hcju/isabelleps huggingface dataset:
gt_GreaterD: assumes "u \<in> Regions.region \<X> I r"   and "Regions.valid_region \<X> k I r"   and "c \<in> \<X>"   and "real (k c) < u c" shows "I c = Regions.intv.Greater (k c)"
ag_pOp_assoc: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "z \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y \<plusminus>\<^bsub>A\<^esub> z = x \<plusminus>\<^bsub>A\<^esub> (y \<plusminus>\<^bsub>A\<^esub> z)"
simps_2: fixes v1 :: "val"   and v2 :: "val"   and t :: "(val \<times> val) list" shows "Tf ((v1, v2) # t) = (T v1 \<rightarrow> T v2) \<sqinter> Tf t"
irreducible_PM_M_PFM: fixes p :: "'b poly" assumes "irreducible p" shows "degree p = 0 \<and> irreducible (coeff p 0) \<or> degree p \<noteq> 0 \<and> irreducible (map_poly to_fract p) \<and> content_ff (map_poly to_fract p) =dff 1"
unifier_comp: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" shows "unifier\<^sub>l\<^sub>s \<sigma> (L\<^sup>C) = unifier\<^sub>l\<^sub>s \<sigma> L"
cases_Total3: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and phi :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wo_rel r"   and "{a, b} \<subseteq> Field r"   and "(a, b) \<in> r - Id \<or> (b, a) \<in> r - Id \<Longrightarrow> phi a b"   and "a = b \<Longrightarrow> phi a b" shows "phi a b"
refl: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and x :: "'a" assumes "linorder_on D cmp"   and "x \<in> D" shows "cmp x x = EQUAL"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.arr f"   and "C.dom f = a"   and "C.cod f = b" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
term_of_pair_injective: fixes p :: "'a \<times> unit"   and q :: "'a \<times> unit" assumes "fst p = fst q" shows "p = q"
en_eq_6: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1"   and y :: "\<kappa>"   and G :: "\<Pi>\<^sub>1" shows "[\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,G\<rbrace> \<^bold>\<equiv> \<^bold>\<box>(\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,G\<rbrace>) in v]"
seq_concat_1: fixes ss1 :: "'a \<times> ('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss2 :: "'a \<times> ('b \<times> 'a) list" assumes "ss1 \<in> seq ars"   and "ss2 \<in> seq ars"   and "lst ss1 = fst ss2" shows "(fst ss1, snd ss1 @ snd ss2) \<in> seq ars"
mult_eq_0_iff: fixes plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and a :: "'m"   and b :: "'m" assumes "class.semiring_no_zero_divisors plus zero times" shows "(times a b = zero) = (a = zero \<or> b = zero)"
ext_irrefl_before_trans_axioms: shows "ext_irrefl_before_trans len_lexext_rev"
gval_fold_not_invalid_all_valid: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "gval (fold gAnd G (Bc True)) s \<noteq> invalid" shows "\<forall>g\<in>set G. gval g s \<noteq> invalid"
non_colinear_A'B'P: fixes A :: "Points"   and B :: "Points"   and P :: "Points"   and A' :: "Points"   and B' :: "Points" assumes "rk {A, B, P} = 3"   and "rk {A, A', P} = 2"   and "rk {B, B', P} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2" shows "rk {A', B', P} = 3"
cfg_onD_action: fixes cfg :: "'s cfg"   and s :: "'s" assumes "cfg \<in> cfg_on s" shows "action cfg \<in> K s"
square_bound_lemma: fixes x :: "real" shows "x < (1 + x) * (1 + x)"
dns1'': fixes x :: "('a \<times> 'a) set" shows "rdom x ; x = x"
rank_invar_meld: fixes bq1 :: "('c, 'd) BinomialTree list"   and bq2 :: "('c, 'd) BinomialTree list" assumes "rank_invar bq1"   and "rank_invar bq2" shows "rank_invar (meld bq1 bq2)"
SPEC_refine: fixes S :: "'b nres"   and R :: "('b \<times> 'c) set"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "S \<le> SPEC (\<lambda>x. \<exists>x'. (x, x') \<in> R \<and> \<Phi> x')" shows "S \<le> \<Down> R (SPEC \<Phi>)"
