hcju/isabelleps huggingface dataset:
simps_3: fixes MI :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U"   and f :: "form"   and g :: "form" shows "FEval MI e (FConj f g) = (FEval MI e f \<and> FEval MI e g)"
S_def: shows "S = {m. coeff p m \<noteq> (0::'a)} \<union> {m. coeff p' m \<noteq> (0::'a)} \<union> {m. coeff (p + p') m \<noteq> (0::'a)}"
u\<^sub>g_def: shows "u\<^sub>g = h \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0"
star_of_def: fixes x :: "'a" shows "star_of x \<equiv> star_n (\<lambda>n. x)"
hyps: fixes v :: "'v" assumes "finite E"   and "finite V"   and "is_trail v xs v'"   and "odd (degree v' G)" shows "int (num_of_odd_nodes (rem_unPath xs G)) = int (num_of_odd_nodes G) + (if odd (degree v G) \<and> v \<noteq> v' then - 2 else 0)"
simps: fixes ant :: "form multiset"   and suc :: "form multiset" shows "succ ( ant \<Rightarrow>* suc) = suc"
mtSet_def: shows "Set{} \<equiv> \<lambda>\<tau>. Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<lfloor>{}\<rfloor>\<rfloor>"
h': shows "h2 \<turnstile> (if document_ptr \<noteq> old_document then Heap_Error_Monad.bind (get_disconnected_nodes old_document) (\<lambda>old_disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes old_document (remove1 child old_disc_nodes)) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disc_nodes. set_disconnected_nodes document_ptr (child # disc_nodes)))) else return ()) \<rightarrow>\<^sub>h h'"
finite_UNIV_complex_def: shows "finite_UNIV = Phantom(complex) False"
trie_odlist_MapOps_def: shows "trie_odlist_MapOps \<equiv> \<lparr>MapOps.empty = empty_trie, lookup = trie_odlist_lookup, update = trie_odlist_update\<rparr>"
lin_red_imp_red: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lin_red F p q" shows "red F p q"
simps: shows "eval_ubx OpAddNumUbx = eval_AddNumUbx"
strad1: fixes x :: "real"   and s :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and e :: "real"   and z :: "real" assumes "\<And>z. \<lbrakk>z \<noteq> x; \<bar>z - x\<bar> < s\<rbrakk> \<Longrightarrow> \<bar>(f z - f x) / (z - x) - f' x\<bar> < e / 2"   and "\<bar>z - x\<bar> < s" shows "\<bar>f z - f x - f' x * (z - x)\<bar> \<le> e / 2 * \<bar>z - x\<bar>"
case_4: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x41 :: "'e list"   and x42 :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" shows "(case \<forall>x41\<langle>\<or>\<noteq>: x42\<rangle>\<^sub>s\<^sub>t of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = f4 x41 x42"
t_gen_vp: fixes v :: "'a \<Rightarrow> ant"   and t :: "'a" assumes "valuation K v"   and "t \<in> carrier K"   and "v t = 1" shows "Vr K v \<diamondsuit>\<^sub>p t = vp K v"
serial_onE: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b" assumes "serial_on A r"   and "x \<in> A" obtains xa :: "'b" where "xa \<in> A" and "(x, xa) \<in> r"
step: fixes k :: "nat"   and X :: "event_data option list set"   and Y :: "event_data option list set"   and z :: "'a" assumes "min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j) \<le> k"   and "Suc k \<le> Monitor.progress \<sigma> P' \<phi> j'"   and "Suc k \<le> Monitor.progress \<sigma> P' \<psi> j'"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) k \<phi>) X"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) k \<psi>) Y"   and "U k z" shows "U (Suc k) (f X Y (\<tau> \<sigma> k) z)"
assms_2: shows "(\<lambda>i. \<Sum>\<^sub>ax\<in>S i. norm (f x)) abs_summable_on I"
vs'_vs: shows "vsubsts n vs' vs = rev [0..<n]"
