hcju/isabelleps huggingface dataset:
arrE: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.arr (fst f) \<and> B.VV.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (fst (snd f))"
mult_divide_eq_ennreal: fixes b :: "ennreal"   and a :: "ennreal" assumes "b \<noteq> 0"   and "b \<noteq> top" shows "a * b / b = a"
Max_wrt_among_of_weak_ranking_Cons: fixes x :: "'b set"   and xs :: "'b set list"   and A :: "'b set" assumes "is_weak_ranking (x # xs)" shows "Max_wrt_among (of_weak_ranking (x # xs)) A = (if x \<inter> A = {} then Max_wrt_among (of_weak_ranking xs) A else x \<inter> A)"
vmu: shows "v = u @\<^sub>v 0\<^sub>v m"
separating_hyperplane_closed_point: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "closed S"   and "z \<notin> S" shows "\<exists>a b. a \<bullet> z < b \<and> (\<forall>x\<in>S. b < a \<bullet> x)"
knows_Spy_bad_privateKey: fixes A :: "agent"   and b :: "bool"   and evs :: "event list" assumes "A \<in> bad" shows "Key (invKey (publicKey b A)) \<in> knows Spy evs"
LIM_offset_zero: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and a :: "'a" assumes "f \<midarrow>a\<rightarrow> L" shows "(\<lambda>h. f (a + h)) \<midarrow>0::'a\<rightarrow> L"
conjugate_moebius: fixes M :: "moebius" shows "conjugate \<circ> moebius_pt M = moebius_pt (conjugate_moebius M) \<circ> conjugate"
Liminf_within: fixes x :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "Liminf (at x within S) f = (SUP e\<in>{0<..}. Inf (f ` (S \<inter> ball x e - {x})))"
h3: shows "all_list_constr [[], [0]] (Suc 0)"
arr_char: fixes f :: "(bool, unit) J.arr" shows "J.arr f = (f = J.MkIde False \<or> f = J.MkIde True)"
simps_2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xs :: "'b Trace"   and x :: "'b"   and ys :: "'c Trace"   and y :: "'c" shows "tZip f (xs \<leadsto> x) (ys \<leadsto> y) = tZip f xs ys \<leadsto> f x y"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "C.arr f"   and "C.cod f = b" shows "b \<cdot>\<^sub>C f = f"
ht_imp_len: fixes l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable" shows "is_hashtable l ht \<Longrightarrow>\<^sub>A is_hashtable l ht * \<up> (0 < length l)"
DBM_reset_sound2: fixes v :: "'c \<Rightarrow> nat"   and c :: "'c"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and u :: "'c \<Rightarrow> 'd" assumes "v c \<le> n"   and "DBM_reset M n (v c) d M'"   and "u \<turnstile>\<^bsub>v,n\<^esub> M'" shows "u c = d"
PiE_I: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x"   and "\<And>x. x \<notin> A \<Longrightarrow> f x = undefined" shows "f \<in> Pi\<^sub>E A B"
freshAbs_substAbs_E1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "freshAbs zs z (A $[Y / y]_ys)"   and "zs \<noteq> ys \<or> z \<noteq> y" shows "freshAbs zs z A"
ft_invar_def: fixes t :: "('e, 'f) FingerTreeStruc" shows "ft_invar t \<equiv> is_leveln_ftree 0 t \<and> is_measured_ftree t"
fib_Cassini_int: fixes n :: "nat" shows "int (fib (Suc (Suc n)) * fib n) - int ((fib (Suc n))\<^sup>2) = - ((- 1) ^ n)"
2: shows "\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<phi>. P \<turnstile> \<phi> \<longrightarrow> Q \<turnstile> \<phi>)"
iter_induct: fixes d :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and x :: "'a::refinement_lattice" assumes "(d::'a::refinement_lattice) \<sqinter> (c::'a::refinement_lattice) ; (x::'a::refinement_lattice) \<sqsubseteq> x" shows "(c::'a::refinement_lattice)\<^sup>\<omega> ; (d::'a::refinement_lattice) \<sqsubseteq> (x::'a::refinement_lattice)"
