hcju/isabelleps huggingface dataset:
eval_poly_monom: fixes aa :: "'a"   and a :: "'a poly"   and n :: "nat"   and x :: "'a" shows "eval_poly (\<lambda>p. poly p aa) (monom a n) x = poly a aa * x ^ n"
A: shows "\<nexists>e. e \<in> arcs G - set p \<and> (tail G e \<in> set (awalk_verts u p) \<or> head G e \<in> set (awalk_verts u p))"
rcs_fixed1: fixes H :: "'a set"   and a :: "'a"   and h :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "h \<in> H" shows "H \<bullet> a = H \<bullet> (h \<cdot> a)"
len_ge2: fixes ts :: "'a tree23s" assumes "not_T ts" shows "2 \<le> len ts"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Cod t)"
card012: shows "card X = 0 \<or> card X = 1 \<or> card X = 2"
cls_val_process_adapted: fixes p :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "trading_strategy p"   and "support_adapt Mkt p" shows "borel_adapt_stoch_proc F (cls_val_process Mkt p)"
vrrestriction_on_vempty: fixes r :: "V" shows "r \<restriction>\<^sup>r\<^sub>\<circ> 0 = 0"
L_ide: fixes l :: "'a \<Rightarrow> 'b"   and \<chi> :: "'a \<Rightarrow> 'j \<Rightarrow> 'b"   and a :: "'a" assumes "\<forall>a. A.ide a \<longrightarrow> limit_cone (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (\<lambda>j. D (j, a)) (l a) (\<chi> a)"   and "A.ide a" shows "L l \<chi> a = l a"
proj_finite_bound: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "lfinite (\<pi>\<^bsub>c\<^esub>inf_llist t)" shows "\<exists>n. \<forall>n'>n. \<not> \<parallel>c\<parallel>\<^bsub>t n'\<^esub>"
map_prod_simp: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "map_prod f g (a, b) = (f a, g b)"
m_2: shows "\<forall>t\<in>S_Min_Lt. t < m"
upper_triangular_zero: fixes n :: "nat" shows "upper_triangular (0\<^sub>m n n)"
r_into_rtrancl: fixes p :: "'b \<times> 'b"   and r :: "('b \<times> 'b) set" assumes "p \<in> r" shows "p \<in> r\<^sup>*"
idem: fixes a :: "'c multiset" shows "a \<union># a = a"
ss0_nth_inA: shows "\<forall>i<length ss0. ss0 ! i \<in> A"
residue_add: fixes s :: "complex set"   and z :: "complex"   and f :: "complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "open s"   and "z \<in> s"   and "f holomorphic_on s - {z}"   and "g holomorphic_on s - {z}" shows "residue (\<lambda>z. f z + g z) z = residue f z + residue g z"
Source_def: fixes T :: "('a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) \<times> 'a) set" shows "Source T \<equiv> source ` T"
lformula_nFNot: fixes \<phi> :: "('a, 'k) aformula" assumes "lformula (FNot \<phi>)" shows "lformula (nFNot \<phi>)"
iT_Plus_neg_Plus_ge_Min_eq: fixes b :: "nat"   and a :: "nat"   and I :: "nat set" assumes "b \<le> a"   and "a \<le> iMin I" shows "I \<oplus>- a \<oplus> b = I \<oplus>- a - b"
Bourbaki_Nakayama: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "S_inductive_set D"   and "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x" shows "\<exists>x0\<in>carrier D. f x0 = x0"
ceq_vector_fraction_iff: fixes x :: "'d"   and u :: "complex"   and v :: "complex"   and a :: "'d" shows "(x = (u / v) *\<^sub>C a) = (if v = 0 then x = (0::'d) else v *\<^sub>C x = u *\<^sub>C a)"
aEval_lowerAtom: fixes A :: "atom"   and x :: "real"   and L :: "real list" assumes "freeIn 0 (fm.Atom A)" shows "aEval A (x # L) = aEval (lowerAtom 0 1 A) L"
tdghm_in_Vset: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V"   and \<beta> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<NN> \<in>\<^sub>\<circ> Vset \<beta>"
