hcju/isabelleps huggingface dataset:
BinOpRed2: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 v\<^sub>1 :: "val"   and bop :: "bop" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> e,s\<rangle> \<rightarrow> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> e',s'\<rangle>"
iffb_iffd_demorgan: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "fb\<^sup>-\<^sub>\<F> \<phi> = (fd\<^sup>-\<^sub>\<F> \<circ> \<partial>\<^sub>F) \<phi>"
is_processT7: fixes P :: "'b process" shows "\<forall>s t. s \<in> D P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> D P"
ins_list_snoc: fixes xs :: "'b list"   and x :: "'b" assumes "Sorted_Less.sorted (xs @ [x])" shows "ins_list x xs = xs @ [x]"
x1ImportantProp3: shows "p x1 \<in> p ` fst ` f"
assms_5: shows "\<forall>A' C' D' F'. B Out A' A \<and> B Out C' C \<and> E Out D' D \<and> E Out F' F \<and> Cong B A' E D' \<and> Cong B C' E F' \<longrightarrow> Cong A' C' D' F'"
Deq_def: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "Deq ic q oc \<equiv> #0 < length<$q> \<and> oc$ = hd<$q> \<and> q$ = tl<$q> \<and> ic$ = $ic"
unique_expression3_1: fixes H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "H \<subseteq> carrier M"   and "f \<in> {l. l \<le> Suc n} \<rightarrow> H"   and "s \<in> {l. l \<le> Suc n} \<rightarrow> carrier R"   and "f (Suc n) \<notin> f ` ({l. l \<le> Suc n} - {Suc n})" shows "\<exists>g m t. g \<in> {l. l \<le> m} \<rightarrow> H \<and> inj_on g {l. l \<le> m} \<and> t \<in> {l. l \<le> m} \<rightarrow> carrier R \<and> l_comb R M (Suc n) s f = l_comb R M m t g \<and> t m = s (Suc n) \<and> g m = f (Suc n)"
assms_2: shows "\<forall>n. \<forall>asset\<in>support_set pf. finite (pf asset n ` space M)"
poly_add_rat_def: fixes r :: "rat"   and p :: "int poly" shows "poly_add_rat r p \<equiv> case quotient_of r of (n, d) \<Rightarrow> poly_mult_rat_main d 1 p \<circ>\<^sub>p [:- n, d:]"
preserves_reflects_arr: fixes f :: "'j" shows "A_B.arr (cones_map f \<chi> f) = J.arr f"
frac_in_carrier_local: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<pp> :: "'c set"   and r :: "'c"   and s :: "'c" assumes "comm_ring R addition multiplication zero unit"   and "\<pp> \<in> comm_ring.spectrum R addition multiplication zero unit"   and "r \<in> R"   and "s \<in> R"   and "s \<notin> \<pp>" shows "quotient_ring.frac (R\<setminus>\<pp>) R addition multiplication zero r s \<in> R \<^bsub>\<pp> addition multiplication zero\<^esub>"
range_image_mset: fixes Ds :: "'b multiset"   and f :: "'c \<Rightarrow> 'b" assumes "set_mset Ds \<subseteq> range f" shows "Ds \<in> range (image_mset f)"
minus_add_distrib: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ab_group_add plus zero minus uminus" shows "uminus (plus a b) = plus (uminus a) (uminus b)"
restore_orgin_paras: fixes xs :: "nat list"   and f :: "recf"   and gs :: "recf list"   and ffp :: "nat"   and anything :: "nat list" shows "{\<lambda>nl. nl = 0 \<up> length xs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - length xs) @ 0 \<up> length gs @ xs @ anything} mv_boxes (Suc (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs)) 0 (length xs) {\<lambda>nl. nl = xs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs) @ anything}"
r_max_eq_r_max_opt: fixes t :: "('b, unit) RBT.rbt" assumes "\<not> RBT.is_empty t" shows "r_max t = r_max_opt t"
cinner_adj_right: fixes x :: "'a"   and G :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and y :: "'b" shows "x \<bullet>\<^sub>C (G* *\<^sub>V y) = (G *\<^sub>V x) \<bullet>\<^sub>C y"
