\<R>] mix_pmf \<alpha> q r"
Delta_def: shows "Delta = snd \<circ> snd \<circ> snd \<circ> Rep_seqauto"
zip_altdef: fixes xs :: "'a list"   and ys :: "'b list" shows "zip xs ys = map (\<lambda>i. (xs ! i, ys ! i)) [0..<min (length xs) (length ys)]"
that: fixes V :: "'b set" assumes "openin (top_of_set U) V"   and "S' retract_of V" shows "thesis"
inf_commute: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = y \<sqinter> x"
Im_Ln_eq_pi_half_2: fixes z :: "complex" assumes "z \<noteq> 0" shows "(Im (Ln z) = - pi / 2) = (Im z < 0 \<and> Re z = 0)"
assms_1: shows "f \<in> M \<rightarrow>\<^sub>M N"
Av_carrier: shows "set Av \<subseteq> carrier_vec d"
weight_mk_lossless: fixes p :: "'a spmf" shows "weight_spmf (mk_lossless p) = (if p = return_pmf None then 0 else 1)"
nuc_bres_closed: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "nucleus f" shows "f (f x \<rightarrow> f y) = f x \<rightarrow> f y"
validn_fresh: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list" assumes "x \<sharp> \<Gamma>" shows "\<nexists>B. (x, B) \<in> set \<Gamma>"
permutations_of_set_image_inj: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "permutations_of_set (f ` A) = map f ` permutations_of_set A"
norm_cauchy_schwarz_eq: fixes x :: "'a"   and y :: "'a" shows "(x \<bullet> y = norm x * norm y) = (norm x *\<^sub>R y = norm y *\<^sub>R x)"
fac_Suc: fixes n :: "nat" shows "Suc n! = Suc n * n!"
x_leq_c_transpose_general: fixes h :: "'a"   and c :: "'a"   and x :: "'a"   and e :: "'a" assumes "forest h"   and "vector c"   and "x[[top]] \<le> forest_components h * e * top"   and "e \<le> c * - c\<^sup>T"   and "c = forest_components h * c" shows "x \<le> c\<^sup>T"
ccw_translate_origin: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "NO_MATCH (0::'a) p" shows "ccw p q r = ccw 0 (q - p) (r - p)"
rbl_mult_app2: fixes bla :: "bool list"   and blb :: "bool list"   and blc :: "bool list" assumes "length bla \<le> length blb" shows "rbl_mult bla (blb @ blc) = rbl_mult bla blb"
res: shows "(if p = 0 then (0::'a, []) else (lead_coeff p, yun_monic_factorization (smult (inverse (lead_coeff p)) p))) = c_bs"
weakCongISym2: fixes P :: "pi"   and Q :: "pi"   and F :: "pi \<Rightarrow> pi" assumes "P \<simeq> Q"   and "\<And>R S. R \<simeq> S \<Longrightarrow> F R \<leadsto>\<guillemotleft>weakBisim\<guillemotright> F S" shows "F P \<simeq> F Q"
intra_kind_def: fixes et :: "('a, 'b, 'c, 'd) edge_kind" shows "intra_kind et \<equiv> (\<exists>f. et = \<Up>f) \<or> (\<exists>Q. et = (Q)\<^sub>\<surd>)"
no_parallel_edge: shows "\<forall>u v. (u, v) \<in> E \<longrightarrow> (v, u) \<notin> E"
cbox_sub_1: shows "cbox c d \<subseteq> box a b"
build_cons: fixes a :: "'b"   and u :: "'b list"   and v :: "nat \<Rightarrow> 'b" shows "(a # u) \<frown> v = a ## u \<frown> v"
simps_3: fixes V :: "'c set"   and t :: "('d, 'c) Term.term"   and S :: "('d, 'c) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (send\<langle>t\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> fv t) S"
Rangep_Range_eq: fixes r :: "('c \<times> 'b) set" shows "Rangep (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> Range r)"
infiter_step_mono: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "mono ((seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice))"
fls_regpart_of_int: fixes i :: "int" shows "fls_regpart (of_int i) = of_int i"
order_refl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.preorder less_eq less" shows "less_eq x x"