P1DQ1: shows "(P1, D, Q1) = diagonalize_2x2_JNF A"
RDirProd_list_nil: shows "RDirProd_list [] = nil_ring"
abs_eq: fixes x :: "('c \<times> nat) list" assumes "eq_onp (\<lambda>x. x \<in> Collect monom_inv) x x" shows "monom_vars_list (Abs_monom x) = map fst x"
Var\<^sub>0_def: fixes n :: "'c" shows "Var\<^sub>0 n \<equiv> monomial (1::'d) (monomial 1 n)"
assoc: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> b \<oplus> c = a \<oplus> (b \<oplus> c)"
init_retains_locals: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and w :: "vals"   and s' :: "state"   and C :: "qtname" assumes "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w, s')"   and "t = In1r (Init C)" shows "locals (snd s) = locals (snd s')"
assms_1: shows "i *\<^sub>R proj2_line_rep l + j *\<^sub>R proj2_line_rep m = 0"
vsum_commute_eqpoll: fixes a :: "V"   and b :: "V" shows "elts (a \<Uplus> b) \<approx> elts (b \<Uplus> a)"
lDown_tDouble: assumes "invar l \<and> invar r \<and> lv = lvl r \<and> lv = Suc (Suc (lvl l)) \<and> sngl r" shows "thesis"
assoc'_naturality: fixes \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a>\<^sup>-\<^sup>1[cod \<mu>, cod \<nu>, cod \<tau>] \<cdot> (\<mu> \<star> \<nu> \<star> \<tau>) = ((\<mu> \<star> \<nu>) \<star> \<tau>) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom \<mu>, local.dom \<nu>, local.dom \<tau>]"
mkArr_def: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" shows "mkArr A B F = (if A \<subseteq> Univ \<and> |A| <o \<AA> \<and> B \<subseteq> Univ \<and> |B| <o \<AA> \<and> F \<in> A \<rightarrow> B then THE f. f \<in> hom (mkIde A) (mkIde B) \<and> Fun f = restrict F A else null)"
rep_eq: fixes x :: "complex" shows "Rep_complex_homo_coords (of_complex_hcoords x) = of_complex_cvec x"
2: shows "t\<^sup>T \<le> v * v\<^sup>T \<sqinter> g"
props': shows "hd ls = 0 \<and> last ls = nmax - 1 \<and> |ls| = m \<and> last (butlast ls) < last ls \<and> increasing ls"
tensor_mat_unitary: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)"   and "unitary m1"   and "unitary m2" shows "unitary (tensor_mat dims vars m1 m2)"
cInf_superset_mono: fixes A :: "'a multiset set"   and B :: "'a multiset set" assumes "A \<noteq> {}"   and "subset_mset.bdd_below B"   and "A \<subseteq> B" shows "Inf B \<subseteq># Inf A"
connected_space_iff_connected_component: fixes X :: "'b topology" shows "connected_space X = (\<forall>x\<in>topspace X. \<forall>y\<in>topspace X. connected_component_of X x y)"
sets_uniform_measure: fixes M :: "'b measure"   and A :: "'b set" shows "sets (uniform_measure M A) = sets M"
homotopy_eqv_rel_frontier_punctured_convex: fixes S :: "'a set"   and a :: "'a"   and T :: "'a set" assumes "convex S"   and "bounded S"   and "a \<in> rel_interior S"   and "convex T"   and "rel_frontier S \<subseteq> T"   and "T \<subseteq> affine hull S" shows "rel_frontier S homotopy_eqv T - {a}"
NetsEq_if_sameP_DD: fixes p :: "((int \<times> int) set set, int) Combinators list"   and u :: "((int \<times> int) set set, int) Combinators"   and v :: "((int \<times> int) set set, int) Combinators"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "allNetsDistinct p"   and "u \<in> set p"   and "v \<in> set p"   and "u = DenyAllFromTo a b"   and "v = DenyAllFromTo c d"   and "x \<in> dom (C u)"   and "x \<in> dom (C v)" shows "a = c \<and> b = d"
T_S': shows "T \<subseteq> \<Union> (S' ` I)"
prefix_code_3: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