hv_inj: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "(Hv x = Hv y) = (x = y)"
rep_eq: fixes x :: "('e, 'c) lgenerator'"   and xa :: "('e, 'd) lgenerator'"   and xb :: "'d" shows "lgenerator (lappend_trans' x xa xb) = append_raw (lgenerator x) (lgenerator xa) xb"
f2'_def: shows "f2' = max (f1 * c) (f2 / e)"
K: shows "expectation (\<lambda>x. \<bar>u 1 x\<bar> * indicat_real {x \<in> space M. K \<le> \<bar>u 1 x\<bar>} x) < rho"
bind_runit: fixes x :: "'c option" shows "x \<bind> Some = x"
induct: fixes P :: "bool list \<Rightarrow> 'b option list \<Rightarrow> bool"   and a0 :: "bool list"   and a1 :: "'b option list" assumes "P [] []"   and "\<And>bs a as. P bs as \<Longrightarrow> P (True # bs) (a # as)"   and "\<And>bs a as. P bs as \<Longrightarrow> P (False # bs) (a # as)"   and "\<And>b bs. P (b # bs) []"   and "\<And>a as. P [] (a # as)" shows "P a0 a1"
mkPoint_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b"   and t :: "'b" assumes "set_category S \<AA>" shows "set_category.mkPoint S \<AA> a t \<equiv> set_category_given_img.mkArr S (set_category.some_img S \<AA>) \<AA> {set_category_data.unity S} (set_category_data.set S (set_category.some_img S \<AA>) a) (\<lambda>_. t)"
is_acc_run_def: fixes r :: "nat \<Rightarrow> 'Q" shows "is_acc_run r \<equiv> is_run r \<and> is_acc r"
map\<^sub>1_def: fixes f :: "'b" shows "map\<^sub>1 f = D.inv (\<Phi>\<^sub>H (\<tau>\<^sub>0 (trg\<^sub>B f), F f)) \<cdot>\<^sub>D H (\<tau>\<^sub>1 f) \<cdot>\<^sub>D \<Phi>\<^sub>H (G f, \<tau>\<^sub>0 (src\<^sub>B f))"
interaction_bounded_by_exec_gpv_bad: fixes gpv :: "('b, 'a, 'c) gpv"   and n :: "nat"   and bad :: "'d \<Rightarrow> bool"   and s :: "'d"   and callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'd) spmf"   and k :: "real" assumes "interaction_bounded_by' (\<lambda>_. True) gpv n"   and "True"   and "\<not> bad s"   and "\<And>s x. \<lbrakk>True; \<not> bad s; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> spmf (map_spmf (bad \<circ> snd) (callee s x)) True \<le> k"   and "0 \<le> k"   and "\<I>_full \<turnstile>g gpv \<surd>" shows "spmf (map_spmf (bad \<circ> snd) (exec_gpv callee gpv s)) True \<le> k * real n"
nprv_cnjI: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F \<phi>"   and "nprv F \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F (cnj \<phi> \<psi>)"
eq_\<C>: shows "\<forall>x\<in>\<C>. mem\<^sub>1 x = mem\<^sub>2 x"
terminates_elim_cases_9: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>DynCom c \<down> s" obtains f :: "'f" where "sb__ = Fault f"   | "sb__ = Stuck"   | s :: "'e" where "sb__ = Normal sc__" and "\<Gamma>\<turnstile>c sc__ \<down> Normal sc__"   | s :: "'e" where "sb__ = Abrupt s"
id: shows "f v = ((+) (1::'a) ^^ Suc k) (g v)"
V1_def: shows "V1 = filter (\<lambda>v. except g v = 0) V"
chaum_ped_\<Sigma>_base_def: fixes \<G> :: "'a cyclic_group" shows "chaum_ped_\<Sigma>_base \<G> \<equiv> prime (order \<G>)"
Step2_lemma: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and l :: "Locs" shows "\<turnstile> \<box>((\<forall>p. ImpNext p) \<and> (\<forall>p. [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> (\<forall>p. $ImpInv rmhist p) \<and> [\<exists>q. Write rmCh mm ires q l]_(mm!l)) \<longrightarrow> \<box>[\<exists>q. Write memCh mm (resbar rmhist) q l]_(mm!l)"
mat_carrier: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "Matrix.mat nr nc f \<in> carrier_mat nr nc"
