hcju/isabelleps huggingface dataset:
ffold_parametric: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f2 :: "'e \<Rightarrow> 'g \<Rightarrow> 'g" assumes "(A ===> B ===> B) f1 f2"   and "comp_fun_idem f1"   and "comp_fun_idem f2" shows "(B ===> rel_fset A ===> B) (ffold f1) (ffold f2)"
simps_1: fixes p :: "fm"   and q :: "fm" shows "islin (And p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
solve_ratfps_def: fixes f :: "complex ratfps" shows "solve_ratfps f = (case quot_of_ratfps f of (p, q) \<Rightarrow> map_option (solve_factored_ratfps' p) (factorize_complex_main (reflect_poly q)))"
rep_eq: fixes x :: "'b coreflexive"   and xa :: "'b coreflexive" shows "Rep_coreflexive (x * xa) = Rep_coreflexive x * Rep_coreflexive xa"
map_replicate_trivial: fixes x :: "'c"   and i :: "nat" shows "map (\<lambda>i. x) [0..<i] = replicate i x"
mult_neg_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a zero"   and "less zero b" shows "less (times a b) zero"
i_cut_bound_2: fixes n :: "'a"   and I :: "'a set"   and t :: "'a" assumes "n \<in> I \<down>\<le> t" shows "n \<le> t"
j_of_in_R: fixes k :: "nat" shows "(f (Suc (j_of k)), f (j_of k)) \<in> R"
set_2: fixes x21 :: "'c"   and x22 :: "'c llist" shows "lset (LCons x21 x22) = insert x21 (lset x22)"
twocoins_def: shows "twocoins = bernoulli_pmf (4 / 10) \<bind> (\<lambda>x. bernoulli_pmf (5 / 10) \<bind> (\<lambda>y. return_pmf (x \<or> y)))"
monic_mult: fixes p :: "'c poly"   and q :: "'c poly" assumes "monic p"   and "monic q" shows "monic (p * q)"
simps_1: fixes uu :: "'a" shows "chainlen \<lbrace> uu \<rbrace> = 0"
box_eq_empty_2: fixes a :: "'c"   and b :: "'c" shows "(cbox a b = {}) = (\<exists>i\<in>Basis. b \<bullet> i < a \<bullet> i)"
dm_subset: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset" assumes "irrefl r"   and "trans r" shows "r \<down>m M \<subseteq> r \<down>m (M -s r \<down>m M)"
pw_bind_nofail: fixes M :: "'d nres"   and f :: "'d \<Rightarrow> 'c nres" shows "nofail (M \<bind> f) = (nofail M \<and> (\<forall>x. inres M x \<longrightarrow> nofail (f x)))"
not_less_iff_gr_or_eq: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(\<not> ko.lt ko x y) = (ko.lt ko y x \<or> x = y)"
assms_2: shows "\<forall>x\<in>{i..<n}. P x"
less_multiset\<^sub>H\<^sub>O: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(M < N) = (M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x)))"
that: fixes k :: "real"   and r :: "real"   and n :: "nat" assumes "0 < k"   and "0 < r"   and "ball \<xi> r \<subseteq> S"   and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> k * cmod (w - \<xi>) ^ n \<le> cmod (f w - f \<xi>)" shows "thesis"
nprv_clear4_1: fixes \<phi>2 :: "'fmla"   and \<phi>3 :: "'fmla"   and \<phi>4 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>1 :: "'fmla" assumes "nprv {\<phi>2, \<phi>3, \<phi>4} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2, \<phi>3, \<phi>4} \<psi>"
calc: shows "polymap (pCons a p) = (\<lambda>v. (a \<cdot>\<cdot> If (v \<in> V) (id v)) (0::'v)) + sum_list (map2 (\<lambda>c k. c \<cdot>\<cdot> (T \<circ> endpow k)) (coeffs p) [0..<Suc (degree p)])"
AnalBinds_lookup: fixes exp :: "exp \<Rightarrow> 'b \<rightarrow> 'a"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>"   and x :: "Vars.var" shows "(ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae) x = (case map_of \<Gamma> x of None \<Rightarrow> \<bottom> | Some e \<Rightarrow> fup\<cdot>(exp e)\<cdot>(ae x))"
