hcju/isabelleps huggingface dataset:
strong_low_bisim_mm_def: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set" shows "abs.strong_low_bisim_mm \<R> \<equiv> sym \<R> \<and> abs.closed_glob_consistent \<R> \<and> (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>A, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>A) \<in> \<R> \<longrightarrow> abs.low_mds_eq mds mem\<^sub>1 mem\<^sub>2 \<and> (\<forall>c\<^sub>1' mds' mem\<^sub>1'. \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>A \<leadsto>\<^sub>A \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>\<^sub>A \<longrightarrow> (\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>A \<leadsto>\<^sub>A \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>\<^sub>A \<and> (\<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>\<^sub>A, \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>\<^sub>A) \<in> \<R>)))"
inter_def: fixes msgs :: "'a \<times> 'a" shows "local.inter msgs = sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. sample_uniform (order \<G>) \<bind> (\<lambda>c. sample_uniform (order \<G>) \<bind> (\<lambda>d. return_spmf (msgs, \<^bold>g [^] a, \<^bold>g [^] b, \<^bold>g [^] c, \<^bold>g [^] d)))))"
reflp: shows "reflp (\<lambda>a b. cmp a b = Equiv)"
congruence_congruence: fixes M1 :: "complex \<times> complex \<times> complex \<times> complex"   and M2 :: "complex \<times> complex \<times> complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "congruence M1 (congruence M2 H) = congruence (M2 *\<^sub>m\<^sub>m M1) H"
Busemann_function_mono: fixes xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b"   and x' :: "'b"   and y' :: "'b" shows "Busemann_function_at xi x y \<le> Busemann_function_at xi x' y' + dist x x' + dist y y'"
in_diff_count: fixes a :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" shows "(a \<in># M - N) = (count N a < count M a)"
interchange_columns_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and n :: "'e"   and m :: "'e" shows "interchange_columns A n m = (\<chi>i j. if j = n then A $h i $h m else if j = m then A $h i $h n else A $h i $h j)"
confluentD: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "confluent R"   and "R\<^sup>*\<^sup>* a b"   and "R\<^sup>*\<^sup>* a c" shows "\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
primitive_part_fract_def: fixes p :: "'b fract poly" shows "primitive_part_fract p = primitive_part (map_poly (fst \<circ> quot_of_fract) (smult (to_fract (Lcm (snd ` quot_of_fract ` set (coeffs p)))) p))"
M: fixes x :: "real" assumes "1 \<le> x" shows "\<bar>R x\<bar> \<le> M"
mat_leqb_eqc_for_LEQ: fixes i :: "nat"   and b :: "rat vec"   and A :: "rat mat"   and c :: "rat vec" assumes "i < dim_vec b"   and "i < dim_row A" shows "mat_leqb_eqc A b c ! i = LEQ (matrix_to_lpolies A ! i) (b $ i)"
alternating_list_append_1: fixes m :: "nat"   and s :: "'a"   and t :: "'a"   and n :: "nat" assumes "gcd_nat.greater_eq m 2" shows "alternating_list m s t @ alternating_list n s t = alternating_list (m + n) s t"
yoneda_lemma_fixed_e_axioms: shows "yoneda_lemma_fixed_e (\<cdot>) (\<cdot>\<^sub>S) \<phi> (Y a') a e"
split_conv: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and a :: "'g"   and b :: "'h" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