STA_sub: fixes A :: "'b set"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and f :: "'b \<Rightarrow> 'b" assumes "A \<turnstile> branch"   and "finite A" shows "f ` A \<turnstile> sub_branch f branch"
distinct_2: fixes condition' :: "'c"   and psi' :: "('a, 'b, 'c) psi"   and psiCase' :: "('a, 'b, 'c) psiCase" shows "\<box> condition' \<Rightarrow> psi' psiCase'  \<noteq> \<bottom>\<^sub>c"
\<E>_E_RF: fixes x :: "'a"   and S :: "'a set"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "x \<in> \<E>\<^bsub>\<Gamma>\<^esub> S" obtains p :: "'a list" and y :: "'a" where "path \<Gamma> x p y" and "y \<in> B \<Gamma>" and "distinct (x # p)" and "\<And>z. z \<in> set p \<Longrightarrow> z \<notin> RF\<^bsub>\<Gamma>\<^esub> S"
simp_el_sing: fixes b :: "'b list"   and G :: "'b list set"   and us :: "'b list list" assumes " b \<in>B G "   and "us \<in> lists G\<^sub>+"   and "concat us = b" shows "us = [b]"
run_altc_nondet: fixes A :: "'c cset"   and f :: "'c \<Rightarrow> ('a, 'm) nondetT" shows "run_nondet (altc_nondet A f) = mergec A (run_nondet \<circ> f)"
RModule_axioms: shows "RModule UNIV (\<cdot>) V"
conjE1_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "conj_PL p q \<in> Cn A" shows "p \<in> Cn A"
of_finite_def: fixes x :: "('a, 'b) IEEE.float" shows "of_finite x = (if is_normal x then Float (normal_mantissa x) (normal_exponent x) else if is_denormal x then Float (denormal_mantissa x) (denormal_exponent TYPE(('a, 'b) IEEE.float)) else 0)"
assms_1: shows "AE x in M. 0 \<le> f x"
zip_nth_conv: fixes xs :: "'a list"   and ys :: "'b list" assumes "length xs = length ys" shows "zip xs ys = map (\<lambda>i. (xs ! i, ys ! i)) [0..<length ys]"
\<psi>T_hom: fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "HRepHom (\<star>) (\<psi> T)"
wf_dec: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type" assumes "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f T" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f \<down>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 T"
quotient_of_coprime: fixes r :: "rat"   and p :: "int"   and q :: "int" assumes "quotient_of r = (p, q)" shows "coprime p q"
indexed_nominal_ts_axioms: shows "indexed_nominal_ts TYPE('idx) (\<turnstile>\<^sub>L) (\<rightarrow>\<^sub>L)"
dghm_proj_fst_is_dghm: fixes i :: "V"   and I :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "\<pi>\<^sub>D\<^sub>G\<^sub>.\<^sub>1 \<AA> \<BB> : \<AA> \<times>\<^sub>D\<^sub>G \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA>"
irreducible\<^sub>d_smultI: fixes c :: "'a"   and p :: "'a poly" assumes "c \<noteq> (0::'a)"   and "irreducible\<^sub>d (smult c p)" shows "irreducible\<^sub>d p"
prems: shows "\<forall>a\<in>set (ab # pi). case a of (a, b) \<Rightarrow> a \<sharp> x \<and> b \<sharp> x"
stream_space_eq_sstart: fixes S :: "'a set"   and M :: "'a stream measure"   and N :: "'a stream measure" assumes "countable S"   and "prob_space M"   and "prob_space N"   and "AE x in M. x \<in> streams S"   and "AE x in N. x \<in> streams S"   and "sets M = sets (stream_space (count_space UNIV))"   and "sets N = sets (stream_space (count_space UNIV))"   and "\<And>xs. \<lbrakk>xs \<noteq> []; xs \<in> lists S\<rbrakk> \<Longrightarrow> emeasure M (sstart S xs) = emeasure N (sstart S xs)" shows "M = N"
equality: fixes r :: "(_) Object"   and r' :: "(_) Object" assumes "RObject.nothing r = RObject.nothing r'"   and "RObject.more r = RObject.more r'" shows "r = r'"