unique_authKeys: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "msg"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and Ka' :: "nat"   and Ta' :: "msg"   and X' :: "msg" assumes "Says Kas A (Crypt Ka \<lbrace>Key authK, Agent Tgs, Ta, X\<rbrace>) \<in> set evs"   and "Says Kas A' (Crypt Ka' \<lbrace>Key authK, Agent Tgs, Ta', X'\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "A = A' \<and> Ka = Ka' \<and> Ta = Ta' \<and> X = X'"
subcringI': fixes R :: "('c, 'd) ring_scheme"   and H :: "'c set" assumes "cring R"   and "subring H R" shows "subcring H R"
axioms_2: fixes x :: "'b \<Longrightarrow> 'c" assumes "wb_lens x" shows "wb_lens_axioms x"
ptensor_vec_def: fixes v1 :: "'a vec"   and v2 :: "'a vec" shows "st2.ptensor_vec v1 v2 = partial_state.tensor_vec st2.dims0 st2.vars1' v1 v2"
ET_target_source: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<subseteq> ET ST"   and "t \<in> TS"   and "target t \<in> States A"   and "A \<in> SAs (HA ST)" shows "source t \<in> States A"
P_def: shows "P \<equiv> Pi\<^sub>M UNIV (\<lambda>_. M)"
triangular_to_jnf_steps_dims_6: fixes A :: "'b mat" shows "dim_col (??.Jordan_Normal_Form_Existence.step_3 A) = dim_col A"
reversepath_simps_3: fixes g :: "real \<Rightarrow> 'b" shows "pathstart (reversepath g) = pathfinish g"
const_res_subseq_prop_1: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "padic_integers p"   and "s \<in> carrier padic_int p\<^bsup>\<omega>\<^esup>" shows "\<forall>m. const_res_subseq k s m k = const_res k s"
lir_fps: fixes cs :: "'b list"   and m :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "cs \<noteq> []"   and "\<And>n. m \<le> n \<Longrightarrow> (\<Sum>k\<le>length cs - 1. cs ! k * f (n + k)) = g (n + (length cs - 1))"   and "last cs \<noteq> (0::'b)" shows "Abs_fps f = (fps_of_poly (lir_fps_numerator m cs f g) + Abs_fps g) / fps_of_poly (lr_fps_denominator cs)"
discrete_imp_closed: fixes e :: "real"   and S :: "'a set" assumes "0 < e"   and "\<forall>x\<in>S. \<forall>y\<in>S. dist y x < e \<longrightarrow> y = x" shows "closed S"
is_path_memb: fixes v :: "'V"   and p :: "('V \<times> 'W \<times> 'V) list"   and v' :: "'V" assumes "is_path v p v'" shows "v \<in> V \<and> v' \<in> V"
comp_termFSwImorph: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) model_scheme"   and h' :: "'c \<Rightarrow> 'f"   and hA' :: "'d \<Rightarrow> 'g"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'f, 'g, 'h) model_scheme" assumes "termFSwImorph h hA MOD"   and "FSwImorph h' hA' MOD MOD'" shows "termFSwImorph (h' \<circ> h) (hA' \<circ> hA) MOD'"
sum_squares_gt_zero_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less zero (plus (times x x) (times y y)) = (x \<noteq> zero \<or> y \<noteq> zero)"
induct: fixes P :: "Vertex \<Rightarrow> bool"   and Vertex :: "Vertex" assumes "P A"   and "P B"   and "P C"   and "P D" shows "P Vertex"
OK_OK_conv: fixes P :: "'b prog"   and T :: "ty"   and T' :: "ty" shows "P \<turnstile> OK T \<le>\<^sub>\<top> OK T' = subtype P T T'"
list_comb_def: shows "list_comb = (\<lambda>(x, y). foldl ($) x y)"
ntsubst_free_to_safe: fixes \<theta> :: "('d + 'e, 'f) trm"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) trm" assumes "dfree \<theta>"   and "\<And>i. dsafe (\<sigma> i)" shows "dsafe (TsubstFO \<theta> \<sigma>)"