Sigma_msetE2: fixes a :: "'b"   and b :: "'c"   and A :: "'b multiset"   and B :: "'b \<Rightarrow> 'c multiset" assumes "(a, b) \<in># Sigma_mset A B" obtains "a \<in># A" and "b \<in># B a"
and_one_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and a one = modulo a (numeral.numeral one plus (num.Bit0 num.One))"
subH2: shows "subgroup (H1 <#> H \<inter> K1) (G\<lparr>carrier := H\<rparr>)"
intersect_segment_xline: fixes prec :: "nat"   and p0 :: "real \<times> real"   and p1 :: "real \<times> real"   and x :: "real"   and m :: "real"   and M :: "real" assumes "intersect_segment_xline prec (p0, p1) x = Some (m, M)" shows "closed_segment p0 p1 \<inter> {p. fst p = x} \<subseteq> {(x, m)..(x, M)}"
vfsequence_at_last: fixes i :: "V"   and x :: "V" assumes "i = vcard xs\<^sub>1" shows "(xs\<^sub>1 #\<^sub>\<circ> x)\<lparr>i\<rparr> = x"
keys_less_poly_mapping_size: fixes k :: "'b"   and m :: "'b \<Rightarrow>\<^sub>0 'c"   and f :: "'b \<Rightarrow> nat"   and g :: "'c \<Rightarrow> nat" assumes "k \<in> keys m" shows "f k + g (lookup m k) < poly_mapping_size f g m"
openin_Int: fixes U :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "openin U S"   and "openin U T" shows "openin U (S \<inter> T)"
mult_cancel_right: fixes f2 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g2 :: "'a \<Rightarrow> 'b"   and f1 :: "'a \<Rightarrow> 'b"   and g1 :: "'a \<Rightarrow> 'b" assumes "f2 \<in> \<Theta>[F](g2)"   and "\<forall>\<^sub>F x in F. g2 x \<noteq> (0::'b)" shows "((\<lambda>x. f1 x * f2 x) \<in> L F (\<lambda>x. g1 x * g2 x)) = (f1 \<in> L F g1)"
2: shows "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
game2_def: shows "game2 = (\<lambda>x y. coin_spmf \<bind> (\<lambda>b. exec_gpv hash_oracle'' (\<A>1 (\<^bold>g [^] x)) ({}, Map.empty) \<bind> (\<lambda>(((msg1, msg2), \<sigma>), s, s_h). TRY assert_spmf (valid_plains msg1 msg2) \<bind> (\<lambda>_. spmf_of_set (nlists UNIV len_plain) \<bind> (\<lambda>h. let cipher = (\<^bold>g [^] y, h [\<oplus>] (if b then msg1 else msg2)) in exec_gpv hash_oracle'' (\<A>2 cipher \<sigma>) (s, s_h) \<bind> (\<lambda>(guess, s', s_h'). return_spmf (guess = b, \<^bold>g [^] (x * y) \<in> s')))) ELSE (coin_spmf \<bind> (\<lambda>b. return_spmf (b, \<^bold>g [^] (x * y) \<in> s))))))"
vsv_appI: fixes a :: "V"   and b :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> VLambda A f" shows "VLambda A f\<lparr>a\<rparr> = b"
not_coin_flip: shows "map_spmf Not coin_spmf = coin_spmf"
commute: fixes a :: "'c"   and b :: "'c" shows "max a b = max b a"
gcd'_dvd1: fixes gcd' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "is_gcd gcd'" shows "gcd' a b dvd a"
stypes_eq: shows "types_t (s_inst \<S> ! i') = types (inst s ! i')"
keys_poly_eval_focus_subset: fixes a :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "keys (poly_eval (\<lambda>x. monomial (a x) 0) (focus X p)) \<subseteq> (\<lambda>t. except t X) ` keys p"
DomainI: fixes a :: "'b"   and b :: "'d"   and r :: "('b \<times> 'd) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
Mp_to_int_poly: fixes f :: "'a mod_ring poly" shows "Mp (to_int_poly f) = to_int_poly f"
mat2matofpoly_iarrays_def: fixes A :: "'d iarray iarray" shows "mat2matofpoly_iarrays A = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>i j. [:A !! i !! j:])"