word_must_wrap: fixes x :: "'a word"   and n :: "'a word" assumes "x \<le> n - 1"   and "n \<le> x" shows "n = 0"
higher_pderiv_minus: fixes n :: "nat"   and p :: "'a poly" shows "(pderiv ^^ n) (- p) = - (pderiv ^^ n) p"
prime_all: shows "list_all (\<lambda>rf. primerec rf (length xs)) (rgs @ rcs)"
exists_split_characterisations_2: shows "(\<exists>x. linear_orderable_1 x) = (\<exists>x. linear_orderable_3 x)"
subtopology_def: fixes U :: "'c topology"   and V :: "'c set" shows "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
coeff_add: fixes p :: "'b poly"   and q :: "'b poly"   and n :: "nat" shows "coeff (p + q) n = coeff p n + coeff q n"
rel_interior_convex_linear_preimage: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'b set" assumes "linear f"   and "convex S"   and "f -` rel_interior S \<noteq> {}" shows "rel_interior (f -` S) = f -` rel_interior S"
floor_exists: fixes x :: "'a" shows "\<exists>z. of_int z \<le> x \<and> x < of_int (z + 1)"
simps_2: fixes n :: "nat"   and s :: "'c stream" shows "sdrop (Suc n) s = sdrop n (stl s)"
subset_mtriples_iff: fixes ts :: "'b triple set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> 'b assn"   and mb :: "'c \<Rightarrow> 'd \<Rightarrow> expr"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> 'b assn"   and ms :: "('c \<times> 'd) set" shows "(ts \<subseteq> {{P} mb-\<succ> {Q} | ms}) = (\<exists>ms'\<subseteq>ms. ts = {{P} mb-\<succ> {Q} | ms'})"
church_rosser: fixes y :: "'a"   and x :: "'a" assumes "y\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
openin_trans_full: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set" assumes "openin (subtopology X U) S"   and "openin X U" shows "openin X S"
Cut_Atom_depth: fixes k :: "'b"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and m :: "nat" assumes "Atom k, \<Gamma> \<Rightarrow> \<Delta> \<down> n"   and "\<Gamma> \<Rightarrow> Atom k, \<Delta> \<down> m" shows "\<Gamma> \<Rightarrow> \<Delta> \<down> n + m"
legal_executionI: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and \<E> :: "('a \<times> ('b, 'a) obs_event action) llist set"   and P :: "'c prog"   and ws :: "nat \<Rightarrow> nat"   and is_justification :: "'c prog \<Rightarrow> ('a \<times> ('b, 'a) obs_event action) llist \<times> (nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> ('b, 'a) justifying_execution) \<Rightarrow> bool"   and J :: "nat \<Rightarrow> ('b, 'a) justifying_execution" assumes "E \<in> \<E>"   and "P \<turnstile> (E, ws) \<surd>"   and "is_justification P (E, ws) J"   and "range (justifying_exec \<circ> J) \<subseteq> \<E>" shows "gen_legal_execution is_justification P \<E> (E, ws)"
orbitp_orbit_eq: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "orbitp f x = (\<lambda>x. x \<in> orbit f x)"
sum_Un: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B" shows "sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
fixpoint_to_comp_pow: fixes m :: "'d \<Rightarrow> nat"   and f :: "'d \<Rightarrow> 'd option"   and x :: "'d"   and y :: "'d option" assumes "fixpoint m f x = y" shows "\<exists>n. option_comp_pow f n x = y"
up_defined: fixes x :: "'b" shows "up\<cdot>x \<noteq> \<bottom>"
prems_5: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) M"
conn2_2: shows "length (map of_int_hom.vec_hom fs') = m"
mod_div_equality_int: fixes n :: "int"   and x :: "int" shows "n div x * x = n - n mod x"