not_prime_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "\<not> normalization_semidom.prime zero times one normalize zero"
prob: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV \<le> 1"
closed_segment_idem: fixes a :: "'b" shows "closed_segment a a = {a}"
inf_sup_aci_5: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = y \<sqinter> x"
rbt_lookup_map: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and t :: "('c, 'e) rbt"   and x :: "'c" assumes "class.linorder less_eq less" shows "ord.rbt_lookup less (RBT_Impl.map f t) x = map_option (f x) (ord.rbt_lookup less t x)"
iso_iff_mono_and_retraction: fixes f :: "'a" shows "local.iso f = (local.mono f \<and> retraction f)"
wlsEnv_imp_goodEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsEnv rho" shows "goodEnv rho"
subrelI: fixes r :: "('c \<times> 'd) set"   and s :: "('c \<times> 'd) set" assumes "\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s" shows "r \<subseteq> s"
assms_2: assumes "\<paragraph>''else''"   and "\<not> bval b s" shows "wp \<pi> c\<^sub>2 Q s"
tanh_real_pos_iff: fixes x :: "real" shows "(0 < tanh x) = (0 < x)"
summset_def: fixes M :: "'a multiset" shows "summset M \<equiv> fold_mset (\<oplus>) \<zero> M"
simps_10: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<nu> :: "'b set" shows "af_letter (\<phi> W\<^sub>n \<psi>) \<nu> = af_letter \<psi> \<nu> or\<^sub>n (af_letter \<phi> \<nu> and\<^sub>n \<phi> W\<^sub>n \<psi>)"
Rep_hierauto_tuple: fixes HA :: "('e, 'f, 'd) hierauto" shows "Rep_hierauto HA = (HAInitValue HA, SAs HA, HAEvents HA, CompFun HA)"
simps: fixes p :: "'a mpoly" shows "is_constant p = Set.is_empty (vars p)"
domain_complex: fixes X :: "'c set set"   and Y :: "'d set set"   and f :: "'c \<Rightarrow> 'd" assumes "ChamberComplexIsomorphism X Y f" shows "ChamberComplex X"
ltl_iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "ltl (iterates f x) = iterates f (f x)"
less_eq_float_def: shows "(\<le>) \<equiv> map_fun real_of_float (map_fun real_of_float id) (\<le>)"
Throw: fixes P :: "'d set"   and A :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and Q :: "'d set" assumes "P \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Throw Q,A"
good_ruleset_alt: fixes rs :: "'b rule list" shows "good_ruleset rs = (\<forall>r\<in>set rs. get_action r = Accept \<or> get_action r = Drop \<or> get_action r = Reject \<or> get_action r = Log \<or> get_action r = Empty)"
SR: shows "(Si, S) \<in> sa_impl_rel_ext unit_rel Rs Rl"
sumlist_Nil: shows "M.sumlist [] = 0\<^sub>v n"
aact''_def: shows "aact'' = aact(a := aact' a)"
vars_snd_rcv_strand_subset2_1: fixes S :: "('d, 'c) strand_step list" shows "\<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S)) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
assms_2: shows "recfn (length xs) (fst (hd (fst c)))"
mult_fract_cancel: fixes c :: "'c"   and a :: "'c"   and b :: "'c" assumes "c \<noteq> (0::'c)" shows "Fraction_Field.Fract (c * a) (c * b) = Fraction_Field.Fract a b"
split_sel: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and prod :: "'c \<times> 'd" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (prod = (fst prod, snd prod) \<longrightarrow> P (f (fst prod) (snd prod)))"
assms_2: shows "(\<Sum>i. f i) \<noteq> \<infinity>"