order_iff_strict: fixes b :: "'g multiset"   and a :: "'g multiset" shows "(b \<subseteq># a) = (b \<subset># a \<or> a = b)"
f: shows "ring_epimorphism f A addA multA zeroA oneA B addB multB zeroB oneB"
fpty_objI: fixes P :: "cld list" shows "find_path_ty P ty_top (Some [])"
run_def: fixes F :: "'s stream \<Rightarrow> bool" shows "run F = HLD (Y - X) suntil (\<lambda>xs. HLD (X \<inter> Y) xs \<and> nxt (\<lambda>xs. HLD Y xs \<and> nxt F xs) xs)"
word_log_esimps_9: fixes x :: "'b word" shows "0 OR x = x"
fsubset: fixes x2 :: "'b fset"   and x1 :: "'b fset" assumes "x2 |\<subset>| x1" shows "\<nexists>e. e |\<in>| x2 \<and> e |\<notin>| x1"
iT_Plus_Max: fixes I :: "nat set"   and k :: "nat" assumes "finite I"   and "I \<noteq> {}" shows "Max (I \<oplus> k) = Max I + k"
b: fixes x :: "complex"   and a :: "complex"   and y :: "complex" shows "(cmod (x - a * y))\<^sup>2 = (cmod x)\<^sup>2 - Re (a * y * cnj x + x * cnj (a * y)) + (cmod (a * y))\<^sup>2"
matches_ipassmt_iface_constrain_srcip_mexpr_case_Iface: fixes ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and p :: "('c, 'd) tagged_packet_scheme"   and \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and ifce :: "iface"   and a :: "action" assumes "ipassmt_sanity_nowildcards ipassmt"   and "\<And>ips. ipassmt (Iface (p_iiface p)) = Some ips \<Longrightarrow> p_src p \<in> ipcidr_union_set (set ips)" shows "matches (common_matcher, \<alpha>) (ipassmt_iface_constrain_srcip_mexpr ipassmt ifce) a p = matches (common_matcher, \<alpha>) (Match (IIface ifce)) a p"
star_inductl: fixes z :: "'b \<times> 'c"   and x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "less_eq_prod (sup_prod z (times_prod' x y)) y" shows "less_eq_prod (times_prod' (prod.qstar x) z) y"
KS_def: shows "KS \<equiv> \<Lambda> k. k\<cdot>lcons\<cdot>lnil"
TOTAL_wrt_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "TOTAL_wrt \<psi> \<equiv> {U. \<exists>s. learn_total \<psi> U s}"
cases: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'c" assumes "r\<^sup>*\<^sup>* a1 a2" obtains a :: "'c" where "a1 = a__" and "a2 = a__"   | a :: "'c" and b :: "'c" and c :: "'c" where "a1 = a" and "a2 = c" and "r\<^sup>*\<^sup>* a b" and "r b c"
predicate2D_vimage2p: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h"   and S :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'e"   and y :: "'f" assumes "R \<le> vimage2p f g S"   and "R x y" shows "S (f x) (g y)"
adjacent_def: fixes v :: "'v"   and v' :: "'v" shows "adjacent v v' \<equiv> \<exists>w. (v, w, v') \<in> E"
one_closed: shows "\<zero>\<^bsub>V\<^esub> \<in> carrier V"
delta0: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and deltaG :: "'c itself \<Rightarrow> real" assumes "class.metric_tree_with_delta dist uniformity open deltaG" shows "deltaG TYPE('c) = 0"
sum_set_upto_conv_sum_list_int: fixes f :: "int \<Rightarrow> 'c"   and i :: "int"   and j :: "int" shows "sum f (set [i..j]) = sum_list (map f [i..j])"
borel_measurable_sum: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure" assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" shows "(\<lambda>x. \<Sum>i\<in>S. f i x) \<in> borel_measurable M"
retraction_is_epi: fixes g :: "'b" assumes "D.retraction g" shows "D.epi g"
cadd_commute: fixes i :: "V"   and j :: "V" shows "i \<oplus> j = j \<oplus> i"
