hcju/isabelleps huggingface dataset:
padd_infinity_l: fixes a :: "'b"   and x :: "'b"   and y :: "'b"   and p :: "'b \<times> 'b \<times> 'b" shows "padd a (x, y, 0::'b) p = p"
vars_mset_subset_size_mset_uniflessI: fixes M :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and N :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "vars_mset M \<subseteq> vars_mset N"   and "size_mset M < size_mset N" shows "(M, N) \<in> unifless"
union_tsI1: fixes c :: "'a"   and A :: "'b \<Rightarrow> 'a set"   and k :: "'b"   and B :: "'b \<Rightarrow> 'a set" assumes "c \<in> A k" shows "c \<in> (A \<Rightarrow>\<union> B) k"
Diff_disjoint: fixes A :: "'k set"   and B :: "'k set" shows "A \<inter> (B - A) = {}"
linearform_def: fixes V :: "'b set"   and f :: "'b \<Rightarrow> real" shows "linearform V f \<equiv> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> f (x + y) = f x + f y) \<and> (\<forall>x a. x \<in> V \<longrightarrow> f (a \<cdot> x) = a * f x)"
integrable_reflect: fixes f :: "'a \<Rightarrow> 'b"   and b :: "'a"   and a :: "'a" shows "((\<lambda>x. f (- x)) integrable_on cbox (- b) (- a)) = (f integrable_on cbox a b)"
empty_def: shows "Tree23_Set.empty = \<langle>\<rangle>"
traverse_tree_tree_iterate: fixes path :: "dir list"   and l :: "'a \<Rightarrow> 'a"   and r :: "'a \<Rightarrow> 'a"   and s :: "'a" shows "traverse_tree path (tree_iterate l r s) = tree_iterate l r (traverse_path l r path s)"
T_subprob: fixes K :: "'c \<Rightarrow> 'c pmf"   and I :: "'c pmf" shows "measure_pmf.random_variable I (subprob_algebra (stream_space (count_space UNIV))) (MC_syntax.T K)"
assms: shows "timpls_transformable_to' TI (Fun f T) (Fun g S)"
countable_E: shows "countable \<^bold>E\<^bsub>\<Psi>\<^esub>"
carrier_polynomial: fixes p :: "'a list" assumes "polynomial K p" shows "polynomial (carrier R) p"
less_imp_encode_less: fixes n :: "nat"   and p :: "nat"   and e :: "nat" assumes "n < p" shows "local.encode e n < local.encode e p"
to_bl_plus_carry: fixes x :: "'a word"   and y :: "'a word" shows "to_bl (x + y) = rev (foldr (\<lambda>(x, y) res car. xor3 x y car # res (carry x y car)) (rev (zip (to_bl x) (to_bl y))) (\<lambda>_. []) False)"
tuple_simps_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "local.dom \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = local.dom h"
CFGExit_wf_axioms: shows "CFGExit_wf sourcenode targetnode kind valid_edge (_Entry_) get_proc get_return_edges procs Main (_Exit_) Def Use ParamDefs ParamUses"
mono_onD: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "Basis.mono_on A f"   and "x \<in> A"   and "y \<in> A"   and "x \<le> y" shows "f x \<le> f y"
g_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" shows "GE.int \<xi> (gT T) = Ik.int \<xi> T"
lu: fixes a :: "'a"   and b :: "'b" assumes "(a, b) \<in> Basis" shows "(la, lb) \<bullet> (a, b) \<le> (ua, ub) \<bullet> (a, b)"
usubst_fml_game_2: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<alpha> :: "game"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set" assumes "Uvariation \<nu> \<omega> U"   and "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg" shows "(\<nu> \<in> game_sem I (the (snd (usubstappp \<sigma> U \<alpha>))) X) = (\<nu> \<in> game_sem (USubst.adjoint \<sigma> I \<omega>) \<alpha> X)"
eval_inverse_monom: fixes monom :: "real \<times> real list"   and basis :: "(real \<Rightarrow> real) list" shows "eval_monom (inverse_monom monom) basis = (\<lambda>x. inverse (eval_monom monom basis x))"