cases: fixes a1 :: "'a stream"   and a2 :: "'a treeFI" assumes "properPath a1 a2" obtains as :: "'a stream" and tr :: "'a treeFI" and tr' :: "'a treeFI" where "a1 = as" and "a2 = tr" and "shd as = lab tr" and "tr' \<in> set (sub tr)" and "properPath (stl as) tr'"
abs_def: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Abs :: "'f set \<Rightarrow> 'g"   and Rep :: "'g \<Rightarrow> 'f set"   and x :: "'f" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_inv_1: fixes l :: "'d strand_label"   and a :: "('e, 'f) stateful_strand_step"   and k :: "'d strand_label"   and b :: "('e, 'f) stateful_strand_step" assumes "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, a) = (k, b)" shows "l = k"
list_Cons_eq_append_cases: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "x # xs = ys @ zs" obtains "ys = []" and "zs = x # xs"   | ys' :: "'b list" where "ys = x # ys'" and "ys' @ zs = xs"
eq_set_tree_empty: fixes t :: "'c tree" shows "(set_tree t = {}) = (t = \<langle>\<rangle>)"
left_unit: fixes a :: "'b" assumes "a \<in> M'" shows "\<one>' \<cdot>' a = a"
B5: shows "strong_reduction_simulation Rel (STCal Source Target) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> (P', Q') \<in> Rel))"
space_as_set: fixes x :: "'d ccsubspace" shows "space_as_set x \<in> {S. closed_csubspace S}"
stream_eq_iff: fixes s :: "'b stream"   and s' :: "'b stream" shows "(s = s') = (\<forall>n. s !! n = s' !! n)"
dg_0_def: shows "dg_0 = [[]\<^sub>\<circ>, []\<^sub>\<circ>, []\<^sub>\<circ>, []\<^sub>\<circ>]\<^sub>\<circ>"
after_exists: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "|~ (\<exists>x. P x)$ = (\<exists>x. P x$)"
the_coext_prop_1: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' (C.cod f') f" shows "\<guillemotleft>x''\<epsilon>.the_coext x' f : x' \<rightarrow>\<^sub>D Go (C.cod f')\<guillemotright>"
Suc: shows "mat n n (mat_powI ze on pl ti n m i)"
list_all2_append1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'b list" shows "list_all2 P (xs @ ys) zs = (\<exists>us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> list_all2 P xs us \<and> list_all2 P ys vs)"
MethodDefs'_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list"   and Cs :: "char list list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr" shows "MethodDefs' P C M Cs mthd \<equiv> (Cs, mthd) \<in> MethodDefs P C M"
intros_28: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (global \<C>)"   and "tg_t (global \<C> ! i) = t"   and "is_mut (global \<C> ! i)" shows "\<C> \<turnstile> [Set_global i] : [t] _> []"
crename_interesting1: fixes a :: "coname"   and b :: "coname"   and c :: "coname"   and M :: "trm" assumes "distinct [a, b, c]" shows "M[a\<turnstile>c>c][c\<turnstile>c>b] = M[c\<turnstile>c>b][a\<turnstile>c>b]"
dterms_choice_disj: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "(p \<in> dterms \<Gamma> (p1 \<oplus> p2)) = (p \<in> dterms \<Gamma> p1 \<or> p \<in> dterms \<Gamma> p2)"
functions_of_def: fixes A :: "'c set"   and B :: "'d multiset" shows "functions_of A B = {f \<in> A \<rightarrow>\<^sub>E set_mset B. image_mset f (mset_set A) = B}"
