hcju/isabelleps huggingface dataset:
rbt_lookup_fold_rbt_insert: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and f :: "'b \<Rightarrow> 'c"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.rbt_lookup less (List.fold (\<lambda>k. ord.rbt_insert less k (f k)) xs t) = ord.rbt_lookup less t ++ map_of (map (\<lambda>k. (k, f k)) xs)"
C: shows "sinks_aux I D (the ` U) (map (inv p) xs) = the ` sinks_aux (con_comp_pol I) (con_comp_map D E p q) U xs"
len_one: assumes "length l = 1" shows "\<exists>n<length (a # l) - 1. (a # l) ! n < x \<and> x < (a # l) ! (n + 1)"
Map_ide_1: fixes a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.ide a" shows "A_B.Map a = A_B.Dom a"
bounded_nat_set_is_finite2: fixes N :: "nat set"   and n :: "nat" assumes "\<forall>i\<in>N. i \<le> n" shows "finite N"
distinct_map_nth: fixes ls :: "'b list"   and inds :: "nat list" assumes "distinct ls"   and "distinct inds"   and "\<And>j. j \<in> set inds \<Longrightarrow> j < length ls" shows "distinct (map ((!) ls) inds)"
djointish: shows "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"
splice_in_shuffles: fixes xs :: "'b list"   and ys :: "'b list" shows "splice xs ys \<in> shuffles xs ys"
assms_1: shows "CT;\<Gamma> \<turnstile> e : C"
e_def: shows "e \<equiv> 63 / 10 ^ 6"
span_eq: fixes S :: "'b set"   and T :: "'b set" shows "(vs1.span S = vs1.span T) = (S \<subseteq> vs1.span T \<and> T \<subseteq> vs1.span S)"
thm_noncont_e_e_4: fixes v :: "i" shows "[NonContingent (L\<^sup>-) in v]"
minus_minus: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.group_add minus plus zero uminus" shows "uminus (uminus a) = a"
simps_23: fixes g1 :: "'b gexp"   and g2 :: "'b gexp"   and v :: "'b aexp"   and va :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (Nor g1 g2) (Eq v va) = False"
basis_reduction_add_row_done: fixes i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant True i fs"   and "i < m"   and "\<mu>_small_row i fs 0" shows "LLL_invariant False i fs"
non_consistency: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> .\<not> p"   and "A \<turnstile> p" shows "A \<turnstile> q"
right_neutral: fixes a :: "'b" shows "a \<squnion> bot = a"
eq_id_iff: fixes f :: "'g \<Rightarrow> 'g" shows "(\<forall>x. f x = x) = (f = id)"
is_cparser_def: fixes p :: "char list \<Rightarrow> char list + 'b \<times> char list" shows "is_cparser p = (\<forall>s r x. p s = Inr (x, r) \<longrightarrow> length r < length s)"
totient_lower_bound: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "\<forall>\<^sub>F n in sequentially. (1 - \<epsilon>) * third_mertens_const * real n / ln (ln (real n)) < real (totient n)"
invar_def: fixes E :: "'b set set"   and k :: "nat"   and C :: "'b set"   and F :: "'b set set" assumes "Bounded_Hypergraph E k" shows "Bounded_Hypergraph.invar E k C F = (F \<subseteq> E \<and> vertex_cover (E - F) C \<and> finite C \<and> (\<exists>M. Bounded_Hypergraph.inv_matching E k C F M))"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "partial_magma.in_hom C (Chn \<mu>) (span_in_category.apex C (Dom \<mu>)) (span_in_category.apex C (Cod \<mu>))"   and "C (Leg0 (Cod \<mu>)) (Chn \<mu>) = Leg0 (Dom \<mu>)"   and "C (Leg1 (Cod \<mu>)) (Chn \<mu>) = Leg1 (Dom \<mu>)" shows "arrow_of_spans_axioms C \<mu>"
lunit'_simps_3: fixes f :: "'b" assumes "D.ide f" shows "trg\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[f] = trg\<^sub>D f"
rt_strictly_fresher_irefl: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "\<not> rt \<sqsubset>\<^bsub>dip\<^esub> rt"
of_nat_0_eq_iff: fixes one :: "'f"   and times :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(zero = semiring_1.of_nat one plus zero n) = (0 = n)"
diff_frac_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R"   and "w \<in> carrier R"   and "y \<noteq> \<zero>"   and "z \<noteq> \<zero>" shows "x \<oslash> y \<ominus> w \<oslash> z = (x \<otimes> z \<ominus> w \<otimes> y) \<oslash> (y \<otimes> z)"
real_of_int_le_2_powr_bitlenI: fixes x :: "int"   and m :: "int"   and n :: "real" assumes "bitlen \<bar>x\<bar> \<le> m"   and "real_of_int m \<le> n" shows "real_of_int x \<le> 2 powr n - 1"
ref_not_x: fixes r :: "'a" assumes "ref = Some r" shows "r \<noteq> fst x"
dom_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "A.dom (g \<cdot>\<^sub>A f) = A.dom f"
deduct_restricted_if_synth: fixes P :: "('fun, 'var) Term.term \<Rightarrow> bool"   and m :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set" assumes "P m"   and "\<forall>t t'. P t \<longrightarrow> t' \<sqsubseteq> t \<longrightarrow> P t'"   and "M \<turnstile>\<^sub>c m" shows "\<langle>M;P\<rangle> \<turnstile>\<^sub>r m"
set_plus_mono2_b: fixes C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and F :: "'b set"   and x :: "'b" assumes "C \<subseteq> D"   and "E \<subseteq> F"   and "x \<in> C + E" shows "x \<in> D + F"
s: shows "s = \<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e))"
2: shows "bprv (imp (neg (exi yy (PPf (Var yy) \<langle>\<phi>R\<rangle>))) (neg (exi yy (cnj (PPf (Var yy) \<langle>\<phi>R\<rangle>) F))))"
inv_joinR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invc l"   and "invc r"   and "invh l"   and "invh r"   and "bheight r \<le> bheight l" shows "invc2 (joinR l x r) \<and> (bheight l \<noteq> bheight r \<and> color l = Black \<longrightarrow> invc (joinR l x r)) \<and> invh (joinR l x r) \<and> bheight (joinR l x r) = bheight l"
supset_circuit_imp_dep: fixes C :: "'a set"   and X :: "'a set" assumes "circuit C \<and> C \<subseteq> X" shows "\<not> indep X"
reachable_trans: fixes q :: "nat \<times> 'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a"   and s :: "nat \<times> 'a" assumes "q \<in> greachable A w V p"   and "s \<in> greachable A w V q" shows "s \<in> greachable A w V p"
Singl: fixes trn :: "('state, 'act, 'out) trans" assumes "validTrans trn" shows "valid2 [trn]"
less_le: fixes x :: "'a"   and y :: "'a" shows "lt x y = (le x y \<and> x \<noteq> y)"
heq: fixes x :: "('a \<times> 'b) \<times> real" assumes "x \<in> C" shows "h x = (snd \<circ> f) x"
CONSTRAINT_D: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "CONSTRAINT P x" shows "P x"
basic_just_sp_pred_axioms: shows "basic_just_sp_pred G dist c s num pred"
trg_dom: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "trg\<^sub>C (C.dom \<mu>) = trg\<^sub>C \<mu>"
s: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and e :: "('a \<times> 'a) set"   and m :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and ov :: "('a \<times> 'a) set"   and d :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set"   and p :: "'a"   and q :: "'a" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> s) = (\<exists>k u v. meets k p \<and> meets p u \<and> meets u v \<and> meets k q \<and> meets q v)"
ID1: fixes x :: "'b" shows "ID\<cdot>x = x"
commute: fixes a :: "'c"   and b :: "'c" shows "min a b = min b a"
floor_zero: shows "\<lfloor>0::'b\<rfloor> = 0"