abs_convergent_prodI: fixes f :: "nat \<Rightarrow> 'e" assumes "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - (1::'e)))" shows "abs_convergent_prod f"
wls_Op_simp: fixes inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and s :: "'sort"   and delta :: "'opSym" assumes "goodInp inp"   and "goodBinp binp" shows "wls s (Op delta inp binp) = (stOf delta = s \<and> wlsInp delta inp \<and> wlsBinp delta binp)"
awalk_ends: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and u' :: "'c"   and v' :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "pre_digraph.awalk G u' p v'" shows "p \<noteq> [] \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
fst_leq_snd_safe_distance: assumes "a\<^sub>e < a\<^sub>o" shows "fst_safe_distance \<le> snd_safe_distance"
simps_11: fixes \<phi> :: "atom fm" shows "clearQuantifiers (AllN 0 \<phi>) = clearQuantifiers \<phi>"
InterpVarE: fixes I :: "('a, 'b, 'c, 'd, 'e) Interpretation_scheme"   and A :: "'a"   and i :: "('c, 'd) IType" assumes "L\<lbrakk>Vx : A \<turnstile> Vx : A\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i" obtains c :: "'c" where "i = IMor (id\<^bsub>iC\<^bsub>I\<^esub>\<^esub> c)" and "L\<lbrakk>\<turnstile> A Type\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IObj c"
Z_inv: shows "Z * Z = 1\<^sub>m 2"
TLA2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and B :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'd" assumes "\<turnstile> P \<longrightarrow> Q"   and "|~ P \<and> \<circle>P \<and> _actrans A f \<longrightarrow> _actrans B g" shows "\<turnstile> \<box>P \<and> \<box>[A]_f \<longrightarrow> \<box>Q \<and> \<box>[B]_g"
borel_measurable_continuous_on_indicator: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "A \<in> sets borel"   and "continuous_on A f" shows "(\<lambda>x. indicat_real A x *\<^sub>R f x) \<in> borel_measurable borel"
asymp_equivI': fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "((\<lambda>x. f x / g x) \<longlongrightarrow> (1::'c)) F" shows "f \<sim>[F] g"
simps_5: fixes b1 :: "bool"   and v :: "'a gexp"   and va :: "'a gexp" shows "local.less_gexp_aux (Bc b1) (Nor v va) = True"
f_setI: shows "(\<lambda>I i. if i < n then sorted_list_of_set I ! i else i) ` {I. I \<subseteq> {0..<m} \<and> card I = n} = local.F_strict"
publicKey_inject: fixes b :: "keymode"   and A :: "agent"   and c :: "keymode"   and A' :: "agent" shows "(publicKey b A = publicKey c A') = (b = c \<and> A = A')"
thirdh: fixes x :: "real" assumes "\<forall>y<x. a * y\<^sup>2 + b * y + c < 0"   and "\<not> a < 0"   and "\<not> 0 < b" shows "c < 0"
rotate_topneg_def: fixes x :: "wall"   and y :: "wall" shows "rotate_topneg x y \<equiv> x = basic [vert, brick.under] \<circ> basic [cap, vert] \<and> y = basic [over, vert] \<circ> basic [vert, cap]"
ExecLenTraceLen: shows "length exec' - 1 = length trace'"
injective_conv_univalent: fixes x :: "'a" shows "injective x = univalent (x\<^sup>T)"
local_diagram1_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and \<beta> :: "'a \<times> ('b \<times> 'a) list"   and \<alpha> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>1 :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>2 :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>3 :: "'a \<times> ('b \<times> 'a) list" shows "local_diagram1 ars \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 = (local_peak ars (\<beta>, \<alpha>) \<and> {\<sigma>1, \<sigma>2, \<sigma>3} \<subseteq> seq ars \<and> lst \<beta> = fst \<sigma>1 \<and> lst \<sigma>1 = fst \<sigma>2 \<and> lst \<sigma>2 = fst \<sigma>3)"