lincomb_vec_index: fixes F :: "complex vec set"   and i :: "nat"   and n :: "nat"   and f :: "complex vec \<Rightarrow> complex" assumes "finite F"   and "i < n"   and "F \<subseteq> carrier_vec n"   and "f \<in> F \<rightarrow> UNIV" shows "module.lincomb (module_cpx_vec n) f F $ i = (\<Sum>v\<in>F. f v * v $ i)"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VVV.arr f = (B.arr (fst f) \<and> B.VV.arr (snd f) \<and> trg (fst f) = src (fst (snd f)))"
BuildSeq_induct: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and a :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "BuildSeq B C s k a"   and "\<And>x. B x \<Longrightarrow> P x"   and "\<And>x y z. \<lbrakk>C x y z; P y; P z\<rbrakk> \<Longrightarrow> P x" shows "P a"
ivl_disj_un_two_touch_4: fixes l :: "'b"   and m :: "'b"   and u :: "'b" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m..u} = {l..u}"
sub: fixes a :: "'a" assumes "a \<in> I" shows "a \<in> A"
worker_wrapper_fusion: fixes wrap :: "'b \<rightarrow> 'a"   and unwrap :: "'a \<rightarrow> 'b"   and work :: "'b"   and body :: "'a \<rightarrow> 'a" assumes "wrap oo unwrap = ID"   and "work = fix\<cdot>(unwrap oo body oo wrap)" shows "(unwrap oo wrap)\<cdot>work = work"
wf_rel_while_option_Some: fixes R :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "wf R"   and "\<And>s. P s \<and> b s \<Longrightarrow> (c s, s) \<in> R"   and "\<And>s. P s \<and> b s \<Longrightarrow> P (c s)"   and "P s" shows "\<exists>t. while_option b c s = Some t"
power_eq_imp_eq_norm: fixes w :: "'a"   and n :: "nat"   and z :: "'a" assumes "w ^ n = z ^ n"   and "0 < n" shows "norm w = norm z"
some_ord0rel: fixes x :: "ord0" shows "(x, SOME y. (x, y) \<in> ord0rel) \<in> ord0rel"
square_eq_iff: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.idom times one plus zero minus uminus" shows "(times a a = times b b) = (a = b \<or> a = uminus b)"
conn2_bb: fixes kid :: "'kid"   and k :: "cnf" assumes "\<parallel>kid\<parallel>\<^bsub>k\<^esub>" shows "bbop (\<sigma>\<^bsub>the_bb\<^esub>k) \<in> ksop (\<sigma>\<^bsub>kid\<^esub>k)"
simps_2: fixes A :: "'b mat"   and i_end :: "nat"   and list :: "(nat \<times> nat) list" shows "??.Jordan_Normal_Form_Existence.identify_blocks_main A (Suc i_end) list = (let i_begin = ??.Jordan_Normal_Form_Existence.identify_block A i_end in ??.Jordan_Normal_Form_Existence.identify_blocks_main A i_begin ((i_begin, i_end) # list))"
elims: fixes n :: "nat"   and fs :: "'a vec list"   and x :: "nat"   and xa :: "nat"   and y :: "'a" assumes "gram_schmidt_fs.\<mu> n fs x xa = y" obtains i :: "nat" and j :: "nat" where "x = i" and "xa = j" and "y = (if j < i then fs ! i \<bullet> gram_schmidt_fs.gso n fs j / \<parallel>gram_schmidt_fs.gso n fs j\<parallel>\<^sup>2 else if i = j then 1::'a else (0::'a))"
compress_interfaces_None_1: fixes \<beta> :: "'c common_primitive \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and ifces :: "iface negation_type list"   and \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('c, 'd) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "compress_interfaces ifces = None" shows "\<not> matches (\<beta>, \<alpha>) (alist_and (NegPos_map IIface ifces)) a p"
is_image_spec_rev: fixes P :: "'p"   and P' :: "'p"   and A :: "'p"   and B :: "'p" assumes "P P' ReflectL A B" shows "P P' ReflectL B A"
finite_deriv: fixes s :: "(nat \<times> form) list" shows "finite (deriv s) = (\<exists>m. f [s] m = [])"