orbit_FOO: fixes a :: "'e"   and g :: "'e \<Rightarrow> 'e"   and g' :: "'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" assumes "a \<in> orbit g a"   and "\<And>x. x \<in> orbit g a \<Longrightarrow> g' (f x) = f (g x)" shows "f ` orbit g a = orbit g' (f a)"
irregularAmicPair_def: fixes n :: "nat"   and m :: "nat" shows "irregularAmicPair n m = (n Amic m \<and> \<not> regularAmicPair n m)"
in_domains_char: fixes a :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" shows "(a \<in> domains f) = ((f \<noteq> Null \<and> Dom f \<in> {False, True} \<and> Cod f \<in> {False, True} \<and> Map f \<in> Collect (G.path (Dom f) (Cod f))) \<and> a = MkIde (Dom f))"
ok3_follow3: fixes r :: "'b mrexp3"   and m :: "bool" assumes "ok3 r" shows "ok3 (follow3 m r)"
monad_catch_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and catch :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "monad_catch return bind fail catch \<equiv> monad_fail return bind fail \<and> monad_catch_axioms return fail catch"
nan_fcompare: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x \<or> IEEE.is_nan y" shows "fcompare x y = Und"
continuous_closed_preimage_constant: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'d" assumes "continuous_on S f"   and "closed S" shows "closed {x \<in> S. f x = a}"
msrevs_1: fixes n :: "nat"   and k :: "nat"   and m :: "nat" assumes "0 < n" shows "(k * n + m) div n = m div n + k"
sigma_sets_top: fixes sp :: "'c set"   and A :: "'c set set" shows "sp \<in> sigma_sets sp A"
correct_subseqs_foldr_impl_axioms: shows "correct_subseqs_foldr_impl f impl invariant"
Lv_z: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<exists>z. Lv K v = ant z"
perp_in_perp_bis_R1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "X \<noteq> A"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.Perp Bet Cong X B C D \<or> Tarski_neutral_dimensionless.Perp Bet Cong A X C D"
inj_on_subset: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'e set" assumes "inj_on f A"   and "B \<subseteq> A" shows "inj_on f B"
reaches_fields: fixes x :: "'a"   and y :: "'a"   and s' :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and s :: "'g process_name \<Rightarrow> ('e, 'g, 'f, 'a) Model.local_state" assumes "(x reaches y) s'"   and "\<forall>r'. \<Union> (ran ` obj_fields ` set_option (sys_heap s' r')) = \<Union> (ran ` obj_fields ` set_option (sys_heap s r'))" shows "(x reaches y) s"
bi_uniqueDr: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and y :: "'f"   and z :: "'f" assumes "bi_unique A"   and "A x y"   and "A x z" shows "y = z"
elims: fixes x :: "'a \<Rightarrow> 'a"   and xa :: "bool"   and xb :: "'a \<times> 'a"   and y :: "'a \<times> 'a" assumes "mod2 x xa xb = y" obtains f :: "'a \<Rightarrow> 'a" and b :: "bool" and a1 :: "'a" and a2 :: "'a" where "x = f" and "xa = b" and "xb = (a1, a2)" and "y = (if b then (a1, f a2) else (f a1, a2))"
subpath_to_frontier: fixes g :: "real \<Rightarrow> 'a"   and S :: "'a set" assumes "path g"   and "pathstart g \<in> closure S"   and "pathfinish g \<notin> S" obtains u :: "real" where "0 \<le> u" and "u \<le> 1" and "g u \<in> frontier S" and "path_image (subpath 0 u g) - {g u} \<subseteq> interior S"
cd_induct: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<And>i. i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k \<Longrightarrow> P i"   and "\<And>k' i'. \<lbrakk>k' cd\<^bsup>\<pi>\<^esup>\<rightarrow> k; P k'; i' icd\<^bsup>\<pi>\<^esup>\<rightarrow> k'\<rbrakk> \<Longrightarrow> P i'" shows "P i"