done_empty_valid_mmuaux'_intro: fixes args :: "Monitor.args"   and cur :: "nat"   and dt :: "nat"   and tp :: "nat"   and tss :: "nat queue"   and len :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and a1_map :: "('a option list, nat) mapping"   and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping"   and "done" :: "'a option list set list"   and done_length :: "nat"   and auxlist :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and dt' :: "nat" assumes "valid_mmuaux' args cur dt (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) auxlist" shows "valid_mmuaux' args cur dt' (tp, tss, len, maskL, maskR, a1_map, a2_map, [], 0) (drop (length done) auxlist)"
part_def: shows "part = partition (p \<circ> snd) (zip [0..<length xs] xs)"
dim_row_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "dim_row (mat nr nc f) = nr"
prems: shows "internal ds (Branching T\<^sub>1 T\<^sub>2)"
sw_involutive: fixes xs :: "'e"   and x :: "'d"   and y :: "'d"   and zs :: "'e"   and z :: "'d" shows "z @zs[x \<and> y]_xs @zs[x \<and> y]_xs = z"
Delta_null_of_null_is_null: fixes B :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "B \<in> sets M"   and "A \<Delta> B \<in> null_sets M"   and "A \<in> null_sets M" shows "B \<in> null_sets M"
cases: fixes x :: "('Var, 'AExp, 'BExp) Stmt" obtains c :: "('Var, 'AExp, 'BExp) Stmt" and uu_ :: "'Var ModeUpd" where "x = c@[uu_]"   | p :: "('Var, 'AExp, 'BExp) Stmt" and q :: "('Var, 'AExp, 'BExp) Stmt" where "x = p__ ;; q__"   | uv_ :: "'BExp" and p :: "('Var, 'AExp, 'BExp) Stmt" and q :: "('Var, 'AExp, 'BExp) Stmt" where "x = Stmt.If uv_ pa__ q"   | uw_ :: "'BExp" and p :: "('Var, 'AExp, 'BExp) Stmt" where "x = While uw_ p"   | v :: "'Var" and va :: "'AExp" where "x = v \<leftarrow> va"   | "x = Skip"   | "x = Stop"
aux2: fixes P :: "pi" shows "bangPred P (P \<parallel> !P)"
Quotient_ltln\<^sub>Q: shows "Quotient (\<sim>\<^sub>Q) abs_ltln\<^sub>Q rep_ltln\<^sub>Q cr_ltln\<^sub>Q"
emeasure_PiM: fixes I :: "'a set"   and A :: "'a \<Rightarrow> real set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (interval_measure (\<lambda>x. x))" shows "emeasure (Pi\<^sub>M I (\<lambda>_. interval_measure (\<lambda>x. x))) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (interval_measure (\<lambda>x. x)) (A i))"
mult_le_mono_hmset: fixes i :: "hmultiset"   and j :: "hmultiset"   and k :: "hmultiset"   and l :: "hmultiset" assumes "i \<le> j"   and "k \<le> l" shows "i * k \<le> j * l"
ffilter_finsert: fixes f :: "'b \<Rightarrow> bool"   and a :: "'b"   and s :: "'b fset" shows "ffilter f (finsert a s) = (if f a then finsert a (ffilter f s) else ffilter f s)"
update_by_pair_id: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and k :: "'a"   and v :: "'d" assumes "oalist_inv_raw x xs"   and "lookup_pair x xs k = v" shows "update_by_pair x (k, v) xs = xs"
rep_eq: fixes f :: "'b \<Rightarrow> real" assumes "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1" shows "measure_pmf (local.embed_pmf f) = density (count_space UNIV) (ennreal \<circ> f)"
tdghm_def: shows "\<NN> = [\<NN>\<lparr>NTMap\<rparr>, \<NN>\<lparr>NTDom\<rparr>, \<NN>\<lparr>NTCod\<rparr>, \<NN>\<lparr>NTDGDom\<rparr>, \<NN>\<lparr>NTDGCod\<rparr>]\<^sub>\<circ>"
assms_14: shows "InvariantVarsF (getF state) F0 Vbl"
is_natural_2: fixes f :: "'a" assumes "arr f" shows "trg (cod f) \<cdot> trg f = trg f"
measure_y_eq: fixes ya :: "'s"   and n :: "nat" shows "measure_pmf.prob N {ya} = K2.T.prob None {\<omega> \<in> space (K2.T None). \<omega> !! n = Some ya}"
inject_1: fixes x2 :: "nat"   and y2 :: "nat" shows "(Card x2 = Card y2) = (x2 = y2)"