less_irrefl: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "class.preorder less_eq less" shows "\<not> less x x"
inf_greatest: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y"   and "less_eq x z" shows "less_eq x (inf y z)"
elims_3: fixes x :: "nat"   and xa :: "floatarith"   and xb :: "real list" assumes "\<not> isDERIV x xa xb" obtains xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xc__" and "xa = Add a__ b__" and "xb = vs__" and "\<not> (isDERIV xc__ a__ vs__ \<and> isDERIV xc__ b__ vs__)"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xca__" and "xa = Mult aa__ ba__" and "xb = vsa__" and "\<not> (isDERIV xca__ aa__ vsa__ \<and> isDERIV xca__ ba__ vsa__)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcb__" and "xa = Minus ab__" and "xb = vsb__" and "\<not> isDERIV xcb__ ab__ vsb__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcc__" and "xa = Inverse ac__" and "xb = vsc__" and "\<not> (isDERIV xcc__ ac__ vsc__ \<and> interpret_floatarith ac__ vsc__ \<noteq> 0)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcd__" and "xa = Cos ad__" and "xb = vsd__" and "\<not> isDERIV xcd__ ad__ vsd__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xce__" and "xa = Arctan ae__" and "xb = vse__" and "\<not> isDERIV xce__ ae__ vse__"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xcf__" and "xa = Min af__ bb__" and "xb = vsf__"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xcg__" and "xa = Max ag__ bc__" and "xb = vsg__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xch__" and "xa = Abs ah__" and "xb = vsh__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xci__" and "xa = Sqrt ai__" and "xb = vsi__" and "\<not> (isDERIV xci__ ai__ vsi__ \<and> 0 < interpret_floatarith ai__ vsi__)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcj__" and "xa = Exp aj__" and "xb = vsj__" and "\<not> isDERIV xcj__ aj__ vsj__"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xck__" and "xa = Powr ak__ b" and "xb = vsk__" and "\<not> (isDERIV xck__ ak__ vsk__ \<and> isDERIV xck__ b vsk__ \<and> 0 < interpret_floatarith ak__ vsk__)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcl__" and "xa = Ln al__" and "xb = vsl__" and "\<not> (isDERIV xcl__ al__ vsl__ \<and> 0 < interpret_floatarith al__ vsl__)"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcm__" and "xa = Floor am__" and "xb = vsm__" and "\<not> (isDERIV xcm__ am__ vsm__ \<and> interpret_floatarith am__ vsm__ \<notin> \<int>)"   | xc :: "nat" and a :: "floatarith" and n :: "nat" and vs :: "real list" where "x = xc" and "xa = a ^\<^sub>e Suc n" and "xb = vs" and "\<not> isDERIV xc a vs"
proots_within_iff: fixes x :: "'a"   and p :: "'a poly"   and s :: "'a set" shows "(x \<in> proots_within p s) = (poly p x = (0::'a) \<and> x \<in> s)"
is_red_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and a :: "'t \<Rightarrow>\<^sub>0 'd" shows "is_red F a = (\<not> relation.is_final (red F) a)"
of_nat_add: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m + n) = plus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
abs_non_negative: fixes x :: "int" shows "0 \<le> Preliminaries.abs x"
R1: shows "\<forall>x y. ((x, y) \<in> r) = (\<exists>u. p_r x y u = 0)"
qe_eq_repeat_eval_augment: fixes amount :: "nat"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "amount \<le> var + 1" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (step_augment qe_eq_repeat IdentityHeuristic amount var L F) (xs @ \<Gamma>))"