order_iff_strict: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" shows "(a \<preceq> b) = (a \<prec> b \<or> a = b)"
wf: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.wellorder less_eq less" shows "wf {(x, y). less x y}"
sat_imp_red_calc_sat: fixes N :: "'f set" assumes "saturated N" shows "reduc_calc.saturated N"
restrict_chart_carrier: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and S :: "'c set"   and x :: "('c, 'd) chart" assumes "submanifold charts k S"   and "x \<in> charts" shows "restrict_chart (manifold.carrier charts) x = x"
split_sel_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "P (case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> option = \<lfloor>the option\<rfloor> \<and> \<not> P (f2 (the option))))"
card_insert_le: fixes A :: "'e set"   and x :: "'e" assumes "finite A" shows "card A \<le> card (insert x A)"
ded_thm_cor_4: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<chi>) in v]"   and "[\<psi> in v]" shows "[\<phi> \<^bold>\<rightarrow> \<chi> in v]"
simps_19: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Br_if i) ts = (if i < length (label \<C>) then type_update ts (to_ct_list (label \<C> ! i @ [T_i32])) (Type (label \<C> ! i)) else Bot)"
FG_advice_inter: fixes Y :: "'b ltln set"   and \<phi> :: "'b ltln"   and S :: "'b ltln set" assumes "Y \<inter> subformulas\<^sub>\<nu> \<phi> \<subseteq> S" shows "\<phi>[Y \<inter> S]\<^sub>\<Sigma>\<^sub>1 = \<phi>[Y]\<^sub>\<Sigma>\<^sub>1"
snoc_2: shows "(u @ [a]) @ v\<^sub>1 =\<^sub>F (u @ [a]) @ v\<^sub>2"
A: shows "\<exists>y r c. y \<in> X \<and> r \<notin> D y \<and> r \<in> D xa \<and> r @ [c] = x"
countable_Collect: fixes A :: "'b set"   and \<phi> :: "'b \<Rightarrow> bool" assumes "countable A" shows "countable {a \<in> A. \<phi> a}"
accessible_on_imp_mem: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b" assumes "accessible_on P A a" shows "a \<in> A"
characterize_\<P>: fixes p :: "(symbol \<times> character list) list" assumes "\<forall>i<length p. \<exists>u. p ! i \<in> \<Z> (charslength (take i p)) u"   and "admissible p" shows "\<exists>u. p \<in> \<P> (charslength p) u"
replicate_append_same: fixes i :: "nat"   and x :: "'a" shows "replicate i x @ [x] = x # replicate i x"
IfFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> val"   and c1 :: "com"   and c2 :: "com" assumes "tbval b s False" shows "(IF b THEN c1 ELSE c2, s) \<rightarrow> (c2, s)"
reflexive_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "reflexive A r = reflexive A r'"
linear_inj_on_span_iff_independent_image: fixes f :: "'d \<Rightarrow> 'e"   and B :: "'d set" assumes "clinear f"   and "cindependent (f ` B)" shows "inj_on f (cspan B) = inj_on f B"
cos_one_sin_zero: fixes x :: "'a" assumes "cos x = (1::'a)" shows "sin x = (0::'a)"
\<alpha>_limit_point_def: fixes x :: "'a"   and p :: "'a" shows "\<alpha>_limit_point x p = ({..0} \<subseteq> existence_ivl0 x \<and> (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity> \<and> (flow0 x \<circ> s) \<longlonglongrightarrow> p))"
flush_all_until_volatile_write_append_volatile_write_commute: fixes sb :: "'b memref list"   and i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and p' :: "'b"   and is' :: "instr list"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and \<D>' :: "'c"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', \<theta>, sb @ [Write\<^sub>s\<^sub>b True a sop v A L R W], \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