rel_stacktraces_box: fixes F1 :: "'c \<Rightarrow> ('dyn, 'd, 'c, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef option"   and xs :: "('c, 'dyn) frame list"   and ys :: "('c, ('dyn, 'ubx1, 'ubx2) unboxed) frame list"   and opt :: "'c option"   and F2 :: "'c \<Rightarrow> ('dyn, 'd, 'c, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef option"   and f :: "'c" assumes "rel_stacktraces F1 xs ys opt"   and "F2 f = map_option Subx.generalize_fundef (F1 f)"   and "\<And>g. f \<noteq> g \<Longrightarrow> F2 g = F1 g"   and "all_same_arities F1 F2" shows "rel_stacktraces F2 xs (Subx.box_stack f ys) opt"
pgen_lossless_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('i, 'j) \<I>"   and gpv :: "('k, 'i, 'j) gpv" shows "pgen_lossless_gpv fail \<I> gpv = (expectation_gpv fail \<I> (\<lambda>_. 1) gpv = 1)"
satC_gC: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" assumes "Ik.wtE \<xi>"   and "GE.satC \<xi> (gC c)" shows "Ik.satC \<xi> c"
int_mono_def: fixes x :: "int" shows "int_mono x \<equiv> 1 \<le> x"
sv_round_def: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and r_decisions :: "process \<Rightarrow> val option" shows "sv_round r S v r_decisions = {(s, s'). r = next_round s \<and> (S \<noteq> {} \<longrightarrow> safe s r v) \<and> d_guard r_decisions (const_map v S) \<and> s' = s\<lparr>next_round := Suc r, votes := (votes s)(r := const_map v S), decisions := decisions s ++ r_decisions\<rparr>}"
in_associates_Hermite_of: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "'b"   and A :: "(('a, 'c) vec, 'b) vec" assumes "ass_function ass"   and "res_function res"   and "is_bezout_ext bezout"   and "\<not> is_zero_row i (Hermite_of A ass res bezout)" shows "Hermite_of A ass res bezout $ i $ (LEAST n. Hermite_of A ass res bezout $ i $ n \<noteq> (0::'a)) \<in> range ass"
NAKTr4: fixes A :: "'b set"   and h :: "nat \<Rightarrow> 'a"   and k :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "ideal R A"   and "h \<in> {j. j \<le> k} \<rightarrow> carrier M"   and "0 < k"   and "h ` {j. j \<le> k} \<subseteq> carrier M"   and "s \<in> {j. j \<le> k} \<rightarrow> A"   and "h k = \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s h j) (k - Suc 0) \<plusminus> s k \<cdot>\<^sub>s h k" shows "(1\<^sub>r\<^bsub>R\<^esub> \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> s k) \<cdot>\<^sub>s h k = \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s h j) (k - Suc 0)"
asymp_equiv_add_right: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "h \<in> o[F](g)" shows "((\<lambda>x. f x + h x) \<sim>[F] g) = (f \<sim>[F] g)"
dim_substandard: fixes d :: "'a set" assumes "d \<subseteq> Basis" shows "dim {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = card d"
Qg_one: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set" assumes "Group G"   and "G \<triangleright> N" shows "\<one>\<^bsub>G / N\<^esub> = N"
cupcake_nsAppend_preserve: fixes ns1 :: "(char list, char list, v) namespace"   and ns2 :: "(char list, char list, v) namespace" assumes "is_cupcake_ns ns1"   and "is_cupcake_ns ns2" shows "is_cupcake_ns (nsAppend ns1 ns2)"
assms_3: shows "x\<^sub>i \<in> rvars (\<T> s')"
subspace: fixes S :: "'c set" assumes "real_vector_space_on S" shows "subspace S"
P2_output_rewrite: fixes s :: "nat"   and r :: "nat"   and u1 :: "nat"   and v1 :: "nat"   and \<alpha> :: "nat" assumes "s < order \<G>" shows "(\<^bold>g [^] (r * u1 + v1), \<^bold>g [^] (r * \<alpha> * u1 + v1 * \<alpha>) \<otimes> inv \<^bold>g [^] u1) = (\<^bold>g [^] (r * ((s + u1) mod order \<G>) + (r * order \<G> - r * s + v1) mod order \<G>), \<^bold>g [^] (r * \<alpha> * ((s + u1) mod order \<G>) + (r * order \<G> - r * s + v1) mod order \<G> * \<alpha>) \<otimes> inv \<^bold>g [^] ((s + u1) mod order \<G> + (order \<G> - s)))"