parenthesis: fixes v :: "'v"   and w :: "'v" assumes "v \<in> dom (finished s)"   and "w \<in> dom (discovered s)"   and "\<delta> s v < \<delta> s w" shows "\<phi> s v < \<delta> s w \<or> \<delta> s w < \<phi> s v \<and> w \<in> dom (finished s) \<and> \<phi> s w < \<phi> s v"
simps_16: fixes x41 :: "nterm"   and x42 :: "nterm"   and x31 :: "name"   and x32 :: "nterm" shows "x41 $\<^sub>n x42 \<noteq> (\<Lambda>\<^sub>n x31. x32)"
INT_lower: fixes a :: "'a"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "a \<in> A" shows "\<Inter> (B ` A) \<subseteq> B a"
select_object\<^sub>S\<^sub>e\<^sub>t_def: shows "select_object\<^sub>S\<^sub>e\<^sub>t = select_object Set{} UML_Set.OclIncluding id"
cross7_skew: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "x \<times>\<^sub>7 y = - (y \<times>\<^sub>7 x)"
rfb_rbd_dual: fixes R :: "('d \<times> 'c) set" shows "fb\<^sub>\<R> R = \<partial>\<^sub>F (bd\<^sub>\<R> (\<smile> R))"
attract: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "y \<le> x"   and "y \<le> z" shows "x \<le> z"
assms: fixes i :: "'i" shows "{x \<in> \<Omega>. P i x} \<in> M"
normalize_rules_2: fixes b :: "bool"   and P :: "assn"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn" assumes "b \<Longrightarrow> <P> f <Q>" shows "<P * \<up> b> f <Q>"
simps_4: fixes nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "disconnected_nodes \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = disconnected_nodes"
intro: fixes G :: "'c set"   and smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and V :: "'a set" assumes "\<And>U. RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V \<Longrightarrow> U = 0 \<or> U = V" shows "IrrFinGroupRepresentation_axioms G smult V"
Suc_1: fixes a :: "'c"   and b :: "'b" assumes "c < k"   and "to_nat a = to_nat b"   and "c \<le> to_nat a"   and "k < min (nrows A) (ncols A)" shows "diagonal_to_Smith_aux A [0..<k] bezout $ from_nat c $ from_nat c dvd diagonal_to_Smith_aux A [0..<k] bezout $ a $ b"
sigma_fin_subalg: shows "sigma_finite_measure (restr_to_subalg M F)"
numeral_Bit0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
h1: shows "MPoly_Type.degree p var = 1 \<longrightarrow> p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\<^sup>2"
iwlsFSb_termFSbImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd" assumes "iwlsFSb MOD" shows "termFSbImorph h hA MOD = ipresCons h hA MOD"
trm_prm_simp_5: fixes \<pi> :: "'b prm"   and A :: "'b trm"   and B :: "'b trm" shows "\<pi> \<cdot> SimplyTyped.Pair A B = SimplyTyped.Pair (\<pi> \<cdot> A) (\<pi> \<cdot> B)"
nlistListRCons: fixes t :: "('a, 'b) FingerTreeStruc"   and xs :: "('a, 'b) Node list" shows "FingerTreeStruc.toList (FingerTreeStruc.rconsNlist t xs) = FingerTreeStruc.toList t @ FingerTreeStruc.nlistToList xs"
has_white_path_toD: fixes x :: "'a"   and y :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "(x has_white_path_to y) s" shows "white y s \<or> x = y"
tauChainResPres: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and x :: "name" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "x \<sharp> \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<Longrightarrow>\<^sup>^\<^sub>\<tau> \<lparr>\<nu>x\<rparr>P'"
wf_sig_iff_exe_wf_sig: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "alist_conds cto"   and "alist_conds tao"   and "exe_osig_conds sa" shows "wf_sig (map_of cto, map_of tao, translate_osig sa) = exe_wf_sig (ExeSignature cto tao sa)"