qim: fixes U :: "'b set"   and q :: "'b \<Rightarrow> 'a" assumes "U \<subseteq> S"   and "continuous_on U q"   and "\<psi> U" shows "\<phi> (q ` U)"
inf_finfun_def: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "inf f g = (\<lambda>(x, y). inf x y) \<circ>$ ($f, g$)"
MtransT_Ex_finTrace: fixes cf :: "('test, 'atom) com \<times> 'state"   and s :: "'state" assumes "cf \<rightarrow>*t s" shows "\<exists>tr. finTrace tr \<and> iconfig tr = cf \<and> fstate tr = s"
real_of_nat_less_inverse_iff: fixes u :: "real"   and n :: "nat" assumes "0 < u" shows "(u < inverse (real (Suc n))) = (real (Suc n) < inverse u)"
leadsTo_slice_project_set: fixes F :: "'a program"   and B :: "'c set"   and CU :: "'a set" assumes "\<forall>y. F \<in> slice B y \<longmapsto> CU" shows "F \<in> project_set h B \<longmapsto> CU"
intro: fixes C\<^sub>2 :: "nat list \<Rightarrow> nat \<Rightarrow> bool"   and B\<^sub>2 :: "nat"   and C\<^sub>1 :: "nat list \<Rightarrow> nat list \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> bool"   and B\<^sub>1 :: "nat list \<Rightarrow> nat" assumes "bounded_gen_check C\<^sub>2 B\<^sub>2"   and "bounded_generate_check_axioms C\<^sub>2 C\<^sub>1 B\<^sub>1" shows "bounded_generate_check C\<^sub>2 B\<^sub>2 C\<^sub>1 B\<^sub>1"
key: shows "key < Array.get h' a ! (Ref.get h' i - 1)"
power_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "num"   and l :: "num" assumes "class.semiring_numeral one times plus" shows "power.power one times (numeral.numeral one plus k) (numeral l) = numeral.numeral one plus (Num.pow k l)"
abs_summable_on_cmult_left: fixes c :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "c \<noteq> (0::'c) \<Longrightarrow> f abs_summable_on A" shows "(\<lambda>x. f x * c) abs_summable_on A"
lcf_nonzero: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "lcf R S X p \<noteq> \<zero>\<^bsub>S\<^esub>"
join_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "dlat.join le x y = (THE sup. dpo.is_sup le x y sup)"
intros_2: fixes v1 :: "'b val"   and r1 :: "'b rexp"   and r2 :: "'b rexp" assumes "\<turnstile> v1 : r1" shows "\<turnstile> val.Left v1 : Plus r1 r2"
wellformed_items_\<J>_\<I>: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.wellformed_items \<RR> Doc (LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc k u) \<and> LocalLexing.wellformed_items \<RR> Doc (LocalLexing.\<I> \<TT> \<RR> \<SS> Lex Sel Doc k)"
cr_wn'': fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat"   and j :: "nat"   and n :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob i \<pi> m"   and "IFC_def.is_cdi prob j \<pi> n"   and "\<not> IFC_def.is_cdi prob m \<pi> n"   and "i \<le> j" shows "m \<le> n"
carrier_subgroup_generated_by_singleton: fixes x :: "'a" assumes "x \<in> carrier G" shows "carrier (subgroup_generated G {x}) = range (([^]) x)"
Sim: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "\<Psi>' \<rhd> S \<leadsto><Rel> T"
dirichlet_inverse_cong_simp: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'b"   and i :: "'b"   and i' :: "'b"   and n' :: "nat" assumes "\<And>m. \<lbrakk>0 < m; m \<le> n\<rbrakk> \<Longrightarrow> f m = f' m"   and "i = i'"   and "n = n'" shows "dirichlet_inverse f i n = dirichlet_inverse f' i' n'"
aposs_le_1: fixes x :: "ant" assumes "0 < x" shows "1 \<le> x"
cs: shows "(t, (\<beta>, u) # xs) \<in> seq lrs"
mult_strict_right_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times a c) (times b c)"
