hcju/isabelleps huggingface dataset:
simps: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "exp0 list"   and a5 :: "'b state \<times> (v list, v) result" shows "evaluate_list a1 a2 a3 a4 a5 = ((\<exists>ck env s. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = [] \<and> a5 = (s, Rval [])) \<or> (\<exists>ck env e es v1 vs s1 s2 s3. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = e # es \<and> a5 = (s3, Rval (v1 # vs)) \<and> evaluate ck env s1 e (s2, Rval v1) \<and> evaluate_list ck env s2 es (s3, Rval vs)) \<or> (\<exists>ck env e es err s s'. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = e # es \<and> a5 = (s', Rerr err) \<and> evaluate ck env s e (s', Rerr err)) \<or> (\<exists>ck env e es v1 err s1 s2 s3. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = e # es \<and> a5 = (s3, Rerr err) \<and> evaluate ck env s1 e (s2, Rval v1) \<and> evaluate_list ck env s2 es (s3, Rerr err)))"
A_2: shows "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c"
mat_of_col_dim_1: fixes v :: "'b vec" shows "dim_row (mat_of_col v) = dim_vec v"
add_less_cancel_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "(c + a < c + b) = (a < b)"
sets_distr: fixes M :: "'d measure"   and N :: "'c measure"   and f :: "'d \<Rightarrow> 'c" shows "sets (distr M N f) = sets N"
seqI': fixes f :: "'d"   and a :: "'d"   and b :: "'d"   and g :: "'d"   and c :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>D c\<guillemotright>" shows "D.seq g f"
out2_bet_out: fixes B :: "'p"   and A :: "'p"   and C :: "'p"   and X :: "'p"   and P :: "'p" assumes "B Out A C"   and "B Out X P"   and "Bet A X C" shows "B Out A P \<and> B Out C P"
sorted_Cons_imp_not_less: fixes cmp :: "'b comparator"   and y :: "'b"   and xs :: "'b list"   and x :: "'b" assumes "Sorting_Algorithms.sorted cmp (y # xs)"   and "x \<in> set xs" shows "compare cmp y x \<noteq> Greater"
assms_1: shows "isDERIV x (fold_const_fa fa) vs"
mult_column_row_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and n :: "'e"   and q :: "'d"   and i :: "'f" shows "mult_column_row A n q i = (\<chi>j. if j = n then A $ i $ j * q else A $ i $ j)"
iprev_mono: fixes n :: "nat"   and I :: "nat set" shows "iprev n I \<le> n"
f5: shows "\<not> (\<exists>(a', b', c')\<in>set b. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
toplevel_summands_deriv: fixes as :: "'b"   and r :: "'b rexp" shows "toplevel_summands (deriv as r) = (\<Union>s\<in>toplevel_summands r. toplevel_summands (deriv as s))"
pdevs_val_filter: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and p :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "e \<in> UNIV \<rightarrow> I"   and "0 \<in> I" obtains e' :: "nat \<Rightarrow> real" where "pdevs_val e (pdevs_of_list (filter p xs)) = pdevs_val e' (pdevs_of_list xs)" and "e' \<in> UNIV \<rightarrow> I"
sorted_domain_raw_def: fixes rep_key_order :: "'b \<Rightarrow> 'a key_order"   and ko :: "'b"   and xs :: "('a \<times> 'c) list \<times> 'b" shows "oalist_raw.sorted_domain_raw rep_key_order ko xs = map fst (oalist_raw.sort_oalist_aux rep_key_order ko xs)"
mono: fixes callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g, 'h, 'i) gpv"   and x :: "('d, 'e, 'f) gpv \<times> 'g" shows "spmf.mono_body (\<lambda>inline1. case x of (gpv, s) \<Rightarrow> the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). curry inline1 (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv)))))"
norm_stack_map: fixes g :: "'a \<Rightarrow> ('dyn, 'ubx1, 'ubx2) unboxed"   and xs :: "'a list" shows "norm_stack (map g xs) = map (Subx.norm_unboxed \<circ> g) xs"