total_order_def: fixes L :: "('a, 'b) gorder_scheme" shows "total_order L \<equiv> partial_order L \<and> total_order_axioms L"
simps_2: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b"   and t :: "'b"   and ts :: "'b list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app f (t # ts) = term.list_comb app (app f t) ts"
norm_axis: fixes i :: "'a"   and x :: "real" shows "norm (axis i x) = \<bar>x\<bar>"
fofu_III_I: assumes "\<exists>k. NCut c s t k \<and> val = NCut.cap c k" shows "isMaxFlow f"
eln_special_values_3: fixes x :: "ennreal" shows "(eln x = \<infinity>) = (x = \<infinity>)"
nat_to_list_of_pos: fixes u :: "nat" assumes "0 < u" shows "nat_to_list u = c_unfold (c_len u) (c_snd (u - 1))"
sub_instr_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr_name :: "sparc_operation"   and result :: "word32"   and rs1_val :: "word32"   and operand2 :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s1)) \<and> t2 = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s2))" shows "low_equal t1 t2"
u: shows "u \<in> additive.generate (R \<inter> {a \<cdot> b |a b. a \<in> \<aa> \<and> b \<in> \<bb>})"
Arg2pi_divide: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0"   and "Arg2pi w \<le> Arg2pi z" shows "Arg2pi (z / w) = Arg2pi z - Arg2pi w"
nonempty_selection: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "PartialMeetContraction Cn selection"   and "K = Cn A" shows "selection K \<phi> \<noteq> {}"
IS_LEFT_TOTAL_def: fixes R :: "('a \<times> 'b) set" shows "IS_LEFT_TOTAL R \<equiv> Domain R = UNIV"
induct: fixes P :: "exp \<Rightarrow> (nat \<times> val) list \<Rightarrow> bool"   and a0 :: "exp"   and a1 :: "(nat \<times> val) list" assumes "\<And>n \<rho>. P (ENat n) \<rho>"   and "\<And>x \<rho>. P (EVar x) \<rho>"   and "\<And>x e \<rho>. (\<And>xa xb xaa xab. (xaa, xab) \<in> fset xb \<Longrightarrow> P e ((x, xaa) # \<rho>)) \<Longrightarrow> P (ELam x e) \<rho>"   and "\<And>e1 e2 \<rho>. \<lbrakk>\<And>x xa xb xc xd. P e1 \<rho>; \<And>x xa xb xc xd. P e2 \<rho>\<rbrakk> \<Longrightarrow> P (EApp e1 e2) \<rho>"   and "\<And>f e1 e2 \<rho>. \<lbrakk>\<And>x xa xb. P e1 \<rho>; \<And>x xa xb. P e2 \<rho>\<rbrakk> \<Longrightarrow> P (EPrim f e1 e2) \<rho>"   and "\<And>e1 e2 e3 \<rho>. \<lbrakk>\<And>x xa. P e1 \<rho>; \<And>x xa. xa = 0 \<Longrightarrow> P e3 \<rho>; \<And>x xa. xa \<noteq> 0 \<Longrightarrow> P e2 \<rho>\<rbrakk> \<Longrightarrow> P (EIf e1 e2 e3) \<rho>" shows "P a0 a1"
is_prefix_append: fixes a :: "'a list"   and b :: "'a list"   and D :: "'a list" shows "is_prefix (a @ b) D = (is_prefix a D \<and> is_prefix b (drop (length a) D))"
least_hmem: fixes a :: "hf" assumes "a \<noteq> 0" shows "least a \<^bold>\<in> a"
assms: fixes x :: "'a"   and y :: "'b" assumes "(x, y) \<in> A" shows "P x y"
eq: shows "0 \<le> real_of (cnj_class.cnj (inner x y) * inner x y)"
principal_le_iff: fixes A :: "'a set"   and B :: "'a set" shows "(principal A \<le> principal B) = (A \<subseteq> B)"
assms_1: shows "\<not> hb (i, Expunge e1 mo i) (ir, Delete is e2)"
right_unit: fixes a :: "'a" assumes "a \<in> I" shows "a + \<zero> = a"
top_le: fixes a :: "'a" assumes "1 \<le> a" shows "a = 1"