simps_2: fixes p :: "nat"   and xs :: "pratt list" shows "valid_cert (Prime p # xs) = (1 < p \<and> valid_cert xs \<and> (\<exists>a. [a ^ (p - 1) = 1] (mod p) \<and> Triple p a (p - 1) \<in> set xs))"
symmetric_point_construction: fixes A :: "'p"   and P :: "'p" shows "\<exists>P'. A Midpoint P P'"
relpowp_Suc_I: fixes n :: "nat"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "(P ^^ n) x y"   and "P y z" shows "(P ^^ Suc n) x z"
special_soundness_def: shows "\<Sigma>_AND.special_soundness \<equiv> \<forall>h e e' a z z'. h \<in> valid_pub_AND \<longrightarrow> e \<in> challenge_space \<longrightarrow> e' \<in> challenge_space \<longrightarrow> e \<noteq> e' \<longrightarrow> check_AND h a e z \<longrightarrow> check_AND h a e' z' \<longrightarrow> lossless_spmf (\<A>ss_AND h (a, e, z) (a, e', z')) \<and> (\<forall>w'\<in>set_spmf (\<A>ss_AND h (a, e, z) (a, e', z')). (h, w') \<in> Rel_AND)"
has_derivative_snd: fixes g :: "'c \<Rightarrow> 'd \<times> 'e"   and g' :: "'c \<Rightarrow> 'd \<times> 'e"   and F :: "'c filter" assumes "(g has_derivative g') F" shows "((\<lambda>x. snd (g x)) has_derivative (\<lambda>x. snd (g' x))) F"
coerce_strict: shows "COERCE('f, 'e)\<cdot>\<bottom> = \<bottom>"
up_inject: fixes x :: "'a"   and y :: "'a" assumes "up\<cdot>x = up\<cdot>y" shows "x = y"
has_lock_upd_lock_implies_has_lock: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and L :: "lock_action"   and t' :: "'b" assumes "has_lock (upd_lock l t L) t'"   and "t \<noteq> t'" shows "has_lock l t'"
cts_def2: shows "cts = cts' @ [ct1, ct2, ct3]"
axioms_2: fixes G :: "('e, 'f) monoid_scheme" assumes "comm_group G" shows "Group.group G"
selectlike_shrinks: fixes X :: "(variable \<Rightarrow> real) set"   and \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set" shows "selectlike X \<nu> V \<subseteq> X"
ipurge_fail_aux_t_out_def: fixes Y :: "('b, 'c) ipurge_rec" shows "ipurge_fail_aux_t_out Y \<equiv> (case ListOp Y of Some ys \<Rightarrow> ys, Set Y)"
discI: fixes aquery :: "'a aquery"   and x2 :: "'a" assumes "aquery = ForwardOrEdit x2" shows "is_ForwardOrEdit aquery"
minus_1_eq: fixes x :: "real" shows "x + - 1 = x - 1"
implicit_dependent_with: fixes U\<^sub>M :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" assumes "module_on U\<^sub>M scale" shows "dependent.with (0::'e) (0::'d) (+) scale = module_on.dependent scale"
Transset_Inf: fixes A :: "V set" assumes "\<And>i. i \<in> A \<Longrightarrow> Transset i" shows "Transset (\<Sqinter> A)"
fold_A_fold: shows "ChamberComplexIsomorphism (opp_fold_A \<turnstile> A) (fold_A \<turnstile> A) fold_A"
iMODb_mod_1: fixes r :: "nat"   and c :: "nat" shows "[ r, mod Suc 0, c ] = [r\<dots>,c]"
mult_cancel: fixes s :: "('b \<times> 'b) set"   and X :: "'b multiset"   and Z :: "'b multiset"   and Y :: "'b multiset" assumes "trans s"   and "irrefl s" shows "((X + Z, Y + Z) \<in> mult s) = ((X, Y) \<in> mult s)"
l_get_owner_document_def: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "l_get_owner_document get_owner_document \<equiv> (\<forall>h ptr. h \<turnstile> ok get_owner_document ptr \<longrightarrow> ptr |\<in>| object_ptr_kinds h) \<and> (\<forall>ptr h. pure (get_owner_document ptr) h)"
keys_flatten_subset: fixes p :: "'c \<Rightarrow>\<^sub>0 'c \<Rightarrow>\<^sub>0 'd" shows "keys (flatten p) \<subseteq> (\<Union>t\<in>keys p. (+) t ` keys (lookup p t))"
preserves_reflects_arr: fixes f :: "'c local.term set" shows "D.arr (E.map f) = \<F>C.arr f"
Im_poly_of_real: fixes p :: "complex poly"   and x :: "real" shows "Im (poly p (complex_of_real x)) = poly (map_poly Im p) x"