divides_prod: fixes A :: "'a mod_ring set"   and g :: "'a mod_ring \<Rightarrow> 'a mod_ring poly"   and f :: "'a mod_ring poly" assumes "\<forall>c1 c2. c1 \<in> A \<and> c2 \<in> A \<and> c1 \<noteq> c2 \<longrightarrow> comm_monoid_mult_class.coprime (g c1) (g c2)"   and "\<forall>c\<in>A. g c dvd f" shows "prod g A dvd f"
lexord_append_leftI_linorder: fixes u :: "'a list"   and v :: "'a list"   and x :: "'a list" assumes "u <lex v" shows "x \<cdot> u <lex x \<cdot> v"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "cycle (x # xs) = (x \<notin> set xs \<and> cycle xs)"
add_opt_def: fixes U\<^sub>1 :: "nat option"   and U\<^sub>2 :: "nat option" shows "U\<^sub>1 \<oplus> U\<^sub>2 \<equiv> case (U\<^sub>1, U\<^sub>2) of (None, b) \<Rightarrow> None | (Some u\<^sub>1, None) \<Rightarrow> None | (Some u\<^sub>1, Some u\<^sub>2) \<Rightarrow> Some (u\<^sub>1 + u\<^sub>2)"
rcosets_part_G: fixes H :: "'a set" assumes "subgroup H (add_monoid G)" shows "\<Union> (rcosets\<^bsub>add_monoid G\<^esub> H) = carrier (add_monoid G)"
det_2: fixes A :: "(('a, 2) vec, 2) vec" shows "det A = A $ 1 $ 1 * A $ 2 $ 2 - A $ 1 $ 2 * A $ 2 $ 1"
null_sets_uniform_measure: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> sets M"   and "emeasure M A \<noteq> \<infinity>" shows "null_sets (uniform_measure M A) = (\<inter>) A -` null_sets M \<inter> sets M"
fmrestrict_fset_twice: fixes S :: "'e fset"   and T :: "'e fset"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmrestrict_fset S (fmrestrict_fset T m) = fmrestrict_fset (S |\<inter>| T) m"
rep_eq: fixes x :: "'b mat_impl" shows "dim_row_impl x = fst (Rep_mat_impl x)"
cqt_further_12: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<not>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>& \<^bold>\<not>(\<^bold>\<exists>\<alpha>. \<psi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) in v]"
val_principalTr1: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "Lv K v \<in> v ` (carrier K - {\<zero>}) \<and> (\<forall>w\<in>v ` carrier K. \<exists>a. w = a * Lv K v) \<and> 0 < Lv K v"
in_idem: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "gcd a (Gcd\<^sub>f\<^sub>i\<^sub>n A) = Gcd\<^sub>f\<^sub>i\<^sub>n A"
exhaust: fixes y :: "'a list" obtains "y = []"   | x21 :: "'a" and x22 :: "'a list" where "y = x21 \<Colon> x22"
type_II_chain_horiz_bound_valid: shows "\<forall>(k, \<gamma>)\<in>two_chain_horizontal_boundary two_chain. valid_path \<gamma>"
normalized_canceled: fixes l :: "(bool \<times> 'c) list" shows "canceled (Cancelation.normalize l)"
distinct_153: fixes x6 :: "Syntax.term"   and x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term" shows "Inj2 x6 \<noteq> Syntax.Let x81 x82 x83"
right_cancel: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "Group.group G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "(y \<otimes>\<^bsub>G\<^esub> x = z \<otimes>\<^bsub>G\<^esub> x) = (y = z)"
subring_def: fixes H :: "'a set"   and R :: "('a, 'b) ring_scheme" shows "subring H R \<equiv> subgroup H (add_monoid R) \<and> submonoid H R"
commute: fixes a :: "'b"   and b :: "'b" shows "sup a b = sup b a"
b_cos: shows "b \<in> rcosets\<^bsub>G\<^esub> kernel G H f"
normal_mantissa_def: fixes x :: "('a, 'b) IEEE.float" shows "normal_mantissa x = (- 1) ^ sign x * (2 ^ LENGTH('b) + int (fraction x))"
evaln_expr_eq: fixes G :: "prog"   and s :: "state"   and t :: "expr"   and n :: "nat"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>s \<midarrow>In1l t\<succ>\<midarrow>n\<rightarrow> (w, s') = (\<exists>v. w = In1 v \<and> G\<turnstile>s \<midarrow>t-\<succ>v\<midarrow>n\<rightarrow> s')"
sfis_nnfis: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "a \<in> sfis f M" shows "a \<in> nnfis f M"
psubst_empty: fixes t :: "trm" shows "[]<t> = t"