limit: shows "(\<lambda>b. (\<Sum>k = a..b. f (real k)) - F (real b) - (\<Sum>i<2 * N + 1. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real b))) \<longlonglongrightarrow> C"
projection_on_orthogonal_complement: fixes M :: "'a set" assumes "closed_csubspace M" shows "projection (orthogonal_complement M) = id - projection M"
all_consts_def: shows "all_consts = heads |\<union>| C"
welltyped_Var: fixes x :: "char list"   and T :: "Unary_PCF.type" shows "\<langle>(x, T)\<rangle> ::: T"
dd0: shows "DD0 \<in> {DD. set_mset DD \<subseteq> N \<and> (\<forall>I. I \<Turnstile>m DD + CC \<longrightarrow> I \<Turnstile> E) \<and> (\<forall>D'. D' \<in># DD \<longrightarrow> D' < D)}"
le_numeral_extra_3: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero zero"
iconfig_def: fixes tr :: "'b list \<times> 'c" shows "iconfig tr \<equiv> hd (fst tr)"
fun_rel_def: fixes A :: "('e \<times> 'g) set"   and B :: "('f \<times> 'h) set" shows "A \<rightarrow> B \<equiv> {(f, f'). \<forall>(a, a')\<in>A. (f a, f' a') \<in> B}"
sees_field_i_i_i_o_i_def: shows "sees_field_i_i_i_o_i \<equiv> \<lambda>x xa xb xd. pred.Pred (\<lambda>xc. x \<turnstile> xa sees xb:xc in xd)"
R2_H2_commute2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R2 (H2 P) = H2 (R2 P)"
enum_all_unit_def: fixes P :: "unit \<Rightarrow> bool" shows "enum_all P = P ()"
smult_eq_iff: fixes b :: "'a"   and a :: "'a"   and p :: "'a poly"   and q :: "'a poly" assumes "b \<noteq> (0::'a)" shows "(Polynomial.smult a p = Polynomial.smult b q) = (Polynomial.smult (a / b) p = q)"
WT_resourceD: fixes \<I> :: "('h, 'i) \<I>"   and res :: "('h, 'i) resource"   and q :: "'h"   and r :: "'i"   and res' :: "('h, 'i) resource" assumes "\<I> \<turnstile>res res \<surd>"   and "q \<in> outs_\<I> \<I>"   and "(r, res') \<in> set_spmf (run_resource res q)" shows "r \<in> responses_\<I> \<I> q \<and> \<I> \<turnstile>res res' \<surd>"
sources_aux_append_nil: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and ys :: "'b list"   and xs :: "'b list" assumes "sources_aux I D U ys = U" shows "sources_aux I D U (xs @ ys) = sources_aux I D U xs"
cases: fixes x :: "'a word \<times> 'a word" obtains a :: "'a word" and b :: "'a word" where "x = (a, b)"
HFun_Sigma_E: fixes H :: "fm set"   and r :: "tm"   and a :: "tm"   and b :: "tm"   and b' :: "tm" assumes "H \<turnstile> HFun_Sigma r"   and "H \<turnstile> HPair a b IN r"   and "H \<turnstile> HPair a b' IN r" shows "H \<turnstile> b EQ b'"
connected_induction: fixes S :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "connected S"   and "\<And>T a. \<lbrakk>openin (top_of_set S) T; a \<in> T\<rbrakk> \<Longrightarrow> \<exists>z. z \<in> T \<and> P z"   and "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. P x \<and> P y \<and> Q x \<longrightarrow> Q y)"   and "a \<in> S"   and "b \<in> S"   and "P a"   and "P b"   and "Q a" shows "Q b"
det30_plus_scaled3: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and x :: "real" shows "coll 0 a (b + x *\<^sub>R a) = coll 0 a b"
P3: shows "Bet A C' B' \<and> Cong C' B' C B"
harm_def: fixes n :: "nat" shows "harm n = (\<Sum>k = 1..n. inverse (of_nat k))"
If: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and P' :: "'b assn"   and c1 :: "stmt"   and c2 :: "stmt"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} e-\<succ> {P'}"   and "\<forall>b. G,A\<turnstile>{P'\<leftarrow>=b} .(if b then c1 else c2). {Q}" shows "G,A\<turnstile>{Normal P} .If(e) c1 Else c2. {Q}"
fps_divide_X_power: fixes a :: "'a fps"   and n :: "nat" shows "a / fps_X ^ n = fps_shift n a"