length_list_of_vec: fixes v :: "'a Matrix.vec" shows "length (list_of_vec v) = dim_vec v"
Integral_def: shows "Integral = (\<lambda>(a, b) f k. \<forall>e>0. \<exists>\<delta>. gauge {a..b} \<delta> \<and> (\<forall>D. fine \<delta> (a, b) D \<longrightarrow> \<bar>rsum D f - k\<bar> < e))"
IH: fixes k :: "nat"   and I :: "'a \<Rightarrow> bool option" assumes "k < length (trace_parallel_plan_strips I \<pi>)" shows "trace_parallel_plan_strips I \<pi> ! k = execute_parallel_plan I (take k \<pi>)"
pmaxsimpchainD_distinct: fixes xs :: "'a set list" assumes "pmaxsimpchain xs" shows "distinct xs"
good_x'x_pickX: shows "qGood (pick X #[[x' \<and> x]]_xs)"
trees_cyl_sub_trees: fixes t :: "'d set tree"   and A :: "'d set set"   and B :: "'d set" assumes "t \<in> trees A"   and "A \<subseteq> Pow B" shows "trees_cyl t \<subseteq> trees B"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "S.arr f"   and "S.cod f = b" shows "b \<cdot>\<^sub>S f = f"
sound_def: fixes R :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" shows "sound R = (\<forall>I. is_interp I \<longrightarrow> (\<forall>i\<ge>0. i < length (fst R) \<longrightarrow> seq_sem I (fst R ! i) = UNIV) \<longrightarrow> seq_sem I (snd R) = UNIV)"
is_path_to_in_verts: fixes G :: "graph"   and p :: "nat list" assumes "is_path G p" shows "hd p \<in> verts G \<and> last p \<in> verts G"
numeral_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (m + n) = plus (numeral.numeral one plus m) (numeral.numeral one plus n)"
preserves_seq: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "seq g f" shows "seq (R g) (R f)"
surj_Sup_pres_id: fixes f :: "'c \<Rightarrow> 'b" assumes "top_pres ((`) f)"   and "Sup_pres f" shows "f \<circ> radj f = id"
map_is_Nil_conv: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "(map f xs = []) = (xs = [])"
obtain_v1_v2_3: shows "disjnt vs (vs' \<union> vs'')"
n_mult_idem: fixes x :: "'a" shows "n x \<cdot> n x = n x"
par_subst_upd_left_out: fixes A :: "'e \<Longrightarrow> 'f"   and x :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f \<Rightarrow> 'f"   and v :: "('g, 'f) uexpr"   and B :: "'h \<Longrightarrow> 'f"   and \<rho> :: "'f \<Rightarrow> 'f" assumes "vwb_lens A"   and "x \<bowtie> A" shows "\<sigma>(x \<mapsto>\<^sub>s v) [A|B]\<^sub>s \<rho> = \<sigma> [A|B]\<^sub>s \<rho>"
A: fixes V :: "'a set" shows "(\<exists>u\<in>sinks_aux I D U xs. \<exists>v\<in>V. (u, v) \<in> I) \<longrightarrow> (\<exists>u\<in>U. \<exists>v\<in>sources_aux I D V xs. (u, v) \<in> I)"
wall_crossings_are_walls: fixes H :: "'a set set set"   and Cs :: "'a set list" assumes "H \<in> set (wall_crossings Cs)"   and "H \<noteq> {}" shows "H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}})"
var_same: fixes v :: "'b"   and t :: "'b trm" shows "[(v, t)] \<doteq> [] = (t = Var v)"
cases: fixes x :: "'a list" obtains "x = []"   | xa :: "'a" where "x = [xa__]"   | xa :: "'a" and y :: "'a" and xs :: "'a list" where "x = xa # y # xs"
all_partial_equivs_on: fixes A :: "'b set" assumes "finite A" shows "all_partial_equivs_on A = {R. partial_equiv A R}"
inequality_case_geq: fixes a :: "real"   and b :: "real"   and c :: "real" shows "(\<exists>x. \<forall>y<x. 0 < a * y\<^sup>2 + b * y + c) = (0 < a \<or> a = 0 \<and> (b < 0 \<or> b = 0 \<and> 0 < c))"
nat_pow_pow: fixes x :: "'a vec"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier_vec n" shows "[m] \<cdot>\<^bsub>V\<^esub> ([n] \<cdot>\<^bsub>V\<^esub> x) = [(n * m)] \<cdot>\<^bsub>V\<^esub> x"