conc_fun_FAIL: fixes R :: "('b \<times> 'c) set" shows "\<Down> R FAIL = FAIL"
quorum_for_def': fixes Q :: "process set"   and v :: "val"   and v_f :: "process \<Rightarrow> val option" shows "quorum_for Q v v_f \<equiv> Q \<in> Quorum \<and> v_f ` Q = {Some v}"
poly_add_hom: fixes hom :: "'a \<Rightarrow> 'b"   and p :: "'a poly"   and q :: "'a poly" assumes "idom_hom hom"   and "hom (lead_coeff p) \<noteq> (0::'b)"   and "hom (lead_coeff q) \<noteq> (0::'b)" shows "map_poly hom (poly_add p q) = poly_add (map_poly hom p) (map_poly hom q)"
induced_automorphism_half_chamber_system_pgallery_map_f: fixes Cs :: "'a set list" assumes "order.greater_eq (f \<turnstile> folding_g.\<C>) (set Cs)"   and "pgallery Cs" shows "pgallery (\
 \<Turnstile> Cs)"
index_valid_mono: fixes as :: "('a \<times> 'b atom) set"   and bs :: "('a \<times> 'b atom) set"   and s :: "('a, 'b) state" assumes "as \<subseteq> bs"   and "index_valid as s" shows "index_valid bs s"
completeness: fixes Cs :: "fterm literal set set" assumes "finite Cs"   and "\<forall>C\<in>Cs. finite C"   and "\<forall>F G. \<not> eval\<^sub>c\<^sub>s F G Cs" shows "\<exists>Cs'. resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
matchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "\<And>P Q a. (P, Q) \<in> Rel \<Longrightarrow> ([a\<frown>a]P, Q) \<in> Rel"   and "Rel \<subseteq> Rel'" shows "[a\<frown>b]P \<leadsto>\<^sup>^<Rel'> [a\<frown>b]Q"
ring_homD: fixes Phi :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c" assumes "Phi \<in> ring_hom P S" shows "ring_hom_cring P S Phi"
functor_axioms: shows "functor B.VV.comp (\<cdot>\<^sub>B) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>)"
ex_gt: shows "\<forall>x. count D x < count C x \<longrightarrow> (\<exists>y>x. count C y < count D y)"
pfa_ensure_def: shows "pfa_ensure \<equiv> \<lambda>(a, n) k. array_ensure a k default \<bind> (\<lambda>a'. return (a', n))"
case_d1: fixes b :: "real"   and a :: "real"   and r :: "real"   and c :: "real" assumes "b < 2 * a * r"   and "a * r\<^sup>2 - b * r + c = 0" shows "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. a * x\<^sup>2 + b * x + c < 0"
elem_list_splits_append: fixes ys :: "'b list"   and y :: "'b"   and zs :: "'b list"   and xs :: "'b list" assumes "(ys, y, zs) \<in> set (list_splits xs)" shows "ys @ [y] @ zs = xs"
not_eventuallyD: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter" assumes "\<not> eventually P F" shows "\<exists>x. \<not> P x"
assms_2: shows "S \<noteq> dSUCCEED"
comm_common_power: fixes r :: "'b list"   and u :: "'b list" assumes "r \<cdot> u = u \<cdot> r" shows "r \<^sup>@ \<^bold>|u\<^bold>| = u \<^sup>@ \<^bold>|r\<^bold>|"
net_1: shows "(f \<longlongrightarrow> l) net"
mon_env_map_loc: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and w :: "('c, 'f) label list list" shows "mon_env fg (map LOC w) = {}"
p_3: fixes i :: "nat" assumes "i < nr"   and "Suc i < nr" shows "f i < f (Suc i) \<or> f (Suc i) = jj"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>S1 S2. S1 \<in>S a1 \<and> S2 \<in>S a2 \<and> (S1, S2) \<in> SRel) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel))"
word_ao_dist: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "(x || y) && z = x && z || y && z"
lexord_fusion_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and g1 :: "('c, 'd) generator"   and g2 :: "('c, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "ord.lexord_fusion less g1 g2 s1 s2 = ord.lexordp less (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