encF_fmla: fixes f :: "'trm \<Rightarrow> 'trm" assumes "f \<in> F" shows "encF f \<in> fmla"
simps_2: fixes clause :: "Literal list"   and formula :: "Literal list list"   and valuation :: "Literal list" shows "formulaTrue (clause # formula) valuation = (clauseTrue clause valuation \<and> formulaTrue formula valuation)"
swap_col_to_front_four_block: fixes A1 :: "'b mat"   and n1 :: "nat"   and m :: "nat"   and A2 :: "'b mat"   and A3 :: "'b mat"   and n2 :: "nat"   and A4 :: "'b mat" assumes "A1 \<in> carrier_mat n1 m"   and "A2 \<in> carrier_mat n1 1"   and "A3 \<in> carrier_mat n2 m"   and "A4 \<in> carrier_mat n2 1" shows "swap_col_to_front (four_block_mat A1 A2 A3 A4) m = four_block_mat A2 A1 A4 A3"
induct: fixes P :: "Formula.formula \<Rightarrow> bool"   and a0 :: "Formula.formula" assumes "\<And>l. P (formula.Ands l)"   and "\<And>v va. P (formula.Pred v va)"   and "\<And>v va vb. P (formula.Let v va vb)"   and "\<And>v va. P (formula.Eq v va)"   and "\<And>v va. P (formula.Less v va)"   and "\<And>v va. P (formula.LessEq v va)"   and "\<And>v. P (formula.Neg v)"   and "\<And>v va. P (formula.Or v va)"   and "\<And>v va. P (formula.And v va)"   and "\<And>v. P (formula.Exists v)"   and "\<And>v va vb vc vd. P (formula.Agg v va vb vc vd)"   and "\<And>v va. P (formula.Prev v va)"   and "\<And>v va. P (formula.Next v va)"   and "\<And>v va vb. P (formula.Since v va vb)"   and "\<And>v va vb. P (formula.Until v va vb)"   and "\<And>v va. P (formula.MatchF v va)"   and "\<And>v va. P (formula.MatchP v va)" shows "P a0"
ipurge_fail_aux_t_aux_set: fixes Y :: "('b, 'c) ipurge_rec" shows "ipurge_fail_aux_t_aux Y \<in> ipurge_fail_aux_t_set Y"
assms_3: shows "h \<turnstile> set_child_nodes ptr children \<rightarrow>\<^sub>h h'"
language_def: fixes M :: "'b nfa" assumes "nfa M" shows "nfa.language M \<equiv> {xs. nfa.nextl M (nfa.init M) xs \<inter> nfa.final M \<noteq> {}}"
simps_2: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'b" assumes "propositional_atoms atom_ordering" shows "complement (Neg A) = Pos A"
has_bottom_def: fixes P :: "'a set" shows "domain.has_bottom P \<equiv> \<exists>z\<in>P. \<forall>x\<in>P. x \<succeq> z"
l_def: shows "l = limsup (\<lambda>n. ereal (root n (norm (f n))))"
rep_in_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "free_monoidal_category.rep C f \<in> f"
edge'_simps_1: fixes x :: "'v option ??.MFMC_Flow_Attainability.flow_attainability.node" shows "??.local.\<Psi>.edge' ??.MFMC_Flow_Attainability.flow_attainability.node.SOURCE x = (x = ??.MFMC_Flow_Attainability.flow_attainability.node.Inner (source \<Psi>))"
sem\<^sub>e\<^sub>s\<^sub>t_c_eq_sem_st: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<A> :: "('fun, 'var) extstrand_step list" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> \<A> = \<lbrakk>M\<^sub>0; to_st \<A>\<rbrakk>\<^sub>c' \<I>"
ti_Lcnv_def: fixes ecnv :: "('a, 'b, 'd) gba_rec_scheme \<Rightarrow> 'c"   and A :: "('a, 'b, 'd) gba_rec_scheme" shows "ti_Lcnv ecnv A \<equiv> \<lparr>igba_L = gba_L A, \<dots> = ecnv A\<rparr>"
below_refl: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.po below" shows "below x x"
prim_list_impl_correct: shows "case prim_list_impl l r of None \<Rightarrow> \<not> G_valid_wgraph_repr l | Some \<pi>i \<Rightarrow> G_valid_wgraph_repr l \<and> (let Gi = G.from_list l in G.invar Gi \<and> P.invar_MST \<pi>i \<and> is_MST (G.\<alpha>w Gi) (component_of (G.\<alpha>g Gi) r) (P.\<alpha>_MST r \<pi>i))"
length_t_wcode_main_first_part_even: shows "is_even (length t_wcode_main_first_part)"