pdec'_2: fixes e :: "nat" shows "pdec2' e = pdec2 e"
wf_atom_mono: fixes tys :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and tys' :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and P :: "ast_problem"   and a :: "'a atom" assumes "tys \<subseteq>\<^sub>m tys'"   and "ast_domain.wf_atom (domain P) tys a" shows "ast_domain.wf_atom (domain P) tys' a"
section_is_mono: fixes g :: "'d" assumes "D.section g" shows "D.mono g"
B: shows "\<forall>c. c \<in> Y \<longrightarrow> (\<forall>t1 u1. sa @ [c] setinterleaves ((t1, u1), insert tick (ev ` A)) \<longrightarrow> ((t1, {}) \<in> F P \<longrightarrow> (u1, {}) \<notin> F Q) \<and> ((t1, {}) \<in> F Q \<longrightarrow> (u1, {}) \<notin> F P))"
prime_equal: fixes p :: "nat"   and q :: "nat"   and x :: "nat" assumes "prime p"   and "prime q"   and "2 ^ x = p * q" shows "p = q"
avl_mkt_bal_r: fixes l :: "'b tree"   and r :: "'b tree"   and n :: "'b" assumes "avl l"   and "avl r"   and "height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1 \<or> height r = height l + 2" shows "avl (mkt_bal_r n l r)"
mod_add_eq_imp_mod_0: fixes n :: "nat"   and k :: "nat"   and m :: "nat" shows "((n + k) mod m = n mod m) = (k mod m = 0)"
n_value_x_1: fixes v :: "'a \<Rightarrow> ant"   and n :: "ant"   and x :: "'a" assumes "valuation K v"   and "0 \<le> n"   and "x \<in> vp K v\<^bsup> Vr K v n\<^esup>" shows "n \<le> n_val K v x"
splitAt_distinct_fst_aux: fixes vs :: "'c list"   and ram :: "'c" assumes "distinct vs" shows "distinct (fst (splitAt ram vs))"
rpc_def: fixes x :: "'b"   and y :: "'b" shows "rpc x y = \<nu>\<^sup>\<natural> (- x \<squnion> y)"
G_def: fixes p :: "'d \<Rightarrow> 'c"   and I :: "'d set" shows "sum' p I \<equiv> if finite {x \<in> I. p x \<noteq> (0::'c)} then sum p {x \<in> I. p x \<noteq> (0::'c)} else (0::'c)"
nonzero_union: fixes xs :: "'a list"   and ps :: "pexpr list"   and qs :: "pexpr list" shows "nonzero xs (List.union ps qs) = (nonzero xs ps \<and> nonzero xs qs)"
Always_LeadsToI: fixes F :: "'b program"   and INV :: "'b set"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> Always INV"   and "F \<in> INV \<inter> A \<longmapsto>w A'" shows "F \<in> A \<longmapsto>w A'"
tuple_simps_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.commutative_square f g h k" shows "C.arr \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle>"
redT_updLns_def: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and ln :: "'c \<Rightarrow>f nat"   and las :: "'c lock_actions" shows "redT_updLns ls t ln las = (\<lambda>(l, n, la). upd_threadRs n l t la) \<circ>$ ($ls, ($ln, las$)$)"
simps: fixes n :: "nat"   and l :: "nat" shows "\<Phi> (n, l) = (if f2' * real l \<le> real n then ai * (real n - f2' * real l) else if real n \<le> f1' * real l \<and> int l0 \<le> \<lfloor>real l / c\<rfloor> then ad * (f1' * real l - real n) else 0)"
homeomorphism_on_inverse: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphism_on X f" shows "homeomorphism_on (f ` X) (inv_into X f)"
lnth_lappend1: fixes n :: "nat"   and xs :: "'c llist"   and ys :: "'c llist" assumes "enat n < llength xs" shows "lnth (lappend xs ys) n = lnth xs n"
eq_fold: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k"   and M :: "'k multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z M = fold_mset f z M"
eval_simps'_2: fixes f :: "'a local.term" assumes "Arr f" shows "src \<lbrace>f\<rbrace> = \<lbrace>Src f\<rbrace>"