\<phi>_ide_simp: fixes f :: "'a" assumes "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"   and "ide f" shows "\<phi> f = (d\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[e\<^sub>1, f \<star> d\<^sub>0, e\<^sub>0]) \<cdot> \<a>[d\<^sub>1, e\<^sub>1, (f \<star> d\<^sub>0) \<star> e\<^sub>0] \<cdot> ((d\<^sub>1 \<star> e\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, d\<^sub>0, e\<^sub>0]) \<cdot> (\<eta>\<^sub>1 \<star> f \<star> \<eta>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src e\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f]"
sums': shows "(\<lambda>k. y\<^sup>2 ^ k / real (2 * (k + n) + 1)) sums (c * (ln x - (\<Sum>k<n. 2 * y ^ (2 * k + 1) / real (2 * k + 1))))"
prime_elem_dvd_multD: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times p (times a b)" shows "dvd.dvd times p a \<or> dvd.dvd times p b"
carrier_vecI: fixes v :: "'c vec"   and n :: "nat" assumes "dim_vec v = n" shows "v \<in> carrier_vec n"
padic_mult_res: fixes p :: "int"   and f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int"   and n :: "nat" shows "padic_mult p f g n = f n \<otimes>\<^bsub>residue_ring (p ^ n)\<^esub> g n"
mod_div_equality_div_eq: fixes a :: "int"   and b :: "int" shows "a div b * b = a - a mod b"
assms_2: shows "alw (\<lambda>xs. \<phi> xs \<longrightarrow> nxt \<phi> xs) xs"
minus_equality: fixes y :: "'a"   and x :: "'a" assumes "y \<oplus> x = \<zero>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus> x = y"
int_binomial: fixes n :: "nat"   and k :: "nat" shows "int (n choose k) = int n gchoose k"
sxalloc_no_jump': fixes G :: "prog"   and s0 :: "state"   and s1 :: "state"   and j :: "jump" assumes "G\<turnstile>s0 \<midarrow>sxalloc\<rightarrow> s1"   and "fst s1 = Some (Jump j)" shows "fst s0 = Some (Jump j)"
count_def: shows "count \<equiv> rec_FingerTreeStruc 0 count_node (\<lambda>_ pr m sf ma. count_digit pr + ma + count_digit sf)"
\<O>_on_emptyset: shows "cring0.sheaf_spec {} = {cring0.\<O>b}"
map_prod_def: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
homotopically_trivial_retraction_gen: fixes s :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and t :: "'b set"   and k :: "'b \<Rightarrow> 'a"   and U :: "'c set"   and Q :: "('c \<Rightarrow> 'b) \<Rightarrow> bool"   and P :: "('c \<Rightarrow> 'a) \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "Retracts s h t k"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P (k \<circ> f)"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q (h \<circ> f)"   and "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"   and "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> s; P f; continuous_on U g; g ` U \<subseteq> s; P g\<rbrakk> \<Longrightarrow> homotopic_with_canon P U s f g"   and "continuous_on U f"   and "f ` U \<subseteq> t"   and "Q f"   and "continuous_on U g"   and "g ` U \<subseteq> t"   and "Q g" shows "homotopic_with_canon Q U t f g"
h: shows "h \<equiv> \<lambda>(s, b). if b then f s else - f s"
set_restriction_Diff: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set"   and B :: "'a set" assumes "set_restriction f" shows "f (A - B) = f A - f B"
all_prerequisites_def: fixes m :: "of_match_field set" shows "all_prerequisites m \<equiv> \<forall>f\<in>m. prerequisites f m"
sys_reg_val_def: fixes reg :: "sys_reg"   and state :: "'b sparc_state" shows "sys_reg_val reg state \<equiv> sys_reg state reg"