example_productF: shows "productF M\<^sub>S M\<^sub>I FAIL\<^sub>I PM\<^sub>I"
compact_lemma: fixes f :: "nat \<Rightarrow> 'b" assumes "bounded (range f)" shows "\<forall>d\<subseteq>Basis. \<exists>l r. strict_mono r \<and> (\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e)"
ctor_induct: fixes phi1 :: "'d IF1 \<Rightarrow> bool"   and phi2 :: "'d IF2 \<Rightarrow> bool"   and a :: "'d IF1"   and b :: "'d IF2" assumes "\<And>x. \<lbrakk>\<And>a. a \<in> F1set2 x \<Longrightarrow> phi1 a; \<And>a. a \<in> F1set3 x \<Longrightarrow> phi2 a\<rbrakk> \<Longrightarrow> phi1 (ctor1 x)"   and "\<And>x. \<lbrakk>\<And>a. a \<in> F2set2 x \<Longrightarrow> phi1 a; \<And>a. a \<in> F2set3 x \<Longrightarrow> phi2 a\<rbrakk> \<Longrightarrow> phi2 (ctor2 x)" shows "phi1 a \<and> phi2 b"
meyer_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.left_kleene_algebra plus times less_eq less one star" shows "times (plus one x) (star (times x x)) = star x"
is_weak_bisimulation_eqvt: fixes R :: "'state \<Rightarrow> 'state \<Rightarrow> bool"   and p :: "perm" assumes "is_weak_bisimulation R" shows "is_weak_bisimulation (p \<bullet> R)"
Top_def: shows "\<top> \<equiv> \<bottom> \<^bold>\<rightarrow> \<bottom>"
distinct_23: fixes x6 :: "ty" shows "Integer \<noteq> x6\<lfloor>\<rceil>"
mkM_def: fixes T :: "('a, 'es, 'as) BEState Trace set" shows "SPR.mkM T \<equiv> \<lparr>worlds = T, relations = \<lambda>a. rel_ext (\<lambda>(t, t'). {t, t'} \<subseteq> T \<and> spr_jview a t = spr_jview a t'), valuation = envVal \<circ> tLast\<rparr>"
nxtAct_eq: fixes n :: "nat"   and n' :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "n \<le> n'"   and "active c (t n')"   and "\<forall>n''\<ge>n. n'' < n' \<longrightarrow> \<not> active c (t n'')" shows "n' = dynamic_component.nxtAct active c t n"
prefixes_tl_only_01: fixes t :: "nat"   and j :: "nat"   and b :: "nat" assumes "local.prefixes t j \<down>= b" shows "\<forall>x>0. e_nth b x = 0 \<or> e_nth b x = 1"
Fun_mkArr'_2: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)" shows "Fun (mkArr A B F) = restrict F A"
below_prod_def: shows "(\<sqsubseteq>) \<equiv> \<lambda>p1 p2. fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2"
Dp_Mp_eq: fixes f :: "int poly" shows "f = pl.Mp f + Polynomial.smult (p ^ l) (pl.Dp f)"
hom_components_keys_disjoint: fixes q1 :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q2 :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "q1 \<in> hom_components p"   and "q2 \<in> hom_components p"   and "q1 \<noteq> q2" shows "keys q1 \<inter> keys q2 = {}"
card_of_Field_ordIso: fixes r :: "('d \<times> 'd) set" assumes "Card_order r" shows "|Field r| =o r"
distinct_5: fixes x1 :: "agent"   and x4 :: "ltkey" shows "cAgent x1 \<noteq> cLtK x4"
le_neq_trans: fixes a :: "'b list"   and b :: "'b list" assumes "a \<le>s b"   and "a \<noteq> b" shows "a <s b"
isCont_cosh: fixes x :: "'a" shows "isCont cosh x"
distinct_1: fixes x21 :: "'x"   and x22 :: "'x list" shows "[] \<noteq> x21 # x22"
inj_on_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
assms_2: shows "norm_instr (body fd2 ! pc) = norm_instr instr"
is_unit_gcd_iff: fixes one :: "'d"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and gcd :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and lcm :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) one = (gcd a b = one)"
mem_permute_iff: fixes p :: "perm"   and x :: "'e"   and X :: "'e set" shows "(p \<bullet> x \<in> p \<bullet> X) = (x \<in> X)"