antisym_conv: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and y :: "'k"   and x :: "'k" assumes "class.order less_eq less"   and "less_eq y x" shows "less_eq x y = (x = y)"
le_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Le C D" shows "A B Le D C"
appNone: fixes i :: "instr"   and P :: "'a prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and et :: "ex_table" shows "app i P mxs T\<^sub>r pc mpc et None = True"
comp_ide_self: fixes a :: "('a, 'a set) Maps.arr" assumes "Maps.ide a" shows "a \<odot> a = a"
abs_eq: fixes x :: "real_alg_3" shows "- Real_Alg_Quotient x = Real_Alg_Quotient (uminus_3 x)"
icard_0_eq: fixes A :: "'a set" shows "(icard A = 0) = (A = {})"
Int_space_eq1: fixes x :: "'b set"   and M :: "'b measure" assumes "x \<in> fmeasurable M" shows "space M \<inter> x = x"
insert_before_locs_def: shows "insert_before_locs \<equiv> l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before_locs set_child_nodes_locs adopt_node_locs set_disconnected_nodes_locs"
pderiv_power_Suc: fixes p :: "'a poly"   and n :: "nat" shows "pderiv (p ^ Suc n) = Polynomial.smult (of_nat (Suc n)) (p ^ n) * pderiv p"
tW_cmpser_is_W_cmpser: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "tW_cmpser G n f" shows "W_cmpser G n f"
cprank_max1_order0: fixes B :: "'a tensor"   and A :: "'a tensor" assumes "cprank_max1 B"   and "order A = 0" shows "cprank_max1 (A \<otimes> B)"
leftmost_notword: fixes i :: "nat"   and a :: "symbol list"   and j :: "nat" assumes "leftmost i a"   and "i < j" shows "\<not> is_word (take j a)"
nonrefl: shows "(x, x) \<notin> condensation_edges g P"
moebius_rotation_dilatation: fixes a :: "complex" assumes "a \<noteq> 0" shows "moebius_rotation_dilatation a = moebius_rotation (arg a) + moebius_dilatation (cmod a)"
coeff_fps_nth_fps_poly_swap1: fixes p :: "'b fps poly"   and m :: "nat"   and n :: "nat" shows "coeff (fps_poly_swap1 p $ m) n = coeff p n $ m"
Comp_preserves_coherent: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Dom t = Cod u"   and "coherent t"   and "coherent u" shows "coherent (t \<^bold>\<cdot> u)"
coeff_monom: fixes a :: "'b"   and m :: "nat"   and n :: "nat" shows "poly.coeff (Polynomial.monom a m) n = (if m = n then a else (0::'b))"
ers_def: fixes E :: "'b set"   and F :: "'b set" shows "(E \<lless>\<^sub>\<S> F) = (\<forall>a. a \<in> E \<longrightarrow> (\<exists>b. b \<in> F \<and> a \<le>\<^sub>\<S> b))"
incrtm0: fixes vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" shows "Itm vs (x # bs) (incrtm0 t) = Itm vs bs t"
map_cong0: fixes x :: "'b literal"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>z. z \<in> set_literal x \<Longrightarrow> f z = g z" shows "map_literal f x = map_literal g x"
prime_power: fixes p :: "nat"   and k :: "nat"   and x :: "nat" assumes "prime p"   and "0 < k" shows "divisor_sigma x (p ^ k) = (if x = 0 then Suc k else (p ^ (Suc k * x) - 1) div (p ^ x - 1))"
card_UNIV_char_def: shows "card_UNIV_class.card_UNIV = Phantom(char) 256"
diff_spmat_def: fixes A :: "(nat \<times> (nat \<times> 'a) list) list"   and B :: "(nat \<times> (nat \<times> 'a) list) list" shows "diff_spmat A B = add_spmat A (minus_spmat B)"
cong_sum: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and a :: "'d" assumes "\<And>x. x \<in> A \<Longrightarrow> [f x = g x] (mod a)" shows "[sum f A = sum g A] (mod a)"