nat_pow_one: fixes G :: "('a, 'b) monoid_scheme"   and n :: "nat" assumes "Group.monoid G" shows "\<one>\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>"
brD: fixes c :: "'a"   and a :: "'b"   and \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" assumes "(c, a) \<in> br \<alpha> I" shows "a = \<alpha> c \<and> I c"
phiCount_def: fixes g :: "'g" shows "other.phiCount g = card ((\<lambda>(n, v). (n, var' g v)) ` dom (phis' g))"
monoid_mult_isom_axioms: shows "monoid_mult_isom (inv (inv hom))"
map_eq_append_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'a list"   and zs :: "'a list" shows "(map f xs = ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
Seq: fixes c1 :: "com"   and s1 :: "char list \<Rightarrow> int option"   and x :: "nat"   and s2 :: "char list \<Rightarrow> int option"   and c2 :: "com"   and y :: "nat"   and s3 :: "char list \<Rightarrow> int option"   and z :: "nat" assumes "(c1, s1) \<Rightarrow>\<^sub>A x \<Down> s2"   and "(c2, s2) \<Rightarrow>\<^sub>A y \<Down> s3"   and "z = x + y" shows "(c1;; c2, s1) \<Rightarrow>\<^sub>A z \<Down> s3"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "l_new_character_data_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"
isovarspar_sum: fixes p :: "'a mpoly"   and q :: "'a mpoly"   and var :: "nat"   and deg :: "nat" shows "isolate_variable_sparse (p + q) var deg = isolate_variable_sparse p var deg + isolate_variable_sparse q var deg"
plus_Integer_def: fixes a :: "Integer"   and b :: "Integer" shows "a + b = (case a of MkI\<cdot>x \<Rightarrow> \<Lambda> (MkI\<cdot>y). MkI\<cdot>(x + y))\<cdot>b"
extreme_boundI: fixes X :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and s :: "'b" assumes "\<And>b. \<lbrakk>bound X less_eq b; b \<in> A\<rbrakk> \<Longrightarrow> less_eq s b"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x s"   and "s \<in> A" shows "extreme_bound A less_eq X s"
delete_append: fixes x :: "'d"   and l1 :: "('d \<times> 'e) list"   and l2 :: "('d \<times> 'e) list" shows "delete x (l1 @ l2) = delete x l1 @ delete x l2"
ed_ou_not_discrete_card: fixes X :: "'a set" assumes "ed_ou X"   and "\<not> discrete X" shows "card {f. CK_nf_pos f} = 4"
simps_6: fixes ux :: "(((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and b :: "bexp"   and C :: "acom" shows "strip ({ux} WHILE b DO C) = WHILE b DO strip C"
assms: shows "raw_match' (Ty a (T # Ts)) (Ty b (U # Us)) subs = Some subs'"
correctCompositionDiffLevelsS7opt: shows "correctCompositionDiffLevels sS7opt"
lub_upper: fixes A :: "'a option set"   and x :: "'a option" assumes "Complete_Partial_Order.chain option_ord A"   and "x \<in> A" shows "option_ord x (flat_lub None A)"
distinct_1: fixes x2 :: "'d" shows "None \<noteq> \<lfloor>x2\<rfloor>"
Amax_geTr: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat" shows "f \<in> {j. j \<le> n} \<rightarrow> Z\<^sub>-\<^sub>\<infinity> \<longrightarrow> (\<forall>j\<in>{j. j \<le> n}. f j \<le> Amax n f)"
Inter_Un_distrib: fixes A :: "'c set set"   and B :: "'c set set" shows "\<Inter> (A \<union> B) = \<Inter> A \<inter> \<Inter> B"
Inf_between_def: fixes N :: "'f set"   and M :: "'f set" shows "no_labels.Inf_between N M = no_labels.Inf_from (N \<union> M) - no_labels.Inf_from (N - M)"
absorb2: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "a + b = b"
iso_finite: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme" assumes "G \<cong> H" shows "finite (carrier G) = finite (carrier H)"