reads_consistent_mem_eq_on_unforwarded_non_volatile_reads: fixes A :: "nat set"   and W :: "nat set"   and m' :: "nat \<Rightarrow> nat"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and pending_write :: "bool"   and \<O> :: "nat set" assumes "\<forall>a\<in>A \<union> W. m' a = m a"   and "unforwarded_non_volatile_reads sb W \<subseteq> A"   and "reads_consistent pending_write \<O> m sb" shows "reads_consistent pending_write \<O> m' sb"
assms_3: fixes X :: "'a" assumes "X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub>" shows "\<eta>1 $$ X = \<eta>2 $$ X"
scaleR_iarray_def: fixes k :: "real"   and A :: "'c iarray" shows "k *\<^sub>R A = IArray.of_fun (\<lambda>i. k *\<^sub>R A !! i) (IArray.length A)"
nan_not_finite: fixes x :: "('a, 'b) IEEE.float" assumes "is_nan x" shows "\<not> is_finite x"
simps_4: fixes \<sigma> :: "nat \<Rightarrow> val"   and oper :: "pdf_operator"   and e :: "expr" shows "expr_sem \<sigma> (oper $$ e) = expr_sem \<sigma> e \<bind> (\<lambda>x. return_val (op_sem oper x))"
H_var1: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "H p x q = (t p \<cdot> x \<cdot> !q = (0::'a))"
Rats_eq_int_div_int': shows "\<rat> = {real_of_int p / real_of_int q |p q. 0 < q}"
simps_4: fixes tst :: "'test"   and c :: "('test, 'atom) com" shows "SC_WbisT (while tst do c) = (if compatTst tst then SC_WbisT c else noWhile (while tst do c) \<and> SC_discr (while tst do c) \<or> SC_siso (while tst do c))"
subspace_span: fixes S :: "'a set" shows "ideal.subspace (More_Modules.ideal S)"
Un_insert_right: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "(A::'b set) \<union> ((a::'b) \<triangleright> (B::'b set)) = a \<triangleright> A \<union> B"
divides_refl: fixes a :: "'a" assumes "a \<in> carrier R" shows "a divides\<^bsub>add_monoid R\<^esub> a"
seqE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VxV.seq g f" obtains "seq (fst g) (fst f)" and "seq (snd g) (snd f)"
ba1_commutative: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
lipschitz: shows "\<forall>t\<in>{t0..T}. L-lipschitz_on D' (f t)"
DataSpace_Data2PData: fixes D :: "'b data" shows "PDataSpace (Data2PData D) = Data.DataSpace D"
data_def: shows "data = {(L, D) |L D. Data L D}"
h1: shows "((\<Sum>i = 0..<d + 1. f i * C ^ (d - i)) * C ^ (d mod 2) \<le> 0) = ((\<Sum>i = 0..<d + 1. f i / C ^ i) \<le> 0)"
sat_n_imp: fixes N :: "'f set" assumes "empty_ord.saturated N" shows "\<exists>q\<in>Q. ground.Inf_from_q q (\<G>_Fset_q q N) \<subseteq> {\<iota>. \<exists>\<iota>'\<in>Inf_from N. \<G>_I_q q \<iota>' \<noteq> None \<and> \<iota> \<in> the (\<G>_I_q q \<iota>')} \<union> Red_I_q q (\<G>_Fset_q q N)"
simps_1: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "eval_tpoly \<alpha> (PVar x) = \<alpha> x"
unat_word_ariths_1: fixes a :: "'b word"   and b :: "'b word" shows "unat (a + b) = (unat a + unat b) mod 2 ^ LENGTH('b)"
lossless_funct: fixes m1 :: "'msg1"   and m2 :: "'msg2" shows "lossless_spmf (funct m1 m2)"
gets_the_opt_map: fixes f :: "'b \<Rightarrow> 'd option"   and g :: "'d \<Rightarrow> 'c option" shows "gets_the (f |> g) = do x \<leftarrow> gets_the f; assert_opt (g x) od"
degcmc': shows "Polynomial_Expression.degree (c +\<^sub>p ~\<^sub>p c') = 0"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_mode type_wf set_mode set_mode_locs"   and "l_get_tag_name type_wf get_tag_name get_tag_name_locs"   and "l_set_mode_get_tag_name_axioms set_mode_locs get_tag_name_locs" shows "l_set_mode_get_tag_name type_wf set_mode set_mode_locs get_tag_name get_tag_name_locs"
strict_coboundedI1: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a < c" shows "inf a b < c"
