hcju/isabelleps huggingface dataset:
fresh_ssubst': fixes V :: "name set"   and i :: "'a"   and t :: "tm" assumes "finite V"   and "atom i \<sharp> t"   and "atom (p' \<bullet> i) \<sharp> t" shows "atom i \<sharp> ssubst t V F'"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>'"
inverse_arrows_def: fixes f :: "'b"   and g :: "'b" shows "B.inverse_arrows f g \<equiv> B.ide (g \<cdot>\<^sub>B f) \<and> B.ide (f \<cdot>\<^sub>B g)"
callee: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "pfinite_gpv \<I>' (callee s x)"
keys_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "keys (tail p) = keys p - {lt p}"
le_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z * x \<le> z * y"
real_distribution_distr: fixes X :: "'a \<Rightarrow> real" assumes "random_variable borel X" shows "real_distribution (distr M borel X)"
real_of_complex_of: shows "real_of \<circ> complex_of = id"
Node_3: shows "\<forall>x y. f (x + y) = f x + f y"
distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring plus times" shows "times a (plus b c) = plus (times a b) (times a c)"
abs_summable_on_finite: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A" shows "f abs_summable_on A"
assms_2: fixes x :: "'a" assumes "nf_inres m x" shows "f x \<le> m'"
assms_2: shows "g \<turnstile> m-ms\<rightarrow>l"
hyps_4: fixes x :: "name"   and M :: "trm" assumes "x:M \<in> \<parallel>(B)\<parallel>" shows "SNa M"
finite_set2: fixes A :: "'b set"   and n :: "nat" assumes "card A = n"   and "finite A" shows "finite {xs. set xs = A \<and> distinct xs}"
thd3_def: shows "thd3 \<equiv> snd \<circ> snd"
simps_2: fixes f :: "'s" shows "local.ground_heads (Sym f) = {f}"
lossless_bind_gpv: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('g, 'd, 'e) gpv"   and f :: "'g \<Rightarrow> ('f, 'd, 'e) gpv" shows "lossless_gpv \<I> (gpv \<bind> f) = (lossless_gpv \<I> gpv \<and> (\<forall>x\<in>results_gpv \<I> gpv. lossless_gpv \<I> (f x)))"
1: fixes \<rho> :: "'a formula" assumes "\<Turnstile> \<Gamma> \<^bold>\<rightarrow> \<rho>"   and "\<Turnstile> \<rho> \<^bold>\<rightarrow> \<Delta>"   and "atoms \<rho> \<subseteq> atoms \<Gamma>"   and "atoms \<rho> \<subseteq> atoms \<Delta>" shows "thesis"
simps: fixes x :: "'b extN"   and y :: "'b extN" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
0: shows "prv (imp (all x (neg \<phi>)) (imp \<phi> fls))"
Fvars_\<phi>L: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c"   and \<phi> :: "'c" assumes "Loeb_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi prv bprv enc S P"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "Fvars (Loeb_Form.\<phi>L var fmla Fvars subst cnj imp bprv enc P \<phi>) = {}"
wordinterval_CIDR_split1_snd: fixes r :: "'b wordinterval"   and s :: "'b prefix_match"   and u :: "'b wordinterval" assumes "wordinterval_CIDR_split1 r = (Some s, u)" shows "u = wordinterval_setminus r (prefix_to_wordinterval s)"
lossless_resample: shows "lossless_spmf (resample A B)"
degree_times_le: fixes A :: "'a mpoly"   and B :: "'a mpoly"   and x :: "nat" shows "MPoly_Type.degree (A * B) x \<le> MPoly_Type.degree A x + MPoly_Type.degree B x"
language\<^sub>b_def: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" shows "language\<^sub>b idx \<phi> = L idx {\<AA>. sat\<^sub>b \<AA> \<phi> \<and> #\<^sub>V \<AA> = idx}"
approximation_preproc_nat_5: fixes i :: "nat"   and j :: "nat" shows "real (i * j) = real i * real j"
sums_Suc_iff: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" shows "(\<lambda>n. f (Suc n)) sums s = f sums (s + f 0)"
kD_unk_or_atleast_one: fixes i :: "nat" shows "paodv i \<TTurnstile> (recvmsg rreq_rrep_sn \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). \<forall>dip\<in>kD (rt \<xi>). \<pi>\<^sub>3 (the (rt \<xi> dip)) = unk \<or> 1 \<le> \<pi>\<^sub>2 (the (rt \<xi> dip)))"
span_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "local.span A \<subseteq> local.span B"
dehomogenize_plus: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "dehomogenize x (p + q) = dehomogenize x p + dehomogenize x q"
prems_2: shows "\<S>\<bullet>\<C> \<turnstile> ves @ [a] : [] _> tvs"
apfst_def: fixes f :: "'d \<Rightarrow> 'f" shows "apfst f = map_prod f id"
sup_absorb2: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x y" shows "sup x y = y"
matrix_to_iarray_row_add: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'c"   and q :: "'a" shows "matrix_to_iarray (row_add A i j q) = row_add_iarray (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j) q"
assms_7: shows "real_euclid.Col (cart2_pt p) (cart2_pt q) (cart2_pt r)"
bind_map_spmf: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c spmf"   and g :: "'b \<Rightarrow> 'a spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
paval_aval_vars: fixes a :: "aexp"   and ps :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int" assumes "vars a \<subseteq> dom ps" shows "paval a ps = aval a (emb ps s)"
ex_pickE: fixes xl :: "Preliminaries.var list"   and al :: "'univ list" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> intT (tpOfV (xl ! i)) (al ! i)" shows "\<exists>\<xi>. wtE \<xi> \<and> (\<forall>i<length xl. \<xi> (xl ! i) = al ! i)"
neg_def: fixes x :: "'a" shows "! x \<equiv> THE y. are_complementary x y"
tmap_eq_tmap_conv_tllist_all2: fixes f :: "'i \<Rightarrow> 'g"   and g :: "'j \<Rightarrow> 'h"   and xs :: "('i, 'j) tllist"   and f' :: "'k \<Rightarrow> 'g"   and g' :: "'l \<Rightarrow> 'h"   and ys :: "('k, 'l) tllist" shows "(tmap f g xs = tmap f' g' ys) = tllist_all2 (\<lambda>x y. f x = f' y) (\<lambda>x y. g x = g' y) xs ys"
Nl_entails_Bl: shows "Nl \<Turnstile>\<inter>\<G>L {Bl}"
\<psi>_def: shows "\<psi> = (\<lambda>f. if arr f then inv_into S \<phi> f else n)"
Lcm_dvd_iff: fixes A :: "'a set"   and x :: "'a" shows "(Lcm A dvd x) = (\<forall>y\<in>A. y dvd x)"
new_document_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and ptr :: "(_) document_ptr"   and getter :: "('n, 'l, 'm, 'u option) RDocument_scheme \<Rightarrow> 'w" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "ptr \<noteq> new_document_ptr" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
poly_smult: fixes a :: "'b"   and p :: "'b poly"   and x :: "'b" shows "poly (smult a p) x = a * poly p x"
alphaAbs_weakestAll_imp_strongestEx: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'"   and "alphaAbs_all_distinct_qAFresh xs x X xs' x' X'" shows "alphaAbs_ex_distinct_qAFresh xs x X xs' x' X'"
fls_subdegree_def: fixes f :: "'a fls" shows "fls_subdegree f \<equiv> if f = 0 then 0 else LEAST n. fls_nth f n \<noteq> (0::'a)"
assms_4: shows "f \<equiv> \<lambda>x. \<integral>\<^sup>+ t. (if g t = x then 1 else 0) * f' t \<partial>M"
iIN_E: fixes x :: "nat"   and n :: "nat"   and d :: "nat" assumes "x \<in> [n\<dots>,d]" obtains "n \<le> x" and "x \<le> n + d"
is_TNil_tSup: fixes A :: "('c, 'b) tllist set" shows "is_TNil (local.tSup A) = (\<forall>x\<in>A. is_TNil x)"
singleton_table_wf_tuple: fixes V :: "nat set"   and i :: "nat"   and x :: "'b option list"   and n :: "nat"   and z :: "'b" assumes "V = {i}"   and "x \<in> singleton_table n i z" shows "wf_tuple n V x"
chk_u_s_if: assumes "chkargs (>\<^sub>t) t s" shows "chkargs (>\<^sub>t) u s"
chine_simps_3: shows "C.cod src_\<mu>.chine = src_\<mu>.cod.apex"
prod_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "hfsetcat.prod a b = hf_to_ide (ide_to_hf a * ide_to_hf b)"
assms_4: fixes x :: "nat" assumes "x \<in> A1" shows "prime x"
Un_upper2: fixes B :: "'k set"   and A :: "'k set" shows "B \<subseteq> A \<union> B"
mono_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<le> f y)"
to_int: fixes y :: "'i"   and z :: "'a mod_ring" assumes "R y z" shows "arith_ops_record.to_int ff_ops y = to_int_mod_ring z"
sigma_algebra_sigma_sets: fixes a :: "'c set set"   and \<Omega> :: "'c set" assumes "a \<subseteq> Pow \<Omega>" shows "sigma_algebra \<Omega> (sigma_sets \<Omega> a)"
rel_setD1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a set"   and B :: "'b set"   and x :: "'a" assumes "rel_set R A B"   and "x \<in> A" shows "\<exists>y\<in>B. R x y"
True: shows "poly p b = 0 \<and> a < b \<and> p \<noteq> 0"
simps: fixes a1 :: "nat"   and a2 :: "nat list set" shows "Form a1 a2 = ((\<exists>xs ys. a1 = 0 \<and> a2 = {xs, ys} \<and> length xs = length ys \<and> xs \<noteq> ys) \<or> (\<exists>k xs ys zs. a1 = 2 * k - 1 \<and> a2 = {xs, ys} \<and> Form_Body k k xs ys zs \<and> 0 < k) \<or> (\<exists>k xs ys zs. a1 = 2 * k \<and> a2 = {xs, ys} \<and> Form_Body (Suc k) k xs ys zs \<and> 0 < k))"
distinct_5: fixes x1 :: "agent"   and x4 :: "key" shows "msg.Agent x1 \<noteq> Key x4"
sym_Union: fixes Theta :: "('b \<times> 'b) set set" assumes "\<And>theta. theta \<in> Theta \<Longrightarrow> sym theta" shows "sym (\<Union> Theta)"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "dual xa - dual x = dual (- (x - xa))"
not_symKeys_pubK: fixes A :: "agent" shows "pubK A \<notin> symKeys"
induct: fixes P :: "'g \<Rightarrow> 'node \<Rightarrow> 'var \<Rightarrow> bool"   and a0 :: "'g"   and a1 :: "'node"   and a2 :: "'var" assumes "\<And>g n v. (\<And>x21 x22. \<lbrakk>\<not> n \<notin> set (\<alpha>n g); v \<notin> defs g n; predecessors g n = x21 # x22; x22 = []\<rbrakk> \<Longrightarrow> P g x21 v) \<Longrightarrow> P g n v" shows "P a0 a1 a2"
pocklington: fixes n :: "nat"   and q :: "nat"   and r :: "nat"   and a :: "nat" assumes "2 \<le> n"   and "n - 1 = q * r"   and "n \<le> q\<^sup>2"   and "[a ^ (n - 1) = 1] (mod n)"   and "\<forall>p. prime p \<and> p dvd q \<longrightarrow> coprime (a ^ ((n - 1) div p) - 1) n" shows "prime n"
I_Step: shows "C M2 M1 \<Omega> V m (Suc (Suc k)) \<subseteq> append_set (append_sets V (lists_of_length (inputs M2) k)) (inputs M2)"
mult_pCons_left: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "pCons a p * q = smult a q + pCons (0::'b) (p * q)"
nn_integral_monotone_convergence_SUP: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "incseq f"   and "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M = (SUP i. integral\<^sup>N M (f i))"
preallocatedE: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and C :: "char list"   and P :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<Rightarrow> char list \<Rightarrow> bool" assumes "preallocated h"   and "C \<in> sys_xcpts"   and "\<And>fs. h (addr_of_sys_xcpt C) = \<lfloor>(C, fs)\<rfloor> \<Longrightarrow> P h C" shows "P h C"
less_induct: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "\<And>x. (\<And>y. x < y \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
create_get: fixes x :: "'d \<Longrightarrow> 'e"   and v :: "'d" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (create\<^bsub>x\<^esub> v) = v"
tensor_lookup_base: fixes v :: "'b list"   and ds :: "nat list"   and e :: "nat list \<Rightarrow> 'b" assumes "length v = prod_list ds"   and "\<And>is. is \<lhd> ds \<Longrightarrow> lookup_base ds v is = e is" shows "tensor_vec_from_lookup ds e = v"
H: fixes b :: "'a"   and a :: "'a" shows "b * a \<le> b"
run_owner_iv_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_owner_iv n s \<equiv> stage_owner_iv n (run_asset_iii n s)"
isCont_Arccos: fixes z :: "complex" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "isCont Arccos z"
mem_degree_aformD: fixes x :: "'a \<times> 'a pdevs"   and XS :: "('a \<times> 'a pdevs) list" assumes "x \<in> set XS" shows "degree_aform x \<le> degree_aforms XS"
summable_rabs_comparison_test: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n"   and "summable g" shows "summable (\<lambda>n. \<bar>f n\<bar>)"
rep_eq: fixes x :: "('c \<times> 'd) fset" shows "ordered_fmap x = ordered_map (fset x)"
conf': shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) (u @ v) = Partial_Cost_Model.config' (rTS h0) ([x, y], rev u @ h) v"
that: fixes e :: "real" assumes "e \<in> {- 1<..<1}"   and "pdevs_val f (pdevs_of_list xs) = e *\<^sub>R sum_list xs" shows "thesis"
assms_2: shows "w2 \<equiv>\<^sub>E r2"
sum_product: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and g :: "'d \<Rightarrow> 'b"   and B :: "'d set" shows "sum f A * sum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
hom_zero: fixes i :: "nat" shows "coeff 0 i = (0::'a)"
attach_CNV_RES: fixes callee :: "'e \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'e, 'c, 'd) gpv"   and s :: "'e"   and res :: "'f \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'f) spmf"   and s' :: "'f" shows "CNV callee s \<rhd> RES res s' = RES (attach_callee callee res) (s, s')"
50: fixes x :: "'a" shows "top \<squnion> - - x = top"
subspace_span: fixes S :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "pmdl.subspace (pmdl S)"
reflect_poly_pCons': fixes p :: "'b poly"   and c :: "'b" assumes "p \<noteq> 0" shows "reflect_poly (pCons c p) = reflect_poly p + monom c (Suc (degree p))"
assms: fixes i :: "int" assumes "\<And>j. \<lbrakk>k \<le> j; j < i\<rbrakk> \<Longrightarrow> P j" shows "P i"
WTrt_elim_cases_1: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and v :: "char list"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> v:=e : T" obtains T :: "ty" and T' :: "ty" where "Ta__ = Void" and "E v = \<lfloor>T\<rfloor>" and "P,E,h,sh \<turnstile> e : T'" and "P \<turnstile> T' \<le> T"
kcomp_assoc: fixes f :: "'a \<Rightarrow> 'd set"   and g :: "'d \<Rightarrow> 'c set"   and h :: "'c \<Rightarrow> 'b set" shows "f \<circ>\<^sub>K g \<circ>\<^sub>K h = f \<circ>\<^sub>K (g \<circ>\<^sub>K h)"
x_axis_def: shows "x_axis \<equiv> abs_circline x_axis_clmat"
adj_cancel2: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "id \<le> g \<circ> f"
of_int_less_numeral_power_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "int"   and x :: "num"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus a) (power.power one times (numeral.numeral one plus x) n) = (a < numeral x ^ n)"
t_de_morgan_var1: fixes x :: "'a"   and y :: "'a" shows "n (n x \<cdot> n y) = n n x + n n y"
assms_3: shows "{x \<in> space N. P x} \<in> sets N"
from_hma_to_hma\<^sub>v: fixes v :: "'b Matrix.vec" assumes "v \<in> carrier_vec CARD('c)" shows "from_hma\<^sub>v (to_hma\<^sub>v v) = v"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f" obtains "\<And>x. d_OUT f x \<le> weight \<Gamma> x" and "\<And>x. d_IN f x \<le> weight \<Gamma> x" and "\<And>x. x \<notin> A \<Gamma> \<Longrightarrow> d_OUT f x \<le> d_IN f x" and "\<And>a. a \<in> A \<Gamma> \<Longrightarrow> d_IN f a = 0" and "\<And>b. b \<in> B \<Gamma> \<Longrightarrow> d_OUT f b = 0" and "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Gamma>\<^esub> \<Longrightarrow> f e = 0"
Smith_normal_form_mat_m0: fixes A :: "'a mat"   and m :: "nat" assumes "A \<in> carrier_mat m 0" shows "Smith_normal_form_mat A"
f8: shows "3 \<le> rk {A, A', B', B, \<gamma>}"
hom_one: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
unat_less_helper: fixes x :: "'b word"   and n :: "nat" assumes "x < word_of_nat n" shows "unat x < n"
i_def: fixes H :: "'a"   and x :: "real" shows "i H x = (1 - x) *\<^sub>R nth_derivative 2 f (X + x *\<^sub>R H) H"
bgauge_existence_lemma: fixes s :: "'a set"   and q :: "real \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<forall>x\<in>s. \<exists>d>0. q d x) = (\<forall>x. \<exists>d>0. x \<in> s \<longrightarrow> q d x)"
in_Inf_multiset_iff: fixes A :: "'b multiset set"   and x :: "'b" assumes "A \<noteq> {}" shows "(x \<in># Inf A) = (\<forall>X\<in>A. x \<in># X)"
bdy: shows "\<forall>s. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' s) bdy {t. return s t \<in> R s t},{t. return s t \<in> A}"
chine_hcomp_ide_ide: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "ide g"   and "ide f"   and "src g = trg f" shows "chine_hcomp g f = Leg0 (Dom g) \<down>\<down> Leg1 (Dom f)"
eq_upto_lexmin_non_empty: fixes C :: "(nat \<Rightarrow> 'a) set"   and i :: "nat" assumes "C \<subseteq> SEQ A"   and "C \<noteq> {}" shows "eq_upto C (lexmin C) i \<noteq> {}"
ChiRel_def: fixes A :: "('a, 'b, 'c) hierauto" shows "ChiRel A \<equiv> {(S, S'). S \<in> HAStates A \<and> S' \<in> HAStates A \<and> S' \<in> Chi A S}"
len: shows "length (row (Matrix_Legacy.transpose nr m) i) = length (col m i)"
map_disc_iff: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b list" shows "(map f a = []) = (a = [])"
length_sequence_of: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "length (sequence_of A enum f) = card A"
C_def: shows "C = real_of_int (B * prod a {..<N}) * sum f {..<N}"
HPhase0Read_Inv2c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "Inv2c s"   and "HPhase0Read s s' p d"   and "Inv2a s" shows "Inv2c s'"
PPs_closed_lcs: fixes s :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t \<in> .[X]" shows "lcs s t \<in> .[X]"
map_fst_zip_take: fixes xs :: "'g list"   and ys :: "'h list" shows "map fst (zip xs ys) = take (min (length xs) (length ys)) xs"
filter_shuffles_disjoint2_2: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "set xs \<inter> set ys = {}"   and "zs \<in> shuffles xs ys" shows "trace_hide zs (set ys) = xs"