id: shows "local.mahler_approximation_main dd c g mm k kk = (if bnd \<le> k then new_mm else local.mahler_approximation_main (dd * dd) c (graeffe_one_step c g) new_mm (Suc k) (2 * kk))"
is_binqueue_meld: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list"   and ys :: "('c, 'd) bintree option list" assumes "is_binqueue l xs"   and "is_binqueue l ys" shows "is_binqueue l (meld xs ys)"
nrows_singleton: fixes j :: "nat"   and i :: "nat"   and e :: "'a" shows "nrows (singleton_matrix j i e) = (if e = (0::'a) then 0 else Suc j)"
uniqueness_algorithm_division_Mp_Rel: fixes g :: "int poly"   and F :: "'a mod_ring poly"   and f :: "int poly"   and G :: "'a mod_ring poly"   and q :: "int poly"   and r :: "int poly"   and Q :: "'a mod_ring poly"   and R :: "'a mod_ring poly" assumes "monic (Mp g)"   and "F = of_int_poly f"   and "G = of_int_poly g"   and "pseudo_divmod (Mp f) (Mp g) = (q, r)"   and "pseudo_divmod F G = (Q, R)" shows "MP_Rel q Q \<and> MP_Rel r R"
lm064: fixes R :: "('c \<times> 'd) set" assumes "runiq R" shows "sum (toFunction R) (Domain R) = sum snd R"
ide_eval_Can: fixes t :: "'a local.term" assumes "Can t" shows "ide \<lbrace>t\<rbrace>"
SUP_upper: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and i :: "'f"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
assms_5: shows "f1 x2 y2 z2 = (0::'a)"
dims0_def: shows "d.dims0 = nths dims d.vars0"
eq: shows "(\<Sum>b | f b \<noteq> (0::'a). f b *s b) = v"
leg1_commutes: shows "gf.cod.leg1 \<cdot> gf.chine = gf.leg1"
Subobjs_Sh: fixes C :: "char list"   and C' :: "char list"   and 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 D :: "char list"   and Cs :: "char list list" assumes "P \<turnstile> C \<preceq>\<^sup>* C'"   and "P \<turnstile> C' \<prec>\<^sub>S D"   and "Subobjs\<^sub>R P D Cs" shows "Subobjs P C Cs"
equivalence_refines_to_adjoint_equivalence: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and \<eta> :: "('a S.E.term, 'a) S.arr" assumes "S.equivalence_map f"   and "\<guillemotleft>g : S.trg f \<rightarrow>\<^sub>S S.src f\<guillemotright>"   and "S.ide g"   and "\<guillemotleft>\<eta> : S.src f \<Rightarrow>\<^sub>S g \<star>\<^sub>S f\<guillemotright>"   and "S.iso \<eta>" shows "\<exists>!\<epsilon>. adjoint_equivalence_in_bicategory (\<cdot>\<^sub>S) (\<star>\<^sub>S) S.\<a> S.\<i> S.src S.trg f g \<eta> \<epsilon>"
dests_vD_inc_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:11, PRreq-:20, PRrep-:7} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> the (dests \<xi> ip) = inc (sqn (rt \<xi>) ip))) \<and> (l = PRerr-:1 \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>) \<and> sqn (rt \<xi>) ip < the (dests \<xi> ip))))"
hcomp_ide_isomorphic: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.ide V f"   and "category.isomorphic V g h"   and "src f = trg g" shows "category.isomorphic V (H f g) (H f h)"
gives_limit_cones: fixes D :: "'j \<Rightarrow> 'c" assumes "diagram (\<cdot>\<^sub>J) (\<cdot>) D" shows "limit_cone (\<cdot>\<^sub>J) (\<cdot>) D (G (J_C.MkIde D)) (J_C.Map (Adj.\<epsilon> (J_C.MkIde D)))"
distinct_115: shows "LessOrEqual \<noteq> ShiftRightSigned"
unitor_coherence: fixes a :: "'a" assumes "ide a" shows "\<l>[a] \<cdot> \
[a, \<I>] = \<r>[a]"
inject_2: fixes x31 :: "'b tree"   and x32 :: "nat"   and x33 :: "nat"   and y31 :: "'b tree"   and y32 :: "nat"   and y33 :: "nat" shows "(Unbal2 x31 x32 x33 = Unbal2 y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
eval_Inv_Can: fixes t :: "'c local.term" assumes "Can t" shows "\<lbrace>Inv t\<rbrace> = D.inv \<lbrace>t\<rbrace>"