tts_AB_transfer: fixes f :: "'f \<Rightarrow> 'e"   and \<UU>A :: "'f set"   and \<UU>B :: "'e set"   and A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'h \<Rightarrow> bool" assumes "f ` \<UU>A \<subseteq> \<UU>B"   and "Domainp A = (\<lambda>x. x \<in> \<UU>A)"   and "bi_unique A"   and "right_total A"   and "Domainp B = (\<lambda>x. x \<in> \<UU>B)"   and "bi_unique B"   and "right_total B" shows "\<exists>rcdt. (A ===> B) f rcdt"
measurable_pair_iff: fixes f :: "'d \<Rightarrow> 'e \<times> 'f"   and M :: "'d measure"   and M1 :: "'e measure"   and M2 :: "'f measure" shows "(f \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2) = (fst \<circ> f \<in> M \<rightarrow>\<^sub>M M1 \<and> snd \<circ> f \<in> M \<rightarrow>\<^sub>M M2)"
simps_21: fixes e :: "('f, 'g, 'h) exp"   and c :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (while (e) c) A = (\<D> e A \<and> \<D> c (A \<squnion> \<A> e))"
hyps: assumes "i \<le> n" shows "x\<^bsup>i\<^esup> \<le> x\<^bsub>0\<^esub>\<^bsup>n\<^esup>"
l_set_tag_name_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_tag_name_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs"
elem_neq_classes_id: shows "(\<lambda>X. X - {elem}) ` (P - {Y}) = P - {Y}"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> ves @ [Callcl cl] : ts _> ts'"
prim_exp_eq: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "primitive u"   and "r \<^sup>@ k = u" shows "u = r"
space_stream_space: fixes M :: "'c measure" shows "space (stream_space M) = streams (space M)"
dist1: shows "norm (min (1 / 2) (min d1 d2) / 2) < d1"
R_def: fixes x :: "bool"   and y :: "bool" shows "R x y = False"
select_convs_1: fixes init :: "'b set"   and trans :: "('b \<times> 'b) set"   and more :: "'c" shows "init \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = init"
weakTransitionTau: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P" obtains P'' :: "('a, 'b, 'c) psi" where "\<Psi> : Q \<rhd> \<tau>.P \<Longrightarrow>\<alpha> \<prec> P''" and "\<Psi> \<rhd> P' \<sim> P''"
\<P>_0_0_Gen: shows "Gen (\<P> 0 0) = {x. wellformed_item x \<and> item_origin x = 0 \<and> item_end x = 0 \<and> derives (item_\<alpha> x) [] \<and> (\<exists>\<gamma>. is_derivation ([item_nonterminal x] @ \<gamma>))}"
subgroup_generated_is_subset: fixes H :: "'a set" shows "subgroup_generated H \<subseteq> G"
no_abs_simps_1: fixes name :: "name" shows "no_abs (const name) = True"
graph_lm_2: fixes y :: "nat"   and f :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "y = f x" shows "(x, y) \<in> graph f"
approx_slp_refine: fixes prec :: "nat"   and fas :: "floatarith list" shows "(nres_of o3 aform_slp prec, approx_slp_spec fas) \<in> nat_rel \<rightarrow> fas_rel \<rightarrow> aforms_rel \<rightarrow> \<langle>\<langle>aforms_rel\<rangle>option_rel\<rangle>nres_rel"
map_cong: fixes xs :: "'e list"   and ys :: "'e list"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "map f xs = map g ys"
fwi_impl'_def: fixes n :: "nat"   and M :: "nat \<times> nat \<Rightarrow> 'b"   and k :: "nat" shows "fwi_impl' n M k = for_rec2 (\<lambda>M. fw_upd' M k) M n n n"
ih: assumes "\<not> is_final (steps0 c A n)" shows "steps0 c (A |+| B) n = steps0 c A n"
assms: shows "is_valid_operator_sas_plus \<Psi> op"
nth_fls_subdegree_zero_iff: fixes f :: "'a fls" shows "(fls_nth f (fls_subdegree f) = (0::'a)) = (f = 0)"
"3_3": shows "wf_tuple n (\<Union> (fv_trm ` set (trm.Var x # ts))) v"