compile_cn_correct: fixes f :: "recf"   and xs :: "nat list"   and gs :: "recf list"   and n :: "nat"   and ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and anything :: "nat list" assumes "terminate f (map (\<lambda>g. rec_exec g xs) gs)"   and "\<And>ap arity fp anything. rec_ci f = (ap, arity, fp) \<Longrightarrow> {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ rec_exec f (map (\<lambda>g. rec_exec g xs) gs) # 0 \<up> (fp - Suc arity) @ anything}"   and "\<forall>g\<in>set gs. terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"   and "rec_ci (Cn n f gs) = (ap, arity, fp)"   and "length xs = n" shows "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_exec (Cn n f gs) xs # 0 \<up> (fp - Suc arity) @ anything}"
local: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and v0s :: "v list"   and n :: "nat"   and j :: "nat" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>" shows "\<lparr>s;v0s;[Local n i vs es]\<rparr> \<leadsto>_ j \<lparr>s';v0s;[Local n i vs' es']\<rparr>"
cf_h_len1: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and f :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "f \<in> pHom R S X, A B Y"   and "\<forall>x\<in>carrier S. f x = h x"   and "pol_coeff S c" shows "fst (cf_h f c) = fst (cf_h h c)"
transformation_by_components_axioms_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and t :: "'d \<Rightarrow> 'e" shows "transformation_by_components_axioms A B F G t \<equiv> (\<forall>a. partial_magma.ide A a \<longrightarrow> partial_magma.in_hom B (t a) (F a) (G a)) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> B (t (partial_magma.cod A f)) (F f) = B (G f) (t (partial_magma.dom A f)))"
Rep_sparse_row_vector_zero: fixes x :: "nat"   and v :: "(nat \<times> 'a) list"   and y :: "nat" assumes "x \<noteq> 0" shows "Rep_matrix (sparse_row_vector v) x y = (0::'a)"
commutative: fixes x :: "'b"   and y :: "'b" assumes "x \<in> \<eta> ` R"   and "y \<in> \<eta> ` R" shows "x +' y = y +' x"
Zorns_po_lemma_nonempty: fixes r :: "('a \<times> 'a) set" assumes "Partial_order r"   and "\<And>C. \<lbrakk>C \<in> Chains r; C \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"   and "r \<noteq> {}" shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
Enc_keys_clean_mono: fixes H :: "msg set"   and G :: "msg set" assumes "Enc_keys_clean H"   and "G \<subseteq> H" shows "Enc_keys_clean G"
reconstruct_poly_defines_mset_of_argument: fixes a :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "a \<noteq> (0::'a)"   and "reconstruct_poly a as = reconstruct_poly a bs" shows "mset as = mset bs"
2: shows "top * e * (w\<^sup>T \<sqinter> - v\<^sup>T)\<^sup>\<star> * w\<^sup>T \<le> top * e * (w\<^sup>T \<sqinter> - v\<^sup>T)\<^sup>\<star> \<squnion> - - v\<^sup>T"
plan_action_enabled_def: fixes \<pi> :: "plan_action"   and M :: "object atom formula set" shows "plan_action_enabled \<pi> M = (wf_plan_action \<pi> \<and> M \<^sup>c\<TTurnstile>\<^sub>= ground_action.precondition (resolve_instantiate \<pi>))"
wary_subst_def: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm" shows "wary_subst \<rho> = (\<forall>x. wary (\<rho> x) \<and> arity_var x \<le> arity (\<rho> x) \<and> local.ground_heads (head (\<rho> x)) \<subseteq> ground_heads_var x)"
mbc_ex: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<exists>x. mbc_cond t n x"
sdom_dom: fixes G :: "nat graph_rec"   and i :: "nat"   and j :: "nat" assumes "cfg_doms G"   and "cfg_doms.strict_dom G i j" shows "cfg_doms.dom G i j"