\<R>\<^sub>1_sym: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option" shows "sym (\<R>\<^sub>1 \<Gamma>)"
arctan_le_zero_iff: fixes x :: "real" shows "(arctan x \<le> 0) = (x \<le> 0)"
ine_exprChannelSingle_def: fixes sP :: "specID"   and ch :: "chanID"   and E :: "Expression" shows "ine_exprChannelSingle sP ch E \<equiv> ch \<in> ins sP \<and> exprChannel ch E \<and> (\<forall>x t. x \<in> ins sP \<and> x \<noteq> ch \<longrightarrow> \<not> exprChannel x E)"
assms: shows "x \<notin> \<real>\<^sub>\<le>\<^sub>0"
arr_member_of_Base: shows "tprg \<turnstile> (Base, fdecl (arr, \<lparr>access = Public, static = True, type = PrimT Boolean.[]\<rparr>)) member_of Base"
monom_a_b: fixes m' :: "nat"   and b :: "nat"   and a :: "'a mod_ring" assumes "monom 1 1 \<in> carrier R"   and "monom 1 1 [^]\<^bsub>R\<^esub> CARD('a) ^ m' = monom 1 1"   and "b < degree f" shows "monom a b [^]\<^bsub>R\<^esub> CARD('a) ^ m' = monom a b"
card_partition: fixes C :: "'b set set"   and k :: "nat" assumes "finite C"   and "finite (\<Union> C)"   and "\<And>c. c \<in> C \<Longrightarrow> card c = k"   and "\<And>c1 c2. \<lbrakk>c1 \<in> C; c2 \<in> C; c1 \<noteq> c2\<rbrakk> \<Longrightarrow> c1 \<inter> c2 = {}" shows "k * card C = card (\<Union> C)"
fimage_vsubset: fixes r :: "V"   and A :: "V"   and B :: "V"   and C :: "V" assumes "r \<subseteq>\<^sub>\<circ> A \<times>\<^sub>\<bullet> B" shows "r `\<^sub>\<bullet> C \<subseteq>\<^sub>\<circ> B"
Minimal_Truth_Soundness_Proof_Repr_axioms: shows "Minimal_Truth_Soundness_Proof_Repr var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv bprv isTrue enc proof prfOf encPf Pf"
noStuckn_def': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Stuck} = (\<not> \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Stuck)"
trm_ord_ground_total: shows "\<forall>x y. ground_term x \<longrightarrow> ground_term y \<longrightarrow> x \<noteq> y \<longrightarrow> (x, y) \<in> trm_ord \<or> (y, x) \<in> trm_ord"
k_reachable: fixes a :: "'a" shows "simAbs ` KBP.KBPAlg.reachable a (set (k_frontier a)) = sim_equiv_class a ` jkbpC"
atLeastatMost_empty_iff2: fixes a :: "'a"   and b :: "'a" shows "({} = {a..b}) = (\<not> a \<le> b)"
max_ipv6_addr_number: shows "max_ipv6_addr = 340282366920938463463374607431768211455"
l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf set_attribute set_attribute_locs"
lab_def_1: shows "\<S>\<bullet>\<C> \<turnstile> vs : t1s _> ts'"
preorder_idempotent: fixes x :: "'a" assumes "top \<le> x \<and> x \<sqinter> x \<le> x" shows "x \<sqinter> x = x"
intros_1: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "(F::'b formula) \<in> (\<Gamma>::'b formula set)" shows "(\<Gamma>::'b formula set) \<turnstile>\<^sub>H (F::'b formula)"
fwi_invariant_aux_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and j'' :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "j'' \<le> j" shows "less_eq (linordered_ab_monoid_add.fwi plus less_eq m n k i j i' j') (linordered_ab_monoid_add.fwi plus less_eq m n k i j'' i' j')"
SeqForm_def: fixes s :: "hf"   and k :: "hf"   and y :: "hf" shows "SeqForm s k y \<equiv> BuildSeq Atomic MakeForm s k y"
distinct_length_2_or_more: fixes a :: "'d"   and b :: "'d"   and xs :: "'d list" shows "distinct (a # b # xs) = (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
spr_ii_sim_r: shows "sim_r SPR.MC spr_ii_simMC spr_ii_sim"
