hcju/isabelleps huggingface dataset:
divide_eq_imp: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero"   and "b = times a c" shows "divide b c = a"
prod_cl: fixes w1 :: "'b list"   and G :: "'b list set"   and w2 :: "'b list" assumes "w1 \<in> G"   and "w2 \<in> \<langle>G\<rangle>" shows "w1 \<cdot> w2 \<in> \<langle>G\<rangle>"
multiplicity_distinct_prime_power: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "p \<noteq> q" shows "factorial_semiring.multiplicity times one p (power.power one times q n) = 0"
Domain_unfold: fixes r :: "('b \<times> 'c) set" shows "Domain r = {x. \<exists>y. (x, y) \<in> r}"
eval_Runit_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = \<r>[\<lbrace>t\<rbrace>]"
length_nat_mono_int: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "length_nat x \<le> length_nat y"
weak_group_morphism_is_iso: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set" assumes "weak_group_morphism f H (add_monoid R)" shows "(\<lambda>x. the_elem (f ` x)) \<in> Group.iso (add_monoid R Mod H) (image_group f (add_monoid R))"
vrat_zero: shows "0\<^sub>\<rat> = []\<^sub>\<circ>"
that: fixes H :: "'a set"   and K :: "'a set" assumes "H \<lhd> B"   and "K \<lhd> B"   and "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}"   and "H <#>\<^bsub>B\<^esub> K = carrier B"   and "f \<in> Group.iso A (subgroup_generated B H)"   and "g \<in> Group.iso (subgroup_generated B K) C" shows "thesis"
one_interval_def: shows "1 \<equiv> Interval (1::'a, 1::'a)"
H: shows "\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O')"
wt_ifz: fixes \<Gamma> :: "(nat \<times> ty) list"   and e1 :: "exp"   and e3 :: "exp"   and B :: "ty"   and e2 :: "exp" assumes "\<Gamma> \<turnstile> e1 : TNat 0"   and "\<Gamma> \<turnstile> e3 : B" shows "\<Gamma> \<turnstile> EIf e1 e2 e3 : B"
some_src_composable: fixes \<tau> :: "'a"   and \<mu> :: "'a" assumes "arr \<tau>" shows "(\<tau> \<star> \<mu> \<noteq> null) = (some_src \<tau> \<star> \<mu> \<noteq> null)"
postdominate_def: fixes n' :: "'node"   and n :: "'node" shows "n' postdominates n \<equiv> valid_node n \<and> valid_node n' \<and> (\<forall>as pex. n -as\<rightarrow>\<^sub>\<iota>* pex \<and> method_exit pex \<longrightarrow> n' \<in> set (sourcenodes as))"
map_bind_spmf: fixes f :: "'i \<Rightarrow> 'h"   and p :: "'j spmf"   and g :: "'j \<Rightarrow> 'i spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
assms: shows "is_some (exhaustive_class.exhaustive f (natural_of_nat n))"
exhaust: fixes y :: "atom" obtains x11 :: "nat" and x12 :: "nat" where "y = Less x11 x12"   | x21 :: "nat" and x22 :: "nat" where "y = Eq x21 x22"
replicate_append_same: fixes i :: "nat"   and x :: "'b" shows "replicate i x @ [x] = x # replicate i x"
discI: fixes list :: "'b list"   and x21 :: "'b"   and x22 :: "'b list" assumes "list = x21 # x22" shows "list \<noteq> []"
WhileFalse: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com" assumes "s \<notin> b" shows "\<Gamma>\<turnstile>While b c \<down> Normal s"
bet_in_angle_bet: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B P"   and "Tarski_neutral_dimensionless.InAngle Bet P A B C" shows "Bet A B C"
get_dom_component_get_scdom_component_is_l_get_dom_component_get_scdom_component: shows "l_get_dom_component_get_scdom_component get_owner_document heap_is_wellformed DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_scdom_component get_dom_component"
exhaust: fixes y :: "Enum.finite_1" obtains "y = a\<^sub>1"
c12_term: shows "(c1;; c2, s) \<Rightarrow> P \<Down> T"
fmlookup_filter: fixes P :: "'d \<Rightarrow> bool"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmlookup (fmfilter P m) x = (if P x then fmlookup m x else None)"
get_tag_name_is_l_get_tag_name: shows "l_get_tag_name ShadowRootClass.type_wf get_tag_name get_tag_name_locs"
is_nan_def: fixes a :: "('c, 'd) IEEE.float" shows "is_nan a = (IEEE.exponent a = emax TYPE(('c, 'd) IEEE.float) \<and> fraction a \<noteq> 0)"
trivial_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and v :: "'val" shows "uninst.trivial u p g v \<equiv> \<exists>v'\<in>uninst.allVars u p g. uninst.isTrivialPhi p g v v'"
extract_matrix_flatten_matrix: fixes A :: "'a mat" shows "extract_matrix (flatten_matrix A) (dim_row A) (dim_col A) = A"
cong_def: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.cong times minus a c b = (\<exists>k. minus a c = times b k)"
cubeImage_def: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" shows "cubeImage twoC \<equiv> twoC ` unit_cube"
monom_zero: fixes n :: "nat" shows "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
assms_2: shows "\<bar>y\<bar> \<le> pi"
BIT_eq_iff: fixes u :: "int"   and b :: "bool"   and v :: "int"   and c :: "bool" shows "(u BIT b = v BIT c) = (u = v \<and> b = c)"
ih_4: shows "InvariantWatchListsUniq (getWatchList state')"
assms_3: shows "\<Union> (Range a1) \<inter> \<Union> (Range a2) = {}"
total_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
UN_disjointed_eq: fixes A :: "nat \<Rightarrow> 'c set" shows "\<Union> (range (disjointed A)) = \<Union> (range A)"
simps_2: fixes x :: "'c \<times> 'd \<times> bool"   and xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool" shows "insert_body (x # xs) e = (if fst x < fst e then e # x # xs else x # insert_body xs e)"
not_empty_rootsE: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "roots p \<noteq> {#}" obtains a :: "'a" where "a \<in> carrier R" and "a \<in># roots p" and "[\<one>, \<ominus> a] \<in> carrier (poly_ring R)" and "[\<one>, \<ominus> a] pdivides p"
arr_char: fixes f :: "'a" shows "Right.arr f = right (src f) f"
lm085: fixes X :: "'a set" shows "X = (\<lambda>x. x \<in> X) -` {True}"
aezfun_setspan_proj_sum_list: fixes A :: "'b set"   and f :: "'c \<Rightarrow> ('a, 'b) aezfun"   and xs :: "'c list" shows "aezfun_setspan_proj A (sum_list (map f xs)) = (\<Sum>x\<leftarrow>xs. aezfun_setspan_proj A (f x))"
fixp_induct: fixes Pa :: "('c llist \<Rightarrow> 'c llist) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "llist.admissible Pa"   and "Pa (\<lambda>ldropWhile. LNil)"   and "\<And>xs. \<lbrakk>Pa xs; llist.le_fun xs (ldropWhile P)\<rbrakk> \<Longrightarrow> Pa (\<lambda>xsa. case xsa of LNil \<Rightarrow> LNil | LCons x xs' \<Rightarrow> if P x then xs xs' else xsa)" shows "Pa (ldropWhile P)"
pre_between_r1: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "pre_between vs ram1 ram2" shows "ram1 \<in> set vs"
ind: shows "abc_steps_l (0, args) (A [+] B) stp = abc_steps_l (0, args) A stp"
simps_3: fixes c :: "'c"   and uv :: "'d" shows "collect_clks (LE c uv) = {c}"
dvds_eq_Idl: fixes l :: "int"   and k :: "int" shows "(l dvd k \<and> k dvd l) = (Idl\<^bsub>\<Z>\<^esub> {k} = Idl\<^bsub>\<Z>\<^esub> {l})"
bprules_4: shows "\<forall>B\<in>P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V}. W B \<le> real c"
Always_Int_distrib: fixes A :: "'b set"   and B :: "'b set" shows "Always (A \<inter> B) = Always A \<inter> Always B"
llength_eq_0: fixes xs :: "'c llist" shows "(llength xs = 0) = lnull xs"
assms_2: fixes theta :: "'a set" assumes "theta \<in> Theta" shows "simul Retr theta"
mod_ring_normalize32: shows "(mod_ring_rel32 ===> mod_ring_rel32) (\<lambda>x. if x = 0 then 0 else 1) normalize"
IH: fixes z :: "complex" assumes "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. f w * g w) z = (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
simps_77: fixes p :: "fm"   and q :: "fm" shows "prep (And p q) = And (prep p) (prep q)"
bad_worst_chain: fixes x :: "'a"   and i :: "nat" assumes "local.bad x" shows "local.bad (local.worst_chain i)"
dimacs_model_to_abs_def: fixes dimacs_M :: "int list"   and M :: "nat \<Rightarrow> bool" shows "dimacs_model_to_abs dimacs_M M \<equiv> fold (\<lambda>l M. if 0 < l then M(nat \<bar>l\<bar> := True) else M(nat \<bar>l\<bar> := False)) dimacs_M M"
setE: fixes x :: "'e"   and xs :: "'e list" assumes "x \<in> set xs" obtains ys :: "'e list" and zs :: "'e list" where "xs = ys @ x # zs"
not_conj_deMorgans: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(\<not> (P \<and> Q)) = (\<not> P \<or> \<not> Q)"
D_impl_def: fixes ode_ops :: "'c \<times> ('d \<times> 'e \<times> 'f \<times> 'g \<times> 'a) \<times> 'h \<times> 'i" shows "D_impl ode_ops = (case ode_ops of (xa, (xb, xc, xd, xe, x), xf, xg) \<Rightarrow> x)"
integrable_subinterval: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "f integrable_on cbox a b"   and "cbox c d \<subseteq> cbox a b" shows "f integrable_on cbox c d"
inv_subset: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a list"   and f' :: "'a \<Rightarrow> 'a list" assumes "invariant P f"   and "\<And>g. P g \<Longrightarrow> set (f' g) \<subseteq> set (f g)" shows "invariant P f'"
sm: fixes \<psi> :: "'a ltl" shows "semi_mojmir \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w"
always_eventually: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter" assumes "\<forall>x. P x" shows "eventually P F"
set_sel_1: fixes a :: "'d list" assumes "a \<noteq> []" shows "hd a \<in> set a"
tauLaw4: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q \<approx> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q"
sumlist_append: fixes xs :: "int vec list"   and ys :: "int vec list" assumes "set xs \<subseteq> carrier_vec n"   and "set ys \<subseteq> carrier_vec n" shows "M.sumlist (xs @ ys) = M.sumlist xs + M.sumlist ys"
less_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.preorder less_eq less"   and "less x y"   and "less y z" shows "less x z"
cs_length_g_one: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "length (cs\<^bsup>\<pi>\<^esup> i) \<noteq> 1" obtains k :: "nat" where "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>\<^esup> k @ [\<pi> i]" and "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
first_summand_not_universe: fixes x :: "i"   and y :: "i" assumes "x \<oplus> y \<noteq> u" shows "x \<noteq> u"
prj_char_4: shows "SPN_fgh.Prj\<^sub>1 = \<lbrakk>\<lbrakk>f\<^sub>0gh\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>"
pow_mono: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a"   and n :: "nat" assumes "SN_Orders.class.ordered_semiring_1 one times plus zero less_eq"   and "less_eq b a"   and "less_eq zero b" shows "less_eq (power.power one times b n) (power.power one times a n) \<and> less_eq zero (power.power one times b n)"
I_subst: fixes \<phi> :: "atom fm"   and t :: "real \<times> real list"   and xs :: "real list" assumes "qfree \<phi>" shows "R.I (subst \<phi> t) xs = R.I \<phi> (eval t xs # xs)"
I_1: fixes i :: "nat" shows "I i \<noteq> k"
satA_transE: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and at :: "('a, 'b) atm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "ModelIkMcalc wtFsym wtPsym arOf resOf parOf \<Phi> infTp intT intF intP"   and "Signature.wtA wtFsym wtPsym arOf resOf parOf at"   and "Struct.satA (ModelIkMcalc.I_intF arOf intT intF) (ModelIkMcalc.I_intP parOf intT intP) \<xi> at" shows "Struct.satA intF intP (ModelIkMcalc.transE intT \<xi>) at"
nob: shows "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real_of_int (c * i) \<noteq> Inum (real_of_int i # bs) e + real_of_int j"
Non_Strict_stable_matches: fixes \<mu> :: "(M \<times> W) set" shows "Non_Strict.stable \<mu> = (\<mu> = {(M1, W2), (M2, W1), (M3, W3)} \<or> \<mu> = {(M1, W3), (M2, W2), (M3, W1)})"
i_append_eq_i_append_conv: fixes xs :: "'d list"   and ys :: "'d list"   and f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" assumes "length xs = length ys" shows "(xs \<frown> f = ys \<frown> g) = (xs = ys \<and> f = g)"
evaln_AbruptIs_1: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and x :: "expr"   and n :: "nat" shows "G\<turnstile>(Some xc, s) \<midarrow>x-\<succ>undefined\<midarrow>n\<rightarrow> (Some xc, s)"
lin_independent_reduce: fixes n :: "'v"   and ms :: "'v list"   and ns :: "'v list" assumes "n \<in> V"   and "set ms \<subseteq> V"   and "R_lin_independent (ms @ n # ns)" shows "R_lin_independent (ms @ ns)"
zdiv_eq_0_iff: fixes i :: "int"   and k :: "int" shows "(i div k = 0) = (k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i)"
add_le_imp_le_diff1: fixes i :: "nat"   and k :: "nat"   and j :: "nat" assumes "i + k \<le> j" shows "i \<le> j - k"
le_Inf_iff: fixes b :: "'b"   and A :: "'b set" shows "(b \<le> \<Sqinter> A) = (\<forall>a\<in>A. b \<le> a)"
ce_rel_lm_1: fixes r :: "(nat \<times> nat) set" shows "ce_set_to_rel (ce_rel_to_set r) = r"
G_def: fixes i :: "nat"   and k :: "nat" shows "G i k = (of_nat i + a gchoose k) * x ^ k * y ^ (i - k)"
nodup_all_tl: fixes x :: "'a"   and l :: "'a list" assumes "nodup_all (x # l)" shows "nodup_all l"
select_convs_2: fixes om :: "'f \<Rightarrow> 'g"   and am :: "'d \<Rightarrow> 'e"   and more :: "'h" shows "\<lparr>om = om, am = am, \<dots> = more\<rparr> \<^bsub>\<a>\<^esub> = am"
psubst_def: shows "psubst \<equiv> \<lambda>\<theta>. lam_rec (lookup \<theta>) (\<lambda>e\<^sub>1 e\<^sub>2. App) (\<lambda>x e. Lam x)"
nrows_not_0: fixes A :: "(('c, 'd) vec, 'e) vec" shows "0 \<noteq> nrows A"
minusinf_uset: fixes p :: "fm"   and vs :: "'b list"   and a :: "'b"   and bs :: "'b list"   and x :: "'b" assumes "islin p"   and "\<not> Ifm vs (a # bs) (minusinf p)"   and "Ifm vs (x # bs) p" shows "\<exists>(c, s)\<in>set (uset p). - Itm vs (a # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<le> x"
t_def: fixes x :: "'m" shows "t x = (\<Sum>i\<in>UNIV. s i * (THE fa. f (X $ i) = (\<Sum>x\<in>UNIV. fa x *s Y $ x)) x)"
arith_special_3: shows "(1::'d) + (1::'d) = (2::'d)"