product_iff: fixes x :: "'b fset"   and A :: "'b fset set"   and B :: "'b fset set" shows "(x \<in> A \<otimes> B) = (\<exists>a\<in>A. \<exists>b\<in>B. x = a |\<union>| b)"
constraints_set_vars_geq_vec_def: fixes start :: "nat"   and b :: "rat vec" shows "set (vars_from_index_geq_vec start b) = {GEQ (lp_monom 1 (i + start)) (b $ i) |i. i \<in> {0..<dim_vec b}}"
ide_char: fixes a :: "'s" shows "ide a = (arr a \<and> local.dom a = a \<and> cod a = a)"
weakly_Pareto_optimal_for_men_def2: fixes \<mu> :: "('m \<times> 'w) set" shows "weakly_Pareto_optimal_for_men \<mu> = (simple \<mu> \<and> (\<forall>\<mu>''. simple \<mu>'' \<and> strictly_preferred_by_a_man \<mu> \<mu>'' \<longrightarrow> strictly_preferred_by_a_man \<mu>'' \<mu>))"
whisker_left: fixes f :: "'a"   and \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.ide f"   and "B.seq \<nu> \<mu>" shows "f \<star>\<^sub>B \<nu> \<cdot>\<^sub>B \<mu> = (f \<star>\<^sub>B \<nu>) \<cdot>\<^sub>B (f \<star>\<^sub>B \<mu>)"
iterates_healthy: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and i :: "nat" assumes "healthy (wp body)" shows "healthy (iterates body G i)"
split_minD: fixes t :: "('b \<times> nat) tree"   and x :: "'b"   and t' :: "('b \<times> nat) tree" assumes "split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "x # Tree2.inorder t' = Tree2.inorder t"
top_stone_phi_pair_def: shows "top \<equiv> Abs_stone_phi_pair stone_phi.pairs_top"
hcomp_simps\<^sub>W\<^sub>C_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "partial_magma.arr V (H \<nu> \<mu>)"
lookup_pair_eq_0: fixes ko :: "'a key_order"   and xs :: "('a \<times> 'b) list"   and k :: "'a" assumes "ko.oalist_inv_raw ko xs" shows "(lookup_pair_ko ko xs k = (0::'b)) = (k \<notin> fst ` set xs)"
evaluation_map_axioms: shows "Coherence.evaluation_map (\<cdot>) src trg (\<cdot>) (\<star>) \<a> \<i> src trg (\<lambda>\<mu>. if arr \<mu> then \<mu> else null)"
exhaust: fixes y :: "PType" obtains "y = P"   | "y = Q"
insert: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "set.F (insert x A) = x \<^bold>* set.F A"
simps_4: fixes a :: "Cooper.num" shows "zsplit0 (Neg a) = (let (i', a') = zsplit0 a in (- i', Neg a'))"
least_fixpoint_same: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "is_least_fixpoint f x" shows "x = \<mu> f"
dom_eqI: fixes a :: "'s"   and f :: "'s" assumes "S.ide a"   and "S.seq f a" shows "S.dom f = a"
natural_hom_image: shows "(\<lambda>a. if a \<in> FreeGroup S then BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {a}) else 0) ` FreeGroup S = Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>"
unity_root_mod_nat: fixes k :: "nat"   and n :: "int" shows "unity_root k (int (nat (n mod int k))) = unity_root k n"
minus_diff_minus: fixes a :: "'c"   and b :: "'c" shows "- a - - b = - (a - b)"
items_mpair: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "items (insert \<lbrace>X, Y\<rbrace> H) = insert \<lbrace>X, Y\<rbrace> (items ({X, Y} \<union> H))"
ndet_def: fixes P :: "'c process"   and Q :: "'c process" shows "P |-| Q \<equiv> Abs_process (F P \<union> F Q, D P \<union> D Q)"
eq_conv_compare: fixes x :: "'a"   and y :: "'a" shows "eq\<cdot>x\<cdot>y = is_EQ\<cdot>(compare\<cdot>x\<cdot>y)"
TAbs: fixes T :: "POPLmark.type"   and t :: "trm" shows "(\<lambda><:T. t) \<in> value"
deg_sum_eq: shows "(\<Sum>i | i < n. degree (m i)) = degree u"