lift_closed_2: fixes ts :: "'a FOL_Fitting.term list" assumes "closedts 0 ts" shows "closedts 0 (liftts ts)"
knowledge_equiv_eq_NS: fixes evs :: "event list"   and A :: "agent"   and B :: "agent" assumes "evs \<in> ns_public" shows "knows A evs \<union> {Key (priEK B), Key (priSK B), Key (shrK B)} = knows B evs \<union> {Key (priEK A), Key (priSK A), Key (shrK A)}"
bij_betw_prime_power_divisors: fixes p :: "nat"   and k :: "nat" assumes "prime p" shows "bij_betw ((^) p) {..k} {d. d dvd p ^ k}"
H: fixes a :: "'a"   and b :: "'a" shows "a * b \<le> b"
ImZ_def: fixes gauss_int :: "gauss_int" shows "ImZ gauss_int = (case gauss_int of Gauss_Int x1 x2 \<Rightarrow> x2)"
rel_comp_def: fixes r1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'g \<Rightarrow> bool"   and r2 :: "'h \<Rightarrow> 'g \<Rightarrow> 'e \<Rightarrow> bool"   and i :: "'f \<times> 'h" shows "rel_comp r1 r2 i \<equiv> r1 (fst i) OO r2 (snd i)"
nontrans1: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and u :: "'b"   and r :: "'b" assumes "class.interval meets \<I>"   and "meets u r" shows "\<nexists>t. meets u t \<and> meets t r"
foldli_foldl: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and \<sigma> :: "'a" shows "foldli xs (\<lambda>_. True) f \<sigma> = foldl (\<lambda>\<sigma> x. f x \<sigma>) \<sigma> xs"
at_phl_while: fixes p :: "'a"   and s :: "'a"   and x :: "'a"   and w :: "'a" assumes "\<lparr>\<tt>\<tt> p \<cdot> \<tt>\<tt> s\<rparr>x\<lparr>x \<cdot> \<tt>\<tt> p\<rparr>" shows "\<lparr>\<tt>\<tt> p\<rparr>(\<tt>\<tt> s \<cdot> x)\<^sup>\<star> \<cdot> \<tt>\<tt> w\<lparr>(\<tt>\<tt> s \<cdot> x)\<^sup>\<star> \<cdot> \<tt>\<tt> w \<cdot> (\<tt>\<tt> p \<cdot> \<tt>\<tt> w)\<rparr>"
subspace_raw_def: shows "subspace \<equiv> module.subspace (*\<^sub>R)"
is_unit_right_imp_coprime: 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 b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "algebraic_semidom.coprime times one a b"
Set_def: fixes t :: "('d, unit) RBT.rbt" shows "RBT_Set.Set t = {x. RBT.lookup t x = Some ()}"
under_incr: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" assumes "trans r"   and "(a, b) \<in> r" shows "under r a \<subseteq> under r b"
cong_mult_self_right: 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 a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (times b a) zero a"
lspasl_empl_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "sep_algebra.sep_empty zero h = (h = zero)"
in_float_timesI: fixes b :: "real"   and a :: "real" assumes "b = a * 2"   and "b \<in> float" shows "a \<in> float"
distinct_butlast_ne_last: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set (butlast xs)" shows "x \<noteq> last xs"
assms_2: shows "\<A> (State k (index vs v)) = the (S v)"
Inf1_E: fixes A :: "('a \<Rightarrow> bool) set"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "(\<Sqinter> A) a" obtains "P a"   | "P \<notin> A"
lookup_inject: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and y :: "'b \<Rightarrow>\<^sub>0 'c" shows "(lookup x = lookup y) = (x = y)"
pdo: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and n :: "nat" assumes "matrix_seq dim f" shows "partial_density_operator (f n)"
binrelchain_join: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a"   and y :: "'a"   and ys :: "'a list" assumes "binrelchain P (xs @ [x])"   and "binrelchain P (y # ys)"   and "P x y" shows "binrelchain P (xs @ x # y # