hcju/isabelleps huggingface dataset:
r1: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "x \<noteq> y" shows "P (xs @ [x]) (ys @ [y])"
eta_star_preserves_term_ok': fixes r :: "Core.term"   and s :: "Core.term"   and \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "r \<rightarrow>\<^sub>\<eta>\<^sup>* s"   and "term_ok' \<Sigma> r" shows "term_ok' \<Sigma> s"
distinct_13: fixes x1 :: "'b"   and x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly" shows "(x1)\<^sub>p \<noteq> CN x81 x82 x83"
countable_INT': fixes X :: "'d set"   and A :: "'d \<Rightarrow> 'e set"   and M :: "'e measure" assumes "countable X"   and "X \<noteq> {}"   and "A ` X \<subseteq> sets M" shows "\<Inter> (A ` X) \<in> sets M"
assms_2: shows "(\<forall>\<^sub>F x in F. - exp (- 1) \<le> f x) \<or> - exp (- 1) < L"
vec_of_basis_enum_add: fixes b1 :: "'a"   and b2 :: "'a" shows "vec_of_basis_enum (b1 + b2) = vec_of_basis_enum b1 + vec_of_basis_enum b2"
add_Real: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "cauchy X"   and "cauchy Y" shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
countable_GE_wtFsym: shows "countable (Collect GE_wtFsym)"
simps: fixes ant :: "'a form multiset"   and suc :: "'a form multiset" shows "seq_size ( ant \<Rightarrow>* suc) = size ant + size suc"
h: shows "steps0 (st, l, r) ap (Suc stp) = (st', l', r')"
prove_nonneg_mono: fixes prnt :: "(nat \<times> nat \<times> char list) list"   and i :: "nat"   and p :: "nat"   and fa :: "floatarith"   and YS :: "(real \<times> real pdevs) list"   and YSS :: "(real \<times> real pdevs) list list" assumes "prove_nonneg prnt i p (slp_of_fas [fa]) (YS # YSS)" shows "prove_nonneg prnt i p (slp_of_fas [fa]) YSS"
MapsToMorDomCod_1: fixes f :: "'b"   and g :: "'b" assumes "f \<approx>> g" shows "f ;; g \<in> mor"
const_least: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and f :: "'e \<Rightarrow> 'd" assumes "wo_rel2 r s"   and "Field r \<noteq> {}"   and "f \<in> FinFunc r s" shows "(wo_rel2.const r s, f) \<in> r ^o s"
ag_p_inv: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c"   and y :: "'c" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> (x \<plusminus>\<^bsub>A\<^esub> y) = -\<^sub>a\<^bsub>A\<^esub> x \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> y"
continuous_on_def: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'e" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
Abs_Betas: fixes s :: "Lambda_Z.term"   and t :: "Lambda_Z.term"   and x :: "name" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* t" shows "Abs x s \<rightarrow>\<^sub>\<beta>\<^sup>* Abs x t"
totatives_eq_empty_iff: fixes n :: "nat" shows "(totatives n = {}) = (n = 0)"
span_breakdown_eq: fixes x :: "'a"   and a :: "'a"   and S :: "'a set" shows "(x \<in> More_Modules.ideal (insert a S)) = (\<exists>k. x - k * a \<in> More_Modules.ideal S)"
SFAccInit\<^sub>1: fixes P :: "nat exp prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and h :: "heap"   and ls :: "val list"   and v' :: "val"   and h' :: "heap"   and ls' :: "val list"   and sh' :: "sheap"   and sfs :: "sfields"   and i :: "init_state"   and v :: "val" assumes "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile>\<^sub>1 \<langle>INIT D ([D],False) \<leftarrow> unit,(h, ls, sh)\<rangle> \<Rightarrow> \<langle>Val v',(h', ls', sh')\<rangle>"   and "sh' D = \<lfloor>(sfs, i)\<rfloor>"   and "sfs F = \<lfloor>v\<rfloor>" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sF{D},(h, ls, sh)\<rangle> \<Rightarrow> \<langle>Val v,(h', ls', sh')\<rangle>"
Gram_Schmidt_def: fixes xs :: "('c, 'd) vec list" shows "Gram_Schmidt xs = foldr Gram_Schmidt_step xs []"
shadow_root_ptr_kinds_def: fixes heap :: "(_) heap" shows "shadow_root_ptr_kinds heap = the |`| cast |`| ffilter is_shadow_root_ptr_kind (object_ptr_kinds heap)"
conf_clinit_shupd': fixes P :: "jvm_prog"   and sh :: "sheap"   and frs :: "frame list"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sfs' :: "sfields" assumes "conf_clinit P sh frs"   and "sh C = \<lfloor>(sfs, i)\<rfloor>" shows "conf_clinit P (sh(C \<mapsto> (sfs', i))) frs"
complete_treeD_del: fixes t :: "'b tree23"   and x :: "'b" assumes "complete t" shows "complete (treeD (del x t))"
siso_transC_indis: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state"   and t :: "'state" assumes "siso c"   and "(c, s) \<rightarrow>c (c', s')"   and "s \<approx> t" shows "\<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>c (c', t')"
finite_trancl: fixes r :: "('a \<times> 'a) set" shows "finite (r\<^sup>+) = finite r"
subcls_asym: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and D :: "char list" assumes "wf_prog wf_md P"   and "(C, D) \<in> (subcls1 P)\<^sup>+" shows "(D, C) \<notin> (subcls1 P)\<^sup>+"
Pareto_strictI': fixes x :: "'alt"   and y :: "'alt"   and i :: "'agent" assumes "\<And>i. i \<in> agents \<Longrightarrow> R i x y"   and "i \<in> agents"   and "\<not> R i y x" shows "x \<prec>[Pareto R] y"
select_convs_10: fixes igWls :: "'f \<Rightarrow> 'b \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'f \<Rightarrow> 'h \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and igAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'h"   and igOp :: "'g \<Rightarrow> ('d \<Rightarrow> 'b option) \<Rightarrow> ('e \<Rightarrow> 'h option) \<Rightarrow> 'b"   and igFresh :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'h \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and igSwapAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'h \<Rightarrow> 'h"   and igSubst :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and igSubstAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'h \<Rightarrow> 'h"   and more :: "'i" shows "igSubst \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igSubst"
p: shows "extreme_bound A (\<sqsubseteq>) {f^n b |. n} p"
uniqueness_of_ctxt: fixes \<Gamma> :: "binding list"   and X :: "tyvrs"   and T :: "ty"   and S :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "TVarB X T \<in> set \<Gamma>"   and "TVarB X S \<in> set \<Gamma>" shows "T = S"
shadow_root_ptr_no_node_ptr_cast: fixes node_ptr :: "(_) node_ptr" shows "\<not> is_shadow_root_ptr_kind (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)"
assms_2: shows "j icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
eval_poly_at_tm_range: fixes p :: "float poly"   and e' :: "nat \<Rightarrow> real"   and e :: "nat \<Rightarrow> real"   and tg :: "taylor_model"   and a :: "float interval list"   and I :: "float interval list"   and prec :: "nat"   and ord :: "nat" assumes "num_params p \<le> 1"   and "e' 0 \<in>\<^sub>i range_tm e tg"   and "develops_at_within e a I"   and "num_params (tm_poly tg) \<le> length I" shows "insertion e' (map_poly real_of_float p) \<in>\<^sub>i range_tm e (eval_poly_at_tm prec ord I a p tg)"
homeomorphic_affine_sets_eq: fixes S :: "'a set"   and T :: "'b set" assumes "affine S"   and "affine T" shows "S homeomorphic T = (aff_dim S = aff_dim T)"
reindex_bij_betw_not_neutral: fixes S' :: "'d set"   and T' :: "'e set"   and h :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'e set"   and g :: "'e \<Rightarrow> 'a" assumes "finite S'"   and "finite T'"   and "bij_betw h (S - S') (T - T')"   and "\<And>a. a \<in> S' \<Longrightarrow> g (h a) = \<^bold>1"   and "\<And>b. b \<in> T' \<Longrightarrow> g b = \<^bold>1" shows "F (\<lambda>x. g (h x)) S = F g T"
max_reg_Times: fixes a1 :: "vname aexp"   and a2 :: "vname aexp" shows "max_reg (Times a1 a2) = max (max_reg a1) (max_reg a2)"
irreducible_imp_prime_elem_gcd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "comm_semiring_1.irreducible times one zero x" shows "comm_semiring_1.prime_elem times one zero x"
ucast_mono_le: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le> y"   and "y < 2 ^ LENGTH('c)" shows "ucast x \<le> ucast y"
vec_cmul: fixes c :: "'a"   and x :: "'a" shows "vec (c * x) = c *s vec x"
q_def: shows "q = Poly (map f [0..<Suc (degree p)])"
cc_lt_c: shows "\<forall>C'. C' \<in># CC \<longrightarrow> C' < C"
abs_eq: fixes xa :: "'c llist \<times> 'b"   and x :: "'c llist \<times> 'b" shows "local.tllist_ord (case xa of (xs, a) \<Rightarrow> tllist_of_llist a xs) (case x of (xs, a) \<Rightarrow> tllist_of_llist a xs) = (case xa of (xs1, b1) \<Rightarrow> \<lambda>(xs2, b2). if lfinite xs1 then b1 = b \<and> lprefix xs1 xs2 \<or> xs1 = xs2 \<and> flat_ord b b1 b2 else xs1 = xs2) x"
pullback_mono_self: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.mono C f" shows "prj0 f f = prj1 f f"
LVar: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and vn :: "lname" shows "G,A\<turnstile>{Normal (\<lambda>s.. P\<leftarrow>In2 (lvar vn s))} LVar vn=\<succ> {P}"
with_def: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" shows "subspace.with plus zero \<equiv> \<lambda>scale S. zero \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. plus x y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. scale c x \<in> S)"
l2_inv7_trans: shows "{l2_inv7 \<inter> l2_inv5 \<inter> l2_inv6} TS.trans l2 {> l2_inv7}"
mat_inverse_1: fixes A :: "'a mat"   and n :: "nat"   and b :: "'b" assumes "A \<in> carrier_mat n n"   and "mat_inverse A = None" shows "A \<notin> Units (ring_mat TYPE('a) n b)"
A_authenticates_B: fixes servK :: "nat"   and T3 :: "nat"   and evs :: "event list"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and A :: "agent"   and Ta :: "nat" assumes "Crypt servK (Number T3) \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbV" shows "Says B A (Crypt servK (Number T3)) \<in> set evs"
Deq1_enabled: shows "\<turnstile> Enabled \<langle>Deq inp q1 mid\<rangle>_vars = ($q1 \<noteq> #[])"
not_is_Var_is_Ind: fixes v :: "hf" assumes "is_Var v" shows "\<not> is_Ind v"
bmod_onel: fixes f :: "'a \<Rightarrow> 'c"   and x :: "'a" shows "\<otimes> \<rho> f \<delta> x = f x"
sigma_finite_subalgebra_axioms: shows "sigma_finite_subalgebra M Invariants"
extended_Gromov_distance_continuous: fixes x :: "'b Gromov_completion" shows "continuous_on UNIV (extended_Gromov_distance x)"
forball_contra1: fixes A :: "'c set"   and P :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'d"   and Q :: "'c \<Rightarrow> bool" assumes "\<forall>y\<in>A. P x y \<longrightarrow> Q y"   and "\<forall>y\<in>A. \<not> Q y" shows "\<forall>y\<in>A. \<not> P x y"
S1_xor_def: fixes A :: "bool \<times> bool"   and out :: "bool" shows "S1_xor A out = return_spmf A"
leg_simps_3: shows "C.arr \<mu>\<nu>_\<pi>.leg1"
setops\<^sub>s\<^sub>s\<^sub>t_pair_image_cons_3: fixes t :: "('fun, 'a) Term.term"   and S :: "('fun, 'a) stateful_strand_step list" shows "pair ` setops\<^sub>s\<^sub>s\<^sub>t (receive\<langle>t\<rangle> # S) = pair ` setops\<^sub>s\<^sub>s\<^sub>t S"
supr_under: fixes r :: "('a \<times> 'a) set"   and a :: "'a" assumes "wo_rel r"   and "a \<in> Field r" shows "a = wo_rel.supr r (under r a)"
vconverse_atE: fixes b :: "V"   and a :: "V" assumes "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>" obtains "r\<lparr>a\<rparr> = b"
gen_shleg_poly_altdef: fixes k :: "nat" assumes "k \<le> n" shows "local.gen_shleg_poly k = (\<Sum>i\<le>k. Polynomial.smult ((- 1) ^ (k - i) * int (k choose i) * int (pochhammer (n - i + 1) i) * int (pochhammer (n - k + i + 1) (k - i))) ([:0, 1:] ^ (n - i) * [:1, - 1:] ^ (n - k + i)))"
VSEquality: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (VSEquality \<phi>) xs = eval \<phi> xs"
permutation_insert_base: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p permutes {0..<n}" shows "permutation_insert n n p = p"
assms_2: shows "(c, b) \<in> A\<^sup>*"
Span_incl: fixes Us :: "'a list" assumes "set Us \<subseteq> carrier R" shows "K <#> set Us \<subseteq> Span K Us"
floor_le_iff: fixes x :: "'d"   and z :: "int" shows "(\<lfloor>x\<rfloor> \<le> z) = (x < of_int z + (1::'d))"
is_proper_suffix_def: fixes u :: "'b list"   and v :: "'b list" shows "is_proper_suffix u v = (\<exists>w. w \<noteq> [] \<and> w @ u = v)"
runit'_naturality: fixes f :: "'d" assumes "arr f" shows "(f \<otimes> \<I>) \<cdot> \<r>\<^sup>-\<^sup>1[local.dom f] = \<r>\<^sup>-\<^sup>1[cod f] \<cdot> f"
frag_cmul_distrib2: fixes c :: "int"   and a :: "'c \<Rightarrow>\<^sub>0 int"   and b :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_cmul c (a + b) = frag_cmul c a + frag_cmul c b"
scaleC: fixes f :: "'a \<Rightarrow> 'b"   and r :: "complex"   and b :: "'a" assumes "clinear f" shows "f (r *\<^sub>C b) = r *\<^sub>C f b"
LSC_invs_2: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset" assumes "F \<^bold>\<or> G, \<Gamma> \<Rightarrow>\<^sub>n" shows "F, \<Gamma> \<Rightarrow>\<^sub>n \<and> G, \<Gamma> \<Rightarrow>\<^sub>n"
int: shows "integrable lborel (\<lambda>x. indicat_real {a..b} x *\<^sub>R f x)"
ty_cmd_pres: fixes \<Gamma>l :: "String.literal \<Rightarrow> unit option"   and \<Gamma>g :: "String.literal \<Rightarrow> unit option"   and c :: "cmd"   and a :: "action + brk_ctd"   and c' :: "cmd" assumes "ty_cmd ((\<Gamma>l, \<Gamma>g), False) c"   and "cfg c a c'" shows "ty_cmd ((\<Gamma>l, \<Gamma>g), False) c'"
set_plus_mono: fixes C :: "'a set"   and D :: "'a set"   and a :: "'a" assumes "order.greater_eq D C" shows "order.greater_eq (a +o D) (a +o C)"
opp_mult_def: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" shows "matrix_dioid.opp_mult x y \<equiv> y \<otimes> x"
Max_ge: fixes A :: "'t set"   and x :: "'t" assumes "finite A"   and "x \<in> A" shows "x \<preceq>\<^sub>t ord_term_lin.Max A"
power2_sum: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "power.power one times (plus x y) 2 = plus (plus (power.power one times x 2) (power.power one times y 2)) (times (times (numeral.numeral one plus (num.Bit0 num.One)) x) y)"
simps_1: fixes Y :: "nat" shows "witness' Y [] = Input Y"
monoid_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" shows "Groups.monoid f z \<equiv> semigroup f \<and> monoid_axioms f z"
norm_triangle_sub: fixes x :: "'a"   and y :: "'a" shows "norm x \<le> norm y + norm (x - y)"
product_nn_integral_fold: fixes I :: "nat set"   and J :: "nat set"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "I \<inter> J = {}"   and "finite I"   and "finite J"   and "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) (\<lambda>x. stock_measure (\<Gamma> x)))" shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) (\<lambda>x. stock_measure (\<Gamma> x))) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>Pi\<^sub>M J (\<lambda>x. stock_measure (\<Gamma> x)) \<partial>Pi\<^sub>M I (\<lambda>x. stock_measure (\<Gamma> x))"
inf_le2: fixes x :: "'k multiset"   and y :: "'k multiset" shows "x \<inter># y \<subseteq># y"
ifte_inject: fixes i :: "'a"   and x :: "'a formula"   and y :: "'a formula"   and x' :: "'a formula"   and y' :: "'a formula"   and S :: "'a set" assumes "ifte (var i) x y = ifte (var i) x' y'"   and "i \<notin> S"   and "x \<in> formulas S"   and "x' \<in> formulas S"   and "y \<in> formulas S"   and "y' \<in> formulas S" shows "x = x' \<and> y = y'"
find_Some_iff: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list"   and x :: "'g" shows "(find P xs = Some x) = (\<exists>i<length xs. P (xs ! i) \<and> x = xs ! i \<and> (\<forall>j<i. \<not> P (xs ! j)))"
wp_def: shows "wp \<equiv> hom_induced (int n) (lsphere n) (equator n) (nsphere n) (upper n) id zp"
open_prod_def: fixes S :: "('d \<times> 'e) set" shows "open S = (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
ub_def: shows "ub = (\<lambda>x. 2 / ln 2 * sqrt x + 8 * ln 2 * x / (ln x)\<^sup>2)"
semi_complete_imp_conversionIff_same_NF: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "semi_complete r" shows "((x, y) \<in> r\<^sup>\<leftrightarrow>\<^sup>*) = (\<forall>u v. (x, u) \<in> r\<^sup>! \<and> (y, v) \<in> r\<^sup>! \<longrightarrow> u = v)"
good_item_simps_43: fixes rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xsa :: "'varSort"   and xa :: "'var" assumes "goodEnv rhoa"   and "rhoa xsa xa = None" shows "Var xsa xa #[rhoa] = Var xsa xa"
assms: shows "A \<le> SPEC (\<lambda>v. P v \<and> Q v)"
less_eq_def: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod x y = (sup_prod x y = y)"
cong_scalar_left: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b"   and d :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (times d b) (times d c) a"
Abs_hyp2_inverse: fixes y :: "proj2" assumes "y \<in> hyp2" shows "Rep_hyp2 (Abs_hyp2 y) = y"
cases: fixes x :: "'a list" obtains x :: "'a" and uu_ :: "'a list" where "xa__ = x # uu_"   | "xa__ = []"
zero_le_of_rat_iff: fixes r :: "rat" shows "((0::'a) \<le> of_rat r) = (0 \<le> r)"
refl: fixes x :: "'a" assumes "x \<in> UNIV" shows "bo x x"
lookup_idx_pm_of_pm: fixes xs :: "'d list"   and f :: "'d \<Rightarrow>\<^sub>0 'c" shows "lookup (idx_pm_of_pm xs f) = (\<lambda>i. lookup f (xs ! i) when i < length xs)"
favorite_in_alts: fixes i :: "'agent" assumes "i \<in> agents" shows "favorite R i \<in> alts"
nths_uminus_vars1: shows "nths dims (- vars1) = nths dims vars2"
6: shows "minarc (v * - v\<^sup>T \<sqinter> g) \<le> v * - v\<^sup>T"
fps_one_code: shows "1 = fps_of_ratfps 1"
is_reducible_lit_def: fixes Ds :: "('b, nat) Term.term literal list list"   and C :: "('b, nat) Term.term literal list"   and L :: "('b, nat) Term.term literal" shows "is_reducible_lit Ds C L = (\<exists>D\<in>set Ds. \<exists>L'\<in>set D. \<exists>\<sigma>. - L = subst_lit L' \<sigma> \<and> subst_cls (mset (remove1 L' D)) \<sigma> \<subseteq># mset C)"
divide_eq_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(divide b c = a) = (if c \<noteq> zero then b = times a c else a = zero)"
multiplicative_sum_divisors: shows "multiplicative_function (\<lambda>n. \<Sum>d | d dvd n. moebius_mu d)"
index_lfp: fixes L :: "'a set"   and Glb :: "'a set \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "Complete_Lattice L Glb" shows "Complete_Lattice.lfp L Glb f \<in> L"
assms_9: shows "contr_pf = qty_sum arb_pf pf"
sub_add_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "sub_add t = (\<forall>P Q s. sound P \<and> sound Q \<longrightarrow> t P s + t Q s \<le> t (\<lambda>s'. P s' + Q s') s)"
learn: shows "learn_cons \<phi> U\<^bsub>LIM-CONS\<^esub> s"
v12_t: shows "(v1, v2) |\<in>| t"
subtract_mask_2: fixes p :: "'b word"   and n :: "nat" shows "p - (p && ~~ mask n) = p && mask n"
hyps: shows "map ((+) n) [i..<j] = [i + n..<j + n]"
dist_fst_le: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "dist (fst x) (fst y) \<le> dist x y"
loc_map_env: fixes w :: "'a list" shows "loc (map ENV w) = []"
card_finite_less_ex: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "card A < card B" shows "\<exists>b\<in>B. b \<notin> A"
bound_of_def: fixes P :: "'a \<Rightarrow> real" shows "bound_of P \<equiv> Sup (range P)"
collision_subset: fixes s :: "real set"   and t :: "real set" assumes "collision s"   and "s \<subseteq> t" shows "collision t"
simps_1: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "vc SKIP Q = True"
strongly_accepts_empty: fixes \<F> :: "nat set set"   and M :: "nat set" shows "strongly_accepts \<F> {} M = (\<forall>N\<subseteq>M. Fpow N \<inter> \<F> = {} \<longrightarrow> finite N)"
zeros: shows "\<forall>i\<in>T - S. [g i = 0::'b] (mod n)"
Dflow_cong: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "Dflow x0 t = Y.Dflow x0 t"
is_lifting_def: fixes f :: "'b \<Rightarrow> 'c"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list" shows "is_lifting f basis basis' = (\<forall>C. eval (f C) = eval C \<and> (is_expansion C basis \<longrightarrow> is_expansion (f C) basis') \<and> trimmed (f C) = trimmed C \<and> fst (dominant_term (f C)) = fst (dominant_term C))"
pr_passwd_says: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and X :: "agent"   and K :: "rat"   and m :: "nat" assumes "(evs, S, A, U) \<in> protocol"   and "Says n run 5 X (Card n) (Crypt (sesK K) (Passwd m)) \<in> set evs" shows "X = User m \<or> X = Spy"
M8: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>[F \<and> G]_v = (\<box>[F]_v \<and> \<box>[G]_v)"
assms: shows "poincare_line_cvec_cmat z w = (A, B, C, D)"
WHILEIT_eq: shows "WHILEIT \<equiv> iWHILEIT (\<bind>) RETURN"