fbox_def: fixes F :: "'b \<Rightarrow> 'c set"   and P :: "'c \<Rightarrow> bool" shows "|F] P = (\<lambda>s. \<forall>s'. s' \<in> F s \<longrightarrow> P s')"
a: shows "ID CENUM('a) = Some (enum_a, enum_all_a, enum_ex_a)"
approximation_preproc_nat_12: fixes n :: "nat"   and m :: "nat" shows "(n = m) = (real n = real m)"
sorted_list_of_set_map_strict: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "strict_mono_on g {0..<n}" shows "sorted_list_of_set (g ` {0..<n}) = map g [0..<n]"
oparleft: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and a :: "'e seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and t :: "'f"   and i :: "nat"   and T :: "('f \<times> 'e seq_action \<times> 'f) set" assumes "((\<sigma>, s), a, \<sigma>', s') \<in> S"   and "\<And>m. a \<noteq> receive m" shows "((\<sigma>, s, t), a, \<sigma>', s', t) \<in> oparp_sos i S T"
\<phi>: fixes x :: "'s \<times> ('c \<Rightarrow> real)"   and y :: "'s \<times> ('c \<Rightarrow> real)" assumes "x \<in> S"   and "x ~ y" shows "\<phi> x = \<phi> y"
th0: shows "(\<Oplus>p\<in>{p. p permutes S}. f (p \<circ> q)) = finsum R (f \<circ> (\<lambda>p. p \<circ> q)) {p. p permutes S}"
same_uptoD: fixes j :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and i' :: "nat"   and j' :: "nat" assumes "local.same_upto j A B"   and "i' < n"   and "j' < j" shows "A $$ (i', j') = B $$ (i', j')"
confine_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "confine f A = (\<lambda>x. if x \<in> A then f x else (0::'b))"
Union_supports_set: fixes X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "\<Union> (supp ` X) supports X"
result: shows "closure ([(init r, init s)], {(post (init r), post (init s))}) = Some ([], N)"
subst_word_singleton: fixes f :: "'b \<Rightarrow> 'a list"   and x :: "'b" shows "subst_word f [x] = f x"
subgraph_isomorphic_trans: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set"   and G\<^sub>3 :: "nat set \<times> nat set set" assumes "G\<^sub>1 \<sqsubseteq> G\<^sub>2"   and "G\<^sub>2 \<sqsubseteq> G\<^sub>3" shows "G\<^sub>1 \<sqsubseteq> G\<^sub>3"
path_shiftpath: fixes g :: "real \<Rightarrow> 'a"   and a :: "real" assumes "path g"   and "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "path (shiftpath a g)"
r_result1_Some': fixes t :: "nat"   and i :: "nat"   and x :: "nat"   and v :: "nat" assumes "eval r_result1 [t, i, x] \<down>= Suc v" shows "eval r_phi [i, x] \<down>= v"
nonempty_CBasis: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "CBasis \<noteq> {}"
degreen_0: fixes p :: "'b poly"   and n :: "nat"   and m :: "nat" assumes "isnpolyh p n"   and "m < n" shows "degreen p m = 0"
1: fixes x :: "'a"   and t :: "('b, 'a) Term.term" assumes "x \<in> fv t" shows "fv (\<theta> x) \<subseteq> fv (t \<cdot> \<theta>)"
cod_char: fixes f :: "'a" shows "S.cod f = (if S.arr f then cod f else null)"
simps_2: fixes e :: "name list"   and k :: "name" shows "trans_tm e (Var k) = lookup e 0 k"
assms_1: fixes n :: "nat" shows "degree (p $ n) \<le> d"
comp_assoc: fixes h :: "'c \<times> 'c"   and g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" shows "(h \<odot> g) \<odot> f = h \<odot> g \<odot> f"
dbm_entry_dbm_min2': fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "dbm_entry_val u None (Some c) (min a b)" shows "dbm_entry_val u None (Some c) a"
Abs_downsets_inverse: fixes y :: "'b set" assumes "y \<in> range \<Down>" shows "Rep_downsets (Abs_downsets y) = y"