thetaParZOL1_converse_ZOretr: shows "thetaParZOL1\<inverse> \<subseteq> ZOretr (thetaParZOL1\<inverse> \<union> ZObis)"
swap_app_Cons: shows "filter p xs @ x # filter (\<lambda>a. \<not> p a) xs <~~> x # filter p xs @ filter (\<lambda>a. \<not> p a) xs"
int_pow_pow: fixes x :: "nat \<Rightarrow> int"   and m :: "int"   and n :: "int" assumes "x \<in> carrier Zp" shows "[m] \<cdot> ([n] \<cdot> x) = [(n * m)] \<cdot> x"
list_append_eq_Cons_cases: fixes ys :: "'a list"   and zs :: "'a list"   and x :: "'a"   and xs :: "'a list" assumes "ys @ zs = x # xs" obtains "ys = []" and "zs = x # xs"   | ys' :: "'a list" where "ys = x # ys'" and "ys' @ zs = xs"
coboundedI1: fixes c :: "'c"   and a :: "'c"   and b :: "'c" assumes "c \<le> a" shows "c \<le> a \<squnion> b"
point_rep_non_existance: fixes x :: "'a" assumes "x \<notin> \<V>" shows "\<B> rep x = 0"
lebesgue_affine_measurable: fixes c :: "'a \<Rightarrow> real"   and t :: "'a" assumes "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0" shows "(\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)) \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
finishedL_def: fixes cl :: "('d, 'e, 'f) cmd list" shows "finishedL cl \<equiv> \<forall>c\<in>set cl. finished c"
ps_chains2_def: fixes S :: "node fset"   and G :: "diagram" shows "ps_chains2 S G \<equiv> mk_ps_chain \<lbrace> initial_ps2 S G \<rbrace> ` lins2 S G"
vec_smult0: fixes as :: "'b list" shows "vec_smult (0::'b) as = vec0 (length as)"
Re_less_pihalf_lemma: fixes z :: "complex" assumes "\<bar>Re z\<bar> < pi / 2" shows "0 < Re ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2)"
Pure: fixes x :: "'d"   and gpv :: "('d, 'e, 'f) gpv"   and \<Gamma> :: "('e, 'f) \<I>" assumes "Pure x \<in> set_spmf (the_gpv gpv)" shows "resultsp_gpv \<Gamma> x gpv"
complex_alt_factorization_exists: fixes p :: "complex poly" assumes "coeff p 0 \<noteq> 0" shows "\<exists>fctrs. is_alt_factorization_of fctrs p"
if_then_else_def: fixes p :: "'a"   and x :: "'a"   and y :: "'a" shows "if p then x else y fi = tp \<cdot> x + at p \<cdot> y"
simps_2: fixes t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and a :: "'b"   and e :: "exp" shows "lift_transform t (Iup a) e = t a e"
normalize_eq_imp_smult: fixes p :: "'b poly"   and q :: "'b poly" assumes "normalize p = normalize q" shows "\<exists>c. c \<noteq> (0::'b) \<and> q = Polynomial.smult c p"
spmf_pred_map: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf" shows "pred_spmf P (map_spmf f p) = pred_spmf (P \<circ> f) p"
is_operator_applicable_in_def: fixes s :: "'a \<Rightarrow> 'b option"   and op :: "('a, 'b) sas_plus_operator" shows "SAS_Plus_Representation.is_operator_applicable_in s op \<equiv> Map.map_of (sas_plus_operator.precondition_of op) \<subseteq>\<^sub>m s"
simps_2: fixes out :: "'g"   and c :: "'h" shows "dest_IO (IO out c) = Some (out, c)"
decomposition_clause_pos_neg: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "C = propositional_atoms.negative_part C \<union> propositional_atoms.positive_part C"
Min_le: fixes A :: "'c Word.word set"   and x :: "'c Word.word" assumes "finite A"   and "x \<in> A" shows "signed.Min A \<le>s x"
game01: shows "run_gpv hash.oracle game0 Map.empty = map_spmf fst (sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> game1 x))"
valof_denormal_of_Float: fixes x :: "Float.float" assumes "is_denormal_Float TYPE(('a, 'b) IEEE.float) x" shows "valof (denormal_of_Float x) = real_of_float x"
bdd_below_uminus: fixes X :: "'a set" shows "bdd_below (uminus ` X) = bdd_above X"
integrable_diff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. f x - g x)"