simps_2: fixes x :: "char list"   and a :: "aexp"   and t :: "char list \<Rightarrow> int option" shows "Fold.fold (x ::= a) t = x ::= afold a t"
distinct_13: fixes x1 :: "agent"   and x81 :: "nat"   and x82 :: "msg" shows "Agent x1 \<noteq> Crypt x81 x82"
infs_infm: fixes P :: "'c \<Rightarrow> bool"   and w :: "'c stream" shows "infs P w = (\<exists>\<^sub>\<infinity>i. P (w !! i))"
comm_semiring_ow_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "comm_semiring_ow U plus times \<equiv> ab_semigroup_add_ow U plus \<and> ab_semigroup_mult_ow U times \<and> comm_semiring_ow_axioms U plus times"
Vinv3_inductive_1: shows "init v_TS \<subseteq> Vinv3"
False: shows "component_of_term (lt q) \<noteq> component_of_term (lt p)"
hyperUn_def: fixes A :: "'b set option"   and B :: "'b set option" shows "A \<squnion> B \<equiv> case A of None \<Rightarrow> None | \<lfloor>A\<rfloor> \<Rightarrow> case B of None \<Rightarrow> None | \<lfloor>B\<rfloor> \<Rightarrow> \<lfloor>A \<union> B\<rfloor>"
inv_greek_def: fixes s :: "(accent \<times> 'b) list" shows "inv_greek s = rev (map inv_letter s)"
dickson_grading_zero: shows "dickson_grading (\<lambda>_. 0)"
simps_2: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and cs :: "('Var, 'AExp, 'BExp) Stmt list"   and c' :: "('Var, 'AExp, 'BExp) Stmt" shows "cxt_to_stmt (c # cs) c' = c' ;; cxt_to_stmt cs c"
epsclo_subset: fixes Q :: "hf set" shows "epsclo Q \<subseteq> nfa.states M"
trancl_unfold_left: fixes r :: "('c \<times> 'c) set" shows "r\<^sup>+ = r O r\<^sup>*"
fun_dual1: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f = g \<circ> \<partial>) = (f \<circ> \<partial> = g)"
ae_disjoint_then_null: fixes A :: "'a set" assumes "A \<in> sets M"   and "\<And>n. 0 < n \<Longrightarrow> A \<inter> (T ^^ n) -` A \<in> null_sets M" shows "A \<in> null_sets M"
card_of_Func_option_mono: fixes A1 :: "'a set"   and A2 :: "'a set"   and B :: "'b set" assumes "A1 \<subseteq> A2"   and "B \<noteq> {}" shows "|Func_option A1 B| \<le>o |Func_option A2 B|"
atMost_iff: fixes i :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c" shows "(i \<in> ord.atMost less_eq k) = less_eq i k"
T_ins3_def: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree" shows "T_ins3 n d x t = time (ins3_tm n d x t)"
distinct_116: fixes x14 :: "Syntax.term"   and x41 :: "Syntax.var"   and x42 :: "Syntax.term" shows "Unroll x14 \<noteq> Rec x41 x42"
SN_pow_imp_SN: fixes n :: "nat"   and A :: "('b \<times> 'b) set" assumes "SN (A ^^ Suc n)" shows "SN A"
sdrop_slength_plus_1: fixes y :: "'r"   and xs :: "[:'s:]"   and ys :: "[:'r:]" assumes "y \<noteq> \<bottom>" shows "sdrop\<cdot>(slength\<cdot>xs + 1)\<cdot>(y :# ys) = sdrop\<cdot>(slength\<cdot>xs)\<cdot>ys"
SeqT: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>t s'" shows "(c1 ;; c2, s) \<rightarrow>c (c2, s')"
bounded_scaleR_comp: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and r :: "real" assumes "bounded (f ` S)" shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` S)"
sum_in_bigo_2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](h)"   and "g \<in> O[F](h)" shows "(\<lambda>x. f x - g x) \<in> O[F](h)"
shared_eq: shows "\<forall>a\<in>X. \<S>' a = \<S> a"
get_rho_pair_char: fixes x :: "'c regular"   and y :: "'c dense Filters.filter" assumes "(x, y) \<in> stone_phi.pairs" shows "up_filter (stone_phi.rho_pair (x, y)) = stone_phi x \<sqinter> y"
obs_ids_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a" shows "IFC_def.obs_ids prob \<pi> = {k. IFC_def.att prob (\<pi> k) \<noteq> None}"
add_nonneg_nonneg: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "(0::'a) \<le> a + b"