get_root_node_si_impl: shows "get_root_node_si = local.a_get_root_node_si"
suppa_sym: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C SuppA D E F " shows "D E F SuppA A B C "
assms_2: shows "ipresIGWlsAbs hA' MOD' MOD''"
simps_5: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "(case \<lfloor>x2\<rfloor> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
zeromaximalideal_fieldI: assumes "maximalideal {\<zero>} R" shows "field R"
simps_4: fixes \<phi> :: "('a, 'b, 'c) formula"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Psubst (? \<phi>) \<sigma> = ? (Fsubst \<phi> \<sigma>)"
Entry_no_phis_inv: fixes v :: "'val" shows "phis' g (Entry g, v) = None"
cases: fixes a :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "a \<in> fsubst" obtains "a = Var"   | \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" and v :: "'a" and t :: "('b, 'a) Term.term" where "a = \<theta>(v := t)" and "\<theta> \<in> fsubst" and "v \<notin> subst_domain \<theta>" and "t \<noteq> Var v"
trans_refl_imp_O_id: fixes r :: "('a \<times> 'a) set" assumes "trans r"   and "refl r" shows "r O r = r"
HcompNml_Trg_Nml: fixes t :: "'a local.term" assumes "Nml t" shows "Trg t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> t = t"
simps_3: fixes obs' :: "('e \<Rightarrow> 'f) \<Rightarrow> 'e \<Rightarrow> 'f"   and init :: "'e set"   and trans :: "('e \<times> 'e) set"   and obs :: "'e \<Rightarrow> 'f"   and more :: "'g" shows "obs_update obs' \<lparr>init = init, trans = trans, obs = obs, \<dots> = more\<rparr> = \<lparr>init = init, trans = trans, obs = obs' obs, \<dots> = more\<rparr>"
transitive: fixes a :: "wall"   and b :: "wall"   and c :: "wall" assumes "a ~ b"   and "b ~ c" shows "a ~ c"
PP_def: shows "PP \<equiv> (B ===> B ===> B) ===> B ===> (B ===> B ===> B) ===> (B ===> B) ===> ((=) ===> B ===> B) ===> rel_set B ===> (=)"
hext_new: fixes h :: "heap"   and a :: "nat"   and x :: "obj" assumes "h a = None" shows "h \<unlhd> h(a \<mapsto> x)"
evaln_elim_cases_27: fixes G :: "prog"   and s :: "st"   and c1 :: "stmt"   and tn :: "qtname"   and vn :: "vname"   and c2 :: "stmt"   and n :: "nat"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (Try c1 Catch(tn vn) c2)\<succ>\<midarrow>n\<rightarrow> (x, s')" obtains s1 :: "state" and s2 :: "state" where "x = \<diamondsuit>" and "G\<turnstile>Norm s \<midarrow>c1\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>sxalloc\<rightarrow> s2" and "if G,s2\<turnstile>catch tn then G\<turnstile>new_xcpt_var vn s2 \<midarrow>c2\<midarrow>n\<rightarrow> s' else s' = s2"
assms: shows "run_one_step d i (s, vs, ves, $Br_if x9) = (s', vs', res)"
cexpr_typing_dist_dens_cexpr: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and dst :: "pdf_dist"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : dist_param_type dst"   and "\<Gamma> \<turnstile>\<^sub>c e2 : dist_result_type dst" shows "\<Gamma> \<turnstile>\<^sub>c dist_dens_cexpr dst e1 e2 : REAL"
cm_all_eq_cm_all': fixes \<Delta> :: "('loc \<times> 't) zmultiset"   and c0 :: "('loc, 't) Propagate.configuration" assumes "\<forall>(loc, t)\<in>#\<^sub>z\<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t" shows "cm_all c0 \<Delta> = cm_all' c0 \<Delta>"
C: shows "offs_none ns ub (x # xs) index key mi ma i"
domain_eq: shows "Domainp (pcr_freeword (=)) = proper_signed_list"
iqB: fixes b :: "'b" assumes "b \<in> B" shows "inv q (q b) = b"