linear_span_iOp_closedTr1: fixes A :: "'b set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "ideal R A"   and "s \<in> {j. j \<le> n} \<rightarrow> A" shows "(\<lambda>x\<in>{j. j \<le> n}. -\<^sub>a\<^bsub>R\<^esub> s x) \<in> {j. j \<le> n} \<rightarrow> A"
T1: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<P> x \<^bold>\<rightarrow> \<^bold>\<diamond>mexistsActB x) w)\<rfloor>"
rep_eq: fixes x :: "'d cset" shows "cBall x = Ball (rcset x)"
UN_simps_5: fixes A :: "'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "(\<Union>x\<in>C. A \<inter> B x) = A \<inter> \<Union> (B ` C)"
prems_9: shows "\<nexists>q. \<psi> = prop\<^sub>r(q) \<or> \<psi> = nprop\<^sub>r(q)"
ts_prop_1: fixes id :: "'bid"   and k :: "cnf" assumes "\<parallel>id\<parallel>\<^bsub>k\<^esub>" shows "id = the_bb"
vmn: shows "V \<subseteq> T (m \<rightarrow> n) \<eta>"
coplanar_perm_4: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D" shows "Coplanar A D B C"
P_def: shows "P = (\<lambda>D A. fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {} \<and> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {})"
zero: fixes i :: "nat" assumes "i < dim_col A" shows "(mat_adjoint A * A) $$ (i, i) \<noteq> (0::'a)"
sg2: shows "st (t + d) = hd (ack (t + d))"
moebius_similarity_only_inf_to_inf: fixes a :: "complex"   and b :: "complex"   and z :: "complex_homo" assumes "a \<noteq> 0"   and "moebius_pt (moebius_similarity a b) z = \<infinity>\<^sub>h" shows "z = \<infinity>\<^sub>h"
support_in_outcomes: shows "\<forall>oc. \<forall>p\<in>\<P> oc. \<forall>a\<in>set_pmf p. a \<in> oc"
roots_mem_iff_is_root: fixes p :: "'a list"   and x :: "'a" assumes "p \<in> carrier (poly_ring R)" shows "(x \<in># roots p) = is_root p x"
C: shows "\<forall>xa. (z xa \<and> u (Suc xa) = u xa + 1 \<or> \<not> z xa \<and> u (Suc xa) = u xa - 1 \<or> u (Suc xa) = 0) \<and> x xa = (u (Suc xa) = 0)"
lock_thread_ok_has_lockE: fixes ls :: "('a, 'b) locks"   and ts :: "'b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option"   and l :: "'a"   and t :: "'b" assumes "lock_thread_ok ls ts"   and "has_lock (ls $ l) t" obtains x :: "'c" and ln' :: "'a \<Rightarrow>f nat" where "ts t = \<lfloor>(x, ln')\<rfloor>"
ipresCons_imp_ipresFreshAll: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'b, 'c, 'd) model_scheme" assumes "ipresCons h hA MOD"   and "igFreshCls MOD"   and "igConsIPresIGWls MOD" shows "ipresFreshAll h hA MOD"
inject: fixes x :: "nat"   and ya :: "nat" shows "(X x = X ya) = (x = ya)"
rctxt_labels: fixes E :: "trm \<Rightarrow> (char list \<times> trm) list"   and t :: "trm"   and l :: "char list"   and t' :: "trm" assumes "E \<in> rctxt"   and "E t\<langle>l\<rangle>\<^sub>? = \<bottom>" shows "E t'\<langle>l\<rangle>\<^sub>? = \<bottom>"
obj_src: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "B.obj (src\<^sub>B \<mu>)"
fail_nondetT_def: shows "fail = nondetM_base.fail_nondet Monad_Overloading.return {#}"
Collect_empty_eq_bot: fixes P :: "'a \<Rightarrow> bool" shows "(Collect P = {}) = (P = bot)"
connected_iff_seq: 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 \<mu> :: "'b"   and \<nu> :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>" shows "(weak_composition.sources V H \<nu> \<inter> weak_composition.targets V H \<mu> \<noteq> {}) = (src \<nu> = trg \<mu>)"
finite_\<alpha>_limit_set_in_compact_imp_unique_fixed_point: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_backward x K"   and "finite (\<alpha>_limit_set x)" obtains y :: "'a" where "\<alpha>_limit_set x = {y}" and "f y = (0::'a)"