induct: fixes P :: "'b \<Rightarrow> (bool \<times> 'b) rexp \<Rightarrow> bool \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "(bool \<times> 'b) rexp"   and a2 :: "bool" assumes "\<And>uu_ uv_. P uu_ One uv_"   and "\<And>uw_ ux_. P uw_ Zero ux_"   and "\<And>c uy_ a m. P c (Atom (uy_, a)) m"   and "\<And>c r s m. \<lbrakk>P c r m; P c s m\<rbrakk> \<Longrightarrow> P c (Plus r s) m"   and "\<And>c r s m. \<lbrakk>P c r m; P c s (final1 r c \<or> m \<and> nullable r)\<rbrakk> \<Longrightarrow> P c (Times r s) m"   and "\<And>c r m. P c r (final1 r c \<or> m) \<Longrightarrow> P c (Star r) m" shows "P a0 a1 a2"
memEqvt: fixes p :: "(name \<times> name) list"   and x :: "'a"   and xs :: "'a list" shows "p \<bullet> x mem xs = p \<bullet> x mem p \<bullet> xs"
stutter_reduced_def: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "\<natural>\<sigma> = \<sigma> \<circ> max_stutter_sampler \<sigma>"
S4Write: fixes p :: "PrIds"   and l :: "Locs"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> Write rmCh mm ires p l \<and> $S4 rmhist p \<and> unchanged (e p, c p, r p) \<and> HNext rmhist p \<longrightarrow> S4 rmhist p$ \<and> unchanged (rmhist!p)"
mk_disjoint_finsert: fixes a :: "'m"   and A :: "'m fset" assumes "a |\<in>| A" shows "\<exists>B. A = finsert a B \<and> a |\<notin>| B"
assms_2: shows "[\<^bold>\<box>\<^bold>\<A>\<phi> in v]"
image_is_empty: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(f ` A = {}) = (A = {})"
prefix_cases: fixes xs :: "'a list"   and ys :: "'a list" obtains "prefix xs ys"   | "strict_prefix ys xs"   | "xs \<parallel> ys"
simps_14: fixes e\<^sub>1 :: "'c exp"   and e\<^sub>2 :: "'c exp" shows "ss_exp (e\<^sub>1;; e\<^sub>2) = (case val_of e\<^sub>1 of None \<Rightarrow> case lass_val_of e\<^sub>1 of None \<Rightarrow> ss_exp e\<^sub>1 | \<lfloor>p\<rfloor> \<Rightarrow> ss_exp e\<^sub>2 | \<lfloor>v\<rfloor> \<Rightarrow> ss_exp e\<^sub>2)"
div_pos_geq: fixes l :: "int"   and k :: "int" assumes "0 < l"   and "l \<le> k" shows "k div l = (k - l) div l + 1"
bigo_sum1: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and A :: "'a \<Rightarrow> 'b set" assumes "\<forall>x y. (0::'c) \<le> h x y"   and "\<exists>c. \<forall>x y. \<bar>f x y\<bar> \<le> c * h x y" shows "(\<lambda>x. sum (f x) (A x)) \<in> O(\<lambda>x. sum (h x) (A x))"
swap_identical_length: fixes i :: "nat"   and j :: "nat"   and t :: "'d list" assumes "i < j"   and "j < length t" shows "length t = length (swap_events i j t)"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>) (\<star>) \<a> \<i> src trg (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B DN cmp\<^sub>D\<^sub>N"
red_uminus: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" shows "red F (- p) (- q)"
assms_1: shows "S \<Turnstile>\<^sub>P eval\<^sub>G \<G> (af\<^sub>G \<phi> (prefix j w))"
assms_2: shows "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
lsl_Sup_closed_var: fixes X :: "'b set" shows "\<nu> (\<Squnion> (\<nu> ` X)) = \<Squnion> (\<nu> ` X)"
z: shows "z \<equiv> mult_rat_1 x (p2, l2, r2)"
scaleR_right: fixes a :: "'a"   and r :: "real"   and b :: "'b" shows "a ** r *\<^sub>R b = r *\<^sub>R (a ** b)"
get_ancestors_si_locs_def: shows "get_ancestors_si_locs = Shadow_DOM.get_parent_locs \<union> get_host_locs"
card_atMost: fixes u :: "nat" shows "card {..u} = Suc u"
det_minor_row': fixes B :: "('c, 'd) sq_matrix"   and i :: "'d"   and j :: "'d" assumes "Square_Matrix.row B i = axis j (1::'c)" shows "Square_Matrix.det (minor B i j) = Square_Matrix.det B"
sup_ereal_def: fixes x :: "ereal"   and y :: "ereal" shows "x \<squnion> y = max x y"
not_numeral_le_neg_numeral: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq (numeral.numeral one plus m) (uminus (numeral.numeral one plus n))"
projection_on_orthogonal_complement: fixes M :: "'b set" assumes "closed_csubspace M" shows "projection (orthogonal_complement M) = id - projection M"