gen_dfs_eq_assert: fixes gds :: "('f, 'e) gen_dfs_struct"   and V0 :: "'f set" shows "gen_dfs_defs.gen_dfs gds V0 = gen_dfs_defs.gen_dfs_assert gds V0"
fmapU_fmapU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'a \<rightarrow> udom\<cdot>'a"   and f :: "udom \<rightarrow> udom"   and g :: "udom \<rightarrow> udom"   and xs :: "udom\<cdot>'a" assumes "class.functor fmapU" shows "fmapU\<cdot>f\<cdot>(fmapU\<cdot>g\<cdot>xs) = fmapU\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
create_gba__name_inj: shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. inj_on name (g_V \<A>))"
assoc: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" shows "a \<inter># b \<inter># c = a \<inter># (b \<inter># c)"
simps_2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> assn"   and a :: "'b"   and as :: "'b list"   and c :: "'c"   and cs :: "'c list" shows "list_assn P (a # as) (c # cs) = P a c * list_assn P as cs"
idem: fixes a :: "'b" shows "a \<squnion> a = a"
foldl1_map: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list"   and h :: "'c \<Rightarrow> 'b" assumes "associative f"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "foldl1 f (map h (xs @ ys)) = f (foldl1 f (map h xs)) (foldl1 f (map h ys))"
dependent_biggerset_general: fixes S :: "'a set" assumes "finite S \<Longrightarrow> dim S < card S" shows "dependent S"
quasiconvex_def: fixes C :: "real"   and X :: "'b set" shows "quasiconvex C X = (0 \<le> C \<and> (\<forall>x\<in>X. \<forall>y\<in>X. \<exists>G. geodesic_segment_between G x y \<and> (\<forall>z\<in>G. infdist z X \<le> C)))"
SPEC_rule: fixes \<Phi> :: "'e \<Rightarrow> bool"   and \<Phi>' :: "'e \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Phi>' x" shows "SPEC \<Phi> \<le> SPEC \<Phi>'"
strict_runit: fixes f :: "'a" assumes "ide f" shows "\<r>[f] = f"
approx_remains_after_same_comp: fixes p :: "'a pmf"   and q :: "'a pmf"   and r :: "'a pmf"   and \<alpha> :: "real" assumes "p \<approx>[\<R>] q"   and "r \<in> local.\<P>"   and "\<alpha> \<in> {0..1}" shows "mix_pmf \<alpha> p r \<approx>[\<R>] mix_pmf \<alpha> q r"
imp_recovery2: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "imp_PL p q \<in> Cn A"   and "imp_PL (imp_PL q p) p \<in> Cn A" shows "q \<in> Cn A"
res_comp1: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "f \<circ> residual f \<circ> f = f"
inject_3: fixes x41 :: "'a"   and x42 :: "('a, 'b) MWLfCom"   and x43 :: "('a, 'b) MWLfCom"   and y41 :: "'a"   and y42 :: "('a, 'b) MWLfCom"   and y43 :: "('a, 'b) MWLfCom" shows "(if x41 then x42 else x43 fi = if y41 then y42 else y43 fi) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
assms_6: shows "a2 = (A - D - cor dsc) / (2 * C)"
Goedel_Form_axioms: shows "Goedel_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi fls prv bprv enc S P"
continuous_open_quotient_map: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e" assumes "continuous_map X X' f"   and "open_map X X' f" shows "quotient_map X X' f = (f ` topspace X = topspace X')"
star_trans: fixes x :: "'b \<times> 'c" shows "less_eq_prod (times_prod' (prod.qstar x) (prod.qstar x)) (prod.qstar x)"
weakly_bisimilar_is_weak_bisimulation: shows "S_transform.is_weak_bisimulation (\<approx>\<cdot>\<^sub>S)"
simps_1: fixes n :: "nat"   and p :: "nat"   and q :: "nat"   and r :: "nat" shows "bn_comp (Suc (Suc n)) p q r = ((Suc (Suc n) * p + r) div q, (Suc (Suc n) * p + r) mod q)"