fold_congs_8: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "nat \<Rightarrow> Literal option"   and f :: "(nat \<Rightarrow> Literal option) \<Rightarrow> nat \<Rightarrow> Literal option"   and f' :: "(nat \<Rightarrow> Literal option) \<Rightarrow> nat \<Rightarrow> Literal option" assumes "r = r'"   and "getWatch1 r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "getWatch1_update f r = getWatch1_update f' r'"
finsum_insert: fixes F :: "'f set"   and a :: "'f"   and f :: "'f \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier R"   and "f a \<in> carrier R" shows "finsum R f (insert a F) = f a \<oplus> finsum R f F"
Hintikka_def: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set" shows "Hintikka A H \<equiv> (((\<forall>i j x. Nom j at i in' H \<longrightarrow> Pro x at j in' H \<longrightarrow> \<not> (\<^bold>\<not> Pro x) at i in' H) \<and> (\<forall>i a. Nom a at i in' H \<longrightarrow> \<not> (\<^bold>\<not> Nom a) at i in' H)) \<and> (\<forall>i p. (\<^bold>\<not> \<^bold>\<not> p) at i in' H \<longrightarrow> p at i in' H) \<and> (\<forall>i p q. (p \<^bold>\<or> q) at i in' H \<longrightarrow> p at i in' H \<or> q at i in' H) \<and> (\<forall>i p q. (\<^bold>\<not> (p \<^bold>\<or> q)) at i in' H \<longrightarrow> (\<^bold>\<not> p) at i in' H \<and> (\<^bold>\<not> q) at i in' H)) \<and> ((\<forall>p i. (\<nexists>a. p = Nom a) \<longrightarrow> (\<^bold>\<diamond> p) at i in' H \<longrightarrow> (\<exists>j. (\<^bold>\<diamond> Nom j) at i in' H \<and> (\<^bold>@ j p) at i in' H)) \<and> (\<forall>i p j. (\<^bold>\<not> (\<^bold>\<diamond> p)) at i in' H \<longrightarrow> (\<^bold>\<diamond> Nom j) at i in' H \<longrightarrow> (\<^bold>\<not> (\<^bold>@ j p)) at i in' H) \<and> (\<forall>a i p. (\<^bold>@ i p) at a in' H \<longrightarrow> p at i in' H)) \<and> (\<forall>a i p. (\<^bold>\<not> (\<^bold>@ i p)) at a in' H \<longrightarrow> (\<^bold>\<not> p) at i in' H) \<and> (\<forall>i p. i \<in> nominals p \<longrightarrow> (\<exists>a. p at a in' H) \<longrightarrow> (\<exists>ps. (ps, i) \<in> H)) \<and> (\<forall>p i j. (\<forall>a. p = Nom a \<or> p = (\<^bold>\<diamond> Nom a) \<longrightarrow> a \<in> A) \<longrightarrow> p at i in' H \<longrightarrow> Nom j at i in' H \<longrightarrow> p at j in' H)"
trans: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "a \<subseteq># b"   and "b \<subseteq># c" shows "a \<subseteq># c"
derives1_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.derives1 \<NN> \<TT> \<RR> u v = (\<exists>x y N \<alpha>. u = x @ [N] @ y \<and> v = x @ \<alpha> @ y \<and> CFG.is_sentence \<NN> \<TT> x \<and> CFG.is_sentence \<NN> \<TT> y \<and> (N, \<alpha>) \<in> \<RR>)"
integral_mono: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x" shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
that: fixes ub :: "real" assumes "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> x < ub"   and "\<forall>x\<ge>ub. \<forall>p\<in>set ps. sgn (poly p x) = sgn_pos_inf p"   and "a < ub" shows "thesis"
IntE: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
diff_numeral_simps_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (numeral.numeral one plus m) (numeral.numeral one plus n) = neg_numeral.sub minus plus one m n"
card_permutations_of_multiset_aux: fixes A :: "'b multiset" shows "card (permutations_of_multiset A) * (\<Prod>x\<in>set_mset A. fact (count A x)) = fact (size A)"
negligible_linear_singular_image: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "linear f"   and "\<not> inj f" shows "negligible (f ` S)"