uminus_sum_list_order2: fixes ss :: "'a list" assumes "\<forall>s\<in>set ss. s + s = (0::'a)" shows "- sum_list ss = sum_list (rev ss)"
diff_fun_space_times: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> diff_fun_space"   and "g \<in> diff_fun_space" shows "f * g \<in> diff_fun_space"
uint64_simps_2: fixes x :: "uint64"   and y :: "uint64" shows "int_of_uint64 (x + y) = (int_of_uint64 x + int_of_uint64 y) mod 18446744073709551616"
Universal_def: shows "TAO_8_Definitions.Universal \<equiv> \<lambda>x. \<lparr>A!,x\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x,F\<rbrace>)"
OpCatCatAx: fixes C :: "('d, 'e, 'f) Category_scheme" assumes "Category_axioms C" shows "Category_axioms (Op C)"
CONS_subseteq_Lim: shows "CONS \<subseteq> LIM"
sep_add_cancelD: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x + z = y + z"   and "x ## z"   and "y ## z" shows "x = y"
zip_append1: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'b list" shows "zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
simps_7: fixes t :: "'c local.term" shows "Cod \<^bold>\<r>\<^bold>[t\<^bold>] = Cod t"
assms_6: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and D :: "nat"   and C :: "nat" assumes "CT;\<Gamma> \<turnstile> e0 : D"   and "P CT \<Gamma> e0 D"   and "CT \<turnstile> D <: C" shows "P CT \<Gamma> (Cast C e0) C"
lborel_affine_euclidean: fixes c :: "'a \<Rightarrow> real"   and t :: "'a" assumes "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0" shows "lborel = density (distr lborel borel (\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j))) (\<lambda>x. ennreal (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
init_fin_\<tau>diverge_RunningI: fixes r :: "('e, 'f, 'c, 'd, 'g, 'h) semantics"   and t :: "'f"   and \<tau>move :: "('e, 'f, 'c, 'd, 'g, 'h) \<tau>moves"   and x :: "'c"   and m :: "'d"   and final :: "'c \<Rightarrow> bool" assumes "\<tau>trsys.\<tau>diverge (r t) \<tau>move (x, m)" shows "\<tau>trsys.\<tau>diverge (multithreaded_base.init_fin final r t) (\<tau>multithreaded.init_fin_\<tau>move \<tau>move) ((Running, x), m)"
LIMSEQ_le_const: fixes X :: "nat \<Rightarrow> 'e"   and x :: "'e"   and a :: "'e" assumes "X \<longlonglongrightarrow> x"   and "\<exists>N. \<forall>n\<ge>N. a \<le> X n" shows "a \<le> x"
additive_principal_def: fixes a :: "ordinal" shows "additive_principal a \<equiv> 0 < a \<and> (\<forall>b<a. b + a = a)"
assms_4: fixes n :: "nat" shows "f n \<in> space\<^sub>N (\<LL> p M)"
p2: shows "\<forall>t>0. nth_derivative k f t 1 = p t / t ^ (2 * k) * exp (- inverse t)"
produce_weaken_type: fixes tn :: "t list"   and cons :: "checker_type"   and tm :: "t list"   and ts :: "t list" assumes "produce (Type tn) cons = Type tm" shows "produce (Type (ts @ tn)) cons = Type (ts @ tm)"
design_defs_6: fixes P :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" shows "H2 P \<equiv> P ;; J"
assms: fixes i :: "'a" assumes "i \<in> I" shows "convex (S i)"
bind_tell: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and tell :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and w :: "'f"   and m :: "'e"   and f :: "'d \<Rightarrow> 'e" assumes "monad_writer return bind tell" shows "bind (tell w m) f = tell w (bind m f)"
DAiniD: fixes p :: "('a, 'b) Combinators list" shows "DenyAll \<in> set (insertDeny p)"
set_mult_ker_hom_2: fixes I :: "'a set" assumes "I \<subseteq> carrier B" shows "g ` (kernel B C g <#>\<^bsub>B\<^esub> I) = g ` I"
method_def: fixes C :: "cname" shows "method C \<equiv> class_rec C methods"