transpose_four_block_mat: fixes A :: "'a mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'a mat"   and nc2 :: "nat"   and C :: "'a mat"   and nr2 :: "nat"   and D :: "'a mat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2" shows "(four_block_mat A B C D)\<^sup>T = four_block_mat A\<^sup>T C\<^sup>T B\<^sup>T D\<^sup>T"
finite_list: fixes A :: "'b set" assumes "finite A" shows "\<exists>l. set l = A \<and> distinct l"
le_bot: fixes a :: "'a" assumes "a \<le> (0::'a)" shows "a = (0::'a)"
prod_mn_less_k: fixes n :: "nat"   and k :: "nat"   and m :: "nat" assumes "0 < n"   and "0 < k"   and "Suc 0 < m"   and "m * n = k" shows "n < k"
V_def: fixes x :: "nat" shows "V x = (if x = 0 then U else carrier - A)"
count_mset_set_leq': fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "count (mset_set A) x \<le> Suc 0"
Fadd_notinE': fixes Fadd :: "('a \<times> 'a) set" assumes "Fadd \<inter> E' = {}" shows "E' - (F' \<union> Fadd) = E' - F'"
cod_dom: fixes f :: "'a2" assumes "C2.arr f" shows "C2.cod (C2.dom f) = C2.dom f"
distinct_2: fixes x21 :: "char list"   and x22 :: "char list list"   and x1 :: "instr" shows "StepC x21 x22 \<noteq> StepI x1"
sel_1: fixes x1 :: "'d" shows "projl (Inl x1) = x1"
lemma_2_10_32: fixes b :: "'a"   and a :: "'a" shows "(((b r\<rightarrow> a) l\<rightarrow> a) r\<rightarrow> b) r\<rightarrow> b r\<rightarrow> a = b r\<rightarrow> a"
elims: fixes x :: "'c Tree"   and y :: "'c \<times> 'c Tree" assumes "hs_remove_max x = y" obtains v :: "'c" and l :: "'c Tree" and r :: "'c Tree" where "x = T v l r" and "y = (v, merge l r)"   | "x = E" and "y = undefined"
simps_1: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "rel_comp_pow r [] x y = False"
pid_system_automaton_alt_def: shows "pid_system_automaton = stutter_extend pid_system_automaton'"
assms_3: shows "InvariantWatchListsUniq (getWatchList state)"
eq_refl: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "x = y" shows "x \<preceq>\<^sub>p y"
P_lnth_Suc: fixes n :: "nat" shows "P $ Suc n = ltl P $ n"
hcomp_char: shows "D'.hcomp = (\<lambda>f g. if D'.arr f \<and> D'.arr g \<and> D'.src f = D'.trg g then f \<star>\<^sub>D g else D'.null)"
calc_ideal_point2_cvec_def: fixes A :: "complex"   and B :: "complex" shows "calc_ideal_point2_cvec A B = (case let discr = Re (cor ((cmod B)\<^sup>2 - (Re A)\<^sup>2)) in (B * (- A + \<i> * cor (sqrt discr)), (cmod B)\<^sup>2) of (x, y) \<Rightarrow> (x, cor y))"
select_convs_2: fixes nodes :: "'c set"   and edges :: "('c \<times> 'c) set"   and more :: "'d" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>C c\<guillemotright>" shows "C.seq g f"
reduce_basis_def: fixes \<alpha> :: "rat"   and fs :: "int vec list" shows "reduce_basis \<alpha> fs = (case fs of [] \<Rightarrow> fs | f # x \<Rightarrow> fs_state (basis_reduction \<alpha> (dim_vec f) fs))"
dp_map1: fixes x :: "'a"   and y :: "'a" assumes "is_direct_product x y" shows "is_map x"
function_mult_l_distr: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a"   and z :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F"   and "z \<in> carrier F" shows "z \<otimes>\<^bsub>F\<^esub> (x \<oplus>\<^bsub>F\<^esub> y) = z \<otimes>\<^bsub>F\<^esub> x \<oplus>\<^bsub>F\<^esub> z \<otimes>\<^bsub>F\<^esub> y"