openin_closedin_eq: fixes U :: "'c topology"   and S :: "'c set" shows "openin U S = (S \<subseteq> topspace U \<and> closedin U (topspace U - S))"
mult_on_union: fixes M :: "'b multiset"   and N :: "'b multiset"   and r :: "('b \<times> 'b) set"   and K :: "'b multiset" assumes "(M, N) \<in> mult r" shows "(K + M, K + N) \<in> mult r"
WTrt_elim_cases_9: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'c expr"   and i :: "'c expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (a\<lfloor>i\<rceil>) T" obtains "J_heap_base.WTrt typeof_addr P h E a (T\<lfloor>\<rceil>)" and "J_heap_base.WTrt typeof_addr P h E i Integer"   | "J_heap_base.WTrt typeof_addr P h E a NT" and "J_heap_base.WTrt typeof_addr P h E i Integer"
LtKeys_payload: fixes NI :: "msg set" assumes "NI \<subseteq> payload" shows "NI \<inter> range LtK = {}"
p_not_div_choose: fixes k :: "nat"   and K :: "nat"   and n :: "nat"   and p :: "nat" assumes "k < K"   and "k \<le> n"   and "\<And>j. \<lbrakk>0 < j; j < K\<rbrakk> \<Longrightarrow> multiplicity p (n - k + (K - j)) = multiplicity p (K - j)"   and "prime p" shows "multiplicity p (n choose k) = 0"
prems_2: shows "insert_seq (Some ref) (xs @ [x])"
HFiniteD: fixes x :: "'a star" assumes "x \<in> HFinite" shows "\<exists>t\<in>\<real>. hnorm x < t"
bool_update_inj: fixes X :: "'a set"   and v :: "'a"   and b :: "bool" assumes "finite X"   and "v \<notin> X" shows "inj_on (fmupd v b) {s. fmdom' s = X}"
sel_3: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real" shows "Im2 (Quat x1 x2 x3 x4) = x3"
llist_less_le_not_le: fixes s :: "'b llist"   and t :: "'b llist" shows "(s < t) = (s \<le> t \<and> \<not> t \<le> s)"
Abs_myopt_inverse: fixes y :: "'d set" assumes "y \<in> {X. finite X \<and> card X \<le> 1}" shows "Rep_myopt (Abs_myopt y) = y"
that: shows "\<forall>x\<in>set_spmf p. S (return_spmf (f x)) (return_spmf (g x))"
fold_insert_remove: fixes A :: "'v set"   and X :: "'v \<Rightarrow> int pmf"   and x :: "'v"   and z :: "'m" assumes "finite A" shows "sample_vars X (insert x A) z = bind (memo (sample_var X) x) (\<lambda>_. sample_vars X (A - {x}) z)"
basis_iff_rank_of: fixes X :: "'a set" assumes "X \<subseteq> \<E>" shows "\<E>.basis X = (\<E>.rank_of X = card X \<and> \<E>.rank_of X = \<E>.rank_of \<E>)"
supporters_nonagent2: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and y :: "'alt"   and x :: "'alt" assumes "is_pref_profile R"   and "y \<notin> alts" shows "supporters R x y = {}"
Tstruct_def: fixes intT :: "'a \<Rightarrow> univ \<Rightarrow> bool" shows "CM.Tstruct intT \<equiv> M.Tstruct intT"
simps: fixes a1 :: "'c llist"   and a2 :: "'c llist" shows "lprefix a1 a2 = ((\<exists>xs. a1 = LNil \<and> a2 = xs) \<or> (\<exists>xs ys x. a1 = LCons x xs \<and> a2 = LCons x ys \<and> lprefix xs ys))"
osum_Well_order: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "Well_order (r +o r')"
dist_triangle2: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.metric_space dist uniformity open" shows "dist x y \<le> dist x z + dist y z"
empty_filter: fixes T :: "'b set set" shows "{} \<in> Filters\<^bsub>T\<^esub>"
dres_ne_bot_basic_5: fixes m1 :: "'a dres"   and m2 :: "'a dres"   and b :: "bool" assumes "m1 \<noteq> dSUCCEED"   and "m2 \<noteq> dSUCCEED" shows "(if b then m1 else m2) \<noteq> dSUCCEED"
image_of_unitor_3: fixes g :: "'a" assumes "ide g" shows "S.UP \<l>\<^sup>-\<^sup>1[g] = S.cmp\<^sub>U\<^sub>P (trg g, g) \<cdot>\<^sub>S (UP.unit (trg g) \<star>\<^sub>S S.UP g)"
Pair_fst_Unity: fixes x :: "'a \<times> unit" shows "(fst x, ()) = x"