member_subc: fixes p :: "('a, 'b) form"   and z :: "('a, 'b) form list"   and c :: "'a"   and s :: "'a FOL_Fitting.term" assumes "p \<in> set z" shows "subc c s p \<in> set (subcs c s z)"
rightUniqueInjectiveOnFirst: fixes P :: "('c \<times> 'd) set" shows "runiq P = inj_on fst P"
progress_mono: fixes j :: "nat"   and j' :: "nat"   and \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula" assumes "j \<le> j'" shows "Monitor.progress \<sigma> \<phi> j \<le> Monitor.progress \<sigma> \<phi> j'"
random_permutation_def: fixes \<sigma> :: "'c \<Rightarrow> 'b option"   and x :: "'c" shows "random_permutation \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (A - ran \<sigma>) \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
m_comm: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "x \<otimes>\<^bsub>K [X]\<^esub> y = y \<otimes>\<^bsub>K [X]\<^esub> x"
m_assoc: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
asymp_equiv_imp_bigomega: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g" shows "f \<in> \<Omega>[F](g)"
emb_lmap: fixes xs :: "'b llist"   and ys :: "'b llist"   and f :: "'b \<Rightarrow> 'c" assumes "emb xs ys" shows "emb (lmap f xs) (lmap f ys)"
lincomb_closed: fixes S :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "S \<subseteq> carrier V"   and "a \<in> S \<rightarrow> carrier K" shows "V.module.lincomb a S \<in> carrier V"
ac: shows "ideal_generated {a} \<subseteq> ideal_generated {c}"
succ_def: fixes i :: "'a" shows "succ i \<equiv> local.suc {i}"
emeasure_eq_measure: fixes A :: "'a set" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
lookup_nodes_of_phis'_not_subst: fixes v :: "'val"   and g :: "'g"   and "next" :: "'node \<times> 'val"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping" assumes "v \<noteq> uninst_code.substitution_code phis g next" shows "Mapping.lookup (uninst_code.nodes_of_phis' g next (uninst_code.substitution_code phis g next) nodes_of_phis) v = (if v = snd next then None else Mapping.lookup nodes_of_phis v)"
law2: fixes I :: "item set" assumes "wellformed_items I" shows "wellformed_items ((Scan T k \<circ> Complete k \<circ> Predict k) I)"
states_of_se_assign1: fixes c :: "('c, 'd) conf"   and v :: "'c"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and c' :: "('c, 'd) conf"   and \<sigma>' :: "'c \<Rightarrow> 'd" assumes "se c (Assign v e) c'"   and "\<sigma>' \<in> states c'" shows "\<exists>\<sigma>\<in>states c. \<sigma>' = \<sigma>(v := e \<sigma>)"
forest_subsets: fixes H :: "('v, 'w) graph" assumes "valid_graph H"   and "edges H \<subseteq> E"   and "nodes H \<subseteq> V" shows "forest H"
strict_implies_not_eq: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a \<subset># b" shows "a \<noteq> b"
stable_elim: fixes r :: "('a \<times> 'a) set"   and A :: "'b set"   and F :: "'b \<Rightarrow> 'c set" assumes "stable r"   and "|A| <o r"   and "\<And>a. a \<in> A \<Longrightarrow> |F a| <o r" shows "|Sigma A F| <o r"
bij_betw_quotient_domain_and_range_permutation_eq_empty: fixes A :: "'c set"   and B :: "'d set" assumes "card A \<noteq> card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B = {}"
no_call: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and c :: "char list" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   and "rs = [Rule MatchAny (Call c)]"   and "s = Undecided"   and "\<Gamma> c = None" shows "False"
Abs_phi_inverse: fixes y :: "'c \<Rightarrow> 'd Filters.filter" assumes "y \<in> {f. bounded_lattice_homomorphism f}" shows "Rep_phi (Abs_phi y) = y"
le_less_linear: fixes x :: "'a list"   and y :: "'a list" shows "x \<le>lex y \<or> y <lex x"