int_pow_closed: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier \<G>" shows "x [^] i \<in> carrier \<G>"
nn_cond_expl_is_cond_exp_tmp: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'c"   and M' :: "'c measure" assumes "\<forall>w\<in>space M. 0 \<le> X w"   and "integrable M X"   and "disc_fct Y"   and "point_measurable M (space M') Y" shows "\<forall>A\<in>sets M'. integrable M (\<lambda>w. expl_cond_expect M Y X w * indicat_real (Y -` A \<inter> space M) w) \<and> LINT w|M. X w * indicat_real (Y -` A \<inter> space M) w = LINT w|M. expl_cond_expect M Y X w * indicat_real (Y -` A \<inter> space M) w"
ennreal_sum_less_top: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "finite I" shows "(sum f I < \<top>) = (\<forall>i\<in>I. f i < \<top>)"
d_OUT_h_plus_not_infty': fixes x :: "'v"   and i :: "nat" assumes "x \<noteq> sink \<Delta>" shows "d_OUT (h_plus i) x \<noteq> \<top>"
min_satisfying_None: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "min_satisfying P l = None \<longrightarrow> (\<forall>x\<in>set l. \<not> P x)"
assms_2: shows "pairwise (\<lambda>x y. disjnt (F x) (F y)) A"
sel'_someE: fixes m :: "'s"   and P :: "'u \<times> 'v \<Rightarrow> bool"   and u :: "'u"   and v :: "'v" assumes "invar m"   and "sel' m P = Some (u, v)" obtains u :: "'u" and v :: "'v" where "\<alpha> m u = Some v" and "P (u, v)"
sup_bot_right: fixes x :: "'a \<times> 'b" shows "sup_prod x bot_prod = x"
create_prism_lens: fixes X :: "'c \<Longrightarrow>\<^sub>\<triangle> 'd" shows "create\<^bsub>prism_lens X\<^esub> = build\<^bsub>X\<^esub>"
shift2_invD_1: fixes d :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "shift2_inv d qs" shows "valid_decomp X qs"
ext_per_left: fixes w :: "'b list"   and p :: "nat" assumes "periodN w p"   and "p \<le> \<^bold>|w\<^bold>|" shows "periodN (take p w \<cdot> w) p"
function_ring_car_memI: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<And>a. a \<in> S \<Longrightarrow> f a \<in> carrier (function_ring (carrier R) R)"   and "\<And>a. a \<notin> S \<Longrightarrow> f a = undefined" shows "f \<in> carrier (function_ring S (function_ring (carrier R) R))"
simps_21: fixes v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "bexp"   and ve :: "'a acom"   and vf :: "'a acom"   and vg :: "'a" shows "({v} WHILE va DO vb {vc} \<le> IF vd THEN ve ELSE vf {vg}) = False"
C: fixes x :: "real"   and y :: "real" assumes "C \<le> x"   and "x \<le> y" shows "f y \<le> f x"
nat_of_natural_inverse: fixes x :: "natural" shows "natural_of_nat (nat_of_natural x) = x"
A': shows "A' = - Ax * MPoly_Type.monom (monomial (Suc 0) x') 1 - Ax * p"
bis_def: fixes B1 :: "'f set"   and B2 :: "'g set"   and s1 :: "'f \<Rightarrow> ('h, 'f, 'g) F1"   and s2 :: "'g \<Rightarrow> ('i, 'f, 'g) F2"   and B1' :: "'j set"   and B2' :: "'k set"   and s1' :: "'j \<Rightarrow> ('h, 'j, 'k) F1"   and s2' :: "'k \<Rightarrow> ('i, 'j, 'k) F2"   and R1 :: "('f \<times> 'j) set"   and R2 :: "('g \<times> 'k) set" shows "bis B1 B2 s1 s2 B1' B2' s1' s2' R1 R2 = ((R1 \<subseteq> B1 \<times> B1' \<and> R2 \<subseteq> B2 \<times> B2') \<and> (\<forall>b1 b1'. (b1, b1') \<in> R1 \<longrightarrow> (\<exists>z\<in>F1in UNIV R1 R2. F1map id fst fst z = s1 b1 \<and> F1map id snd snd z = s1' b1')) \<and> (\<forall>b2 b2'. (b2, b2') \<in> R2 \<longrightarrow> (\<exists>z\<in>F2in UNIV R1 R2. F2map id fst fst z = s2 b2 \<and> F2map id snd snd z = s2' b2')))"