numadd_nz: fixes a :: "MIR.num"   and b :: "MIR.num" assumes "nozerocoeff a"   and "nozerocoeff b" shows "nozerocoeff (numadd (a, b))"
assms: fixes s :: "'a" assumes "s \<in> S" shows "\<exists>t\<in>T. d s \<le> d t"
leD: fixes x :: "('d, 'e) Term.term"   and y :: "('d, 'e) Term.term" assumes "x \<unrhd> y" shows "(y, x) \<notin> {\<rhd>}"
pairwise_compat: fixes c1 :: "('c, 'c) chart"   and c2 :: "('c, 'c) chart"   and k :: "enat" assumes "c1 \<in> charts_eucl"   and "c2 \<in> charts_eucl" shows "k-smooth_compat c1 c2"
fmrestrict_set_add_distrib: fixes A :: "'c set"   and m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "fmrestrict_set A (m ++\<^sub>f n) = fmrestrict_set A m ++\<^sub>f fmrestrict_set A n"
lm038: fixes a :: "('c \<times> 'd set) set"   and N :: "'c set" assumes "a \<in> allAllocations N {}" shows "pseudoAllocation a = {}"
table_of_mapf_SomeD: fixes f :: "'c \<Rightarrow> 'a"   and t :: "('b \<times> 'c) list"   and k :: "'b"   and z :: "'a" assumes "table_of (map (\<lambda>(k, x). (k, f x)) t) k = Some z" shows "? y:table_of t k: z = f y"
variation_cases_2: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "y < 0" shows "variation x y = - 1"
\<S>_of_mds_def: fixes mds :: "Mode \<Rightarrow> Eg1.var set" shows "\<S>_of_mds mds \<equiv> (mds AsmNoWrite, mds AsmNoReadOrWrite)"
absorb2: fixes b :: "'d upred"   and a :: "'d upred" assumes "a \<sqsubseteq> b" shows "(a \<and> b) = b"
adds_term_def: fixes u :: "'a"   and v :: "'a" shows "punit.adds_term u v = (punit.component_of_term u = punit.component_of_term v \<and> punit.pp_of_term u adds punit.pp_of_term v)"
inverse_transformation_axioms: shows "inverse_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<tau>"
induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree"   and a1 :: "'b"   and a2 :: "('b \<times> nat) tree" assumes "\<And>A a B b uu_ C. P A a \<langle>B, (b, uu_), C\<rangle>"   and "\<And>a b. P a b \<langle>\<rangle>" shows "P a0 a1 a2"
inject_1: fixes x1 :: "char list"   and y1 :: "char list" shows "(Forward x1 = Forward y1) = (x1 = y1)"
succs: fixes g' :: "'c"   and succs :: "'c \<Rightarrow> 'c list"   and g :: "'c"   and g'' :: "'c" assumes "g' \<in> set (succs g)"   and "g' [succs]\<rightarrow>* g''" shows "g [succs]\<rightarrow>* g''"
fpxs_const_add: fixes x :: "'b"   and y :: "'b" shows "fpxs_const (x + y) = fpxs_const x + fpxs_const y"
\<HH>_def: shows "\<HH> = smc_SemiCAT \<alpha>\<lparr>Comp\<rparr>\<lparr>\<GG>\<FF>\<rparr>"
bisimInputPres: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a" assumes "\<And>Tvec. length xvec = length Tvec \<Longrightarrow> \<Psi> \<rhd> P[xvec::=Tvec] \<sim> Q[xvec::=Tvec]" shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<sim> M\<lparr>\<lambda>*xvec N\<rparr>.Q"
set_rel_sngD: fixes a :: "'d"   and b :: "'e"   and R :: "('d \<times> 'e) set" assumes "({a}, {b}) \<in> \<langle>R\<rangle>set_rel" shows "(a, b) \<in> R"
lActive_greatest: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat"   and n :: "nat" assumes "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<forall>n'>n. \<not> \<parallel>c\<parallel>\<^bsub>t n'\<^esub>" shows "i \<le> \<langle>c \<and> t\<rangle>"
betw_3n: shows "\<forall>n n' n''. n < n' \<and> n' < n'' \<longrightarrow> ord (f n) (f n') (f n'')"
pre_subdivFace_face_in_f': fixes f :: "face"   and v :: "nat"   and ls :: "nat option list"   and a :: "nat" assumes "pre_subdivFace_face f v ls"   and "Some a \<in> set ls" shows "a \<in> \<V> f"
valid_plan_then_lookup_Some: fixes \<pi>s :: "char list list" assumes "valid_plan \<pi>s" shows "\<forall>\<pi>\<in>set \<pi>s. lookup_operator \<pi> \<noteq> None"