circle_top_edge_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real" assumes "circle i j d" shows "circle.circle_top_edge d = (- 1, \<lambda>t. (x_coord t * d, d * circle_y (x_coord t)))"
fcomp_norm_dflt_init: fixes x :: "('a \<Rightarrow> 'b Heap) \<times> ('c \<Rightarrow> 'd nres)"   and P :: "'c \<Rightarrow> bool"   and R :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and T :: "'d \<Rightarrow> 'b \<Rightarrow> assn"   and S :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)" assumes "x \<in> [P]\<^sub>a R \<rightarrow> T"   and "hrp_imp R S" shows "x \<in> [P]\<^sub>a S \<rightarrow> T"
omega_sum_less_iff_cases: fixes m :: "nat"   and n :: "nat"   and ms :: "nat list"   and ns :: "nat list" assumes "0 < m"   and "0 < n" shows "(omega_sum (m # ms) < omega_sum (n # ns)) = (length ms < length ns \<or> length ms = length ns \<and> m < n \<or> length ms = length ns \<and> m = n \<and> omega_sum ms < omega_sum ns)"
induct: fixes P :: "'b list \<Rightarrow> ty list \<Rightarrow> 'c val list \<Rightarrow> ('b, 'd, 'c) exp \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "ty list"   and a2 :: "'c val list"   and a3 :: "('b, 'd, 'c) exp" assumes "\<And>V Vs T Ts v vs e. P Vs Ts vs e \<Longrightarrow> P (V # Vs) (T # Ts) (v # vs) e"   and "\<And>e. P [] [] [] e"   and "\<And>v va c d. P [] (v # va) c d"   and "\<And>b v va d. P [] b (v # va) d"   and "\<And>v va c d. P (v # va) [] c d"   and "\<And>a v va d. P a [] (v # va) d"   and "\<And>v va b d. P (v # va) b [] d"   and "\<And>a v va d. P a (v # va) [] d" shows "P a0 a1 a2 a3"
Suc: shows "(cospoly (Suc n) has_real_derivative - sinpoly n x) (at x)"
w\<sigma>: shows "weight (t \<cdot> \<sigma>) \<le> weight (Fun f ss \<cdot> \<sigma>)"
SUP_sup_continuous_ennreal: fixes f :: "ennreal \<Rightarrow> 'a"   and I :: "'b set"   and g :: "'b \<Rightarrow> ennreal" assumes "sup_continuous f"   and "I \<noteq> {}" shows "(\<Squnion>i\<in>I. f (g i)) = f (\<Squnion> (g ` I))"
coalg_F2set3: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and a :: "'e" assumes "coalg B1 B2 s1 s2"   and "a \<in> B2" shows "F2set3 (s2 a) \<subseteq> B2"
find_crossing_edge_on_path: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "path g u p v"   and "\<not> P u"   and "P v" obtains u' :: "'b" and v' :: "'b" where "(u', v') \<in> set p" and "\<not> P u'" and "P v'"
restrict_mono_map: fixes A :: "nat \<Rightarrow> bool option"   and B :: "nat \<Rightarrow> bool option"   and x :: "nat set"   and C :: "nat set" assumes "A \<subseteq>\<^sub>s B" shows "A \<ominus>\<^bsub>x\<^esub> C \<subseteq>\<^sub>s B \<ominus>\<^bsub>x\<^esub> C"
extends_refl: fixes S :: "'b list" shows "S \<lesssim> S"
nl3: fixes l' :: "real"   and r' :: "real"   and lr :: "real"   and r :: "real" assumes "1 \<le> l'"   and "1 \<le> r'"   and "1 \<le> lr"   and "1 \<le> r" shows "4 * (l' + r')\<^sup>2 * (lr + r) * (lr + r' + r) \<le> (l' + lr + r') * (l' + lr + r' + r) ^ 3"
backward_terminating_iff1: fixes x :: "'a" assumes "path x" shows "backward_terminating x = (has_start_points x \<or> x = 0)"
Liminf_eq: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "Liminf F f = Liminf F g"
hyps: assumes "set ws \<subseteq> carrier_vec n" shows "lincomb f (set ws) = lincomb_list (\<lambda>i. if \<exists>j<i. ws ! i = ws ! j then 0::'a else f (ws ! i)) ws"
uminus_set_def: fixes A :: "'d set" shows "- A = Collect (- (\<lambda>x. x \<in> A))"
tranclp_rtranclp_absorb: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "R\<^sup>+\<^sup>+\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
intros_1: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "r \<in> complex_rotations"
l_def: shows "l = 47388813395531028639296492901910937 / 82101866951584879688289000000000000"
simps_16: fixes M :: "char list"   and n :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" shows "succs (Invoke M n) \<tau> pc = (if fst \<tau> ! n = NT then [] else [pc + 1])"
simps_2: shows "TernaryFalse \<noteq> TernaryTrue"
inverse_distrib: fixes times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.ab_group_mult times one divide inverse" shows "inverse (times a b) = times (inverse a) (inverse b)"
is_stmt_rews_2: fixes x :: "Term.var" shows "\<not> is_stmt (In2 x)"
findnth_correct: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires" shows "\<exists>stp l' r'. steps0 (Suc 0, l, r) (findnth n) stp = (Suc (2 * n), l', r') \<and> inv_locate_a (as, lm) (n, l', r') ires"
insert_def: fixes a :: "'b"   and B :: "'b set" shows "(a::'b) \<triangleright> (B::'b set) = {x::'b. x = a} \<union> B"
jkbpCn_tFirst_init_inv: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> jkbpC\<^bsub>n\<^esub>" shows "tFirst t \<in> set envInit"
\<phi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "S.UP \<circ> inl \<in> C.hom b a \<rightarrow> Hom.set (b, a)"
matrix_add_rdistrib: fixes B :: "(('a, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and C :: "(('a, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "(B + C) ** A = B ** A + C ** A"
equal: fixes equal :: "'aq \<Rightarrow> 'aq \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
P_sym: fixes t1 :: "dag"   and t2 :: "dag" shows "P t1 t2 var = P t2 t1 var"
lUse: shows "lUse \<equiv> lift_Use Use Entry Exit H L"
round_of_int: fixes n :: "int" shows "round (of_int n) = n"
ex_fresh_gt: fixes A :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'b" assumes "\<exists>g. surj g"   and "finite A"   and "i \<notin> A" shows "\<exists>j. j \<notin> f ` A"
ANR_from_Un_Int_local: fixes S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set (S \<union> T)) S"   and "closedin (top_of_set (S \<union> T)) T"   and "ANR (S \<union> T)"   and "ANR (S \<inter> T)" shows "ANR S"
elims_3: fixes x :: "atom list \<times> 'a"   and xa :: "atom list \<times> 'a" assumes "\<not> x \<approx>abs_lst xa" obtains bs :: "atom list" and xb :: "'a" and cs :: "atom list" and y :: "'a" where "x = (bs, xb)" and "xa = (cs, y)" and "\<nexists>p. (bs, xb) \<approx>lst (=) supp p (cs, y)"
coreflexive_one_closed: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b" assumes "class.non_associative_left_semiring one times sup less_eq less bot" shows "less_eq one one"
var_rename_def: fixes n :: "nat"   and v :: "'b \<times> nat" shows "var_rename n v \<equiv> Var (fst v, snd v + Suc n)"
aprimedivisor_nat_1: fixes n :: "nat" assumes "n \<noteq> Suc 0" shows "prime (aprimedivisor n)"
U_def: fixes i :: "nat" shows "U i = inv_chart (\<psi> i) ` ball (0::'b) 1"
singleton_times_right_hmset: fixes N :: "hmultiset"   and M :: "hmultiset" shows "N * \<omega>^ M = HMSet (image_mset ((+) M) (hmsetmset N))"
xfer_fair_run_aux2: fixes cs :: "nat \<Rightarrow> 'c" assumes "C.is_fair_run cs" obtains as :: "nat \<Rightarrow> 'a" where "A.is_fair_run as" and "\<forall>i. R (as i) (cs i)"
disj_nb: fixes p :: "fm"   and q :: "fm" assumes "bound0 p"   and "bound0 q" shows "bound0 (Ferrack.disj p q)"
Range_of_constraint_conv: fixes A :: "('b \<times> 'a) set"   and C :: "'a set" shows "Range (A \<inter> UNIV \<times> C) = Range A \<inter> C"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "Cd d X = greatest (Pd d) (X \<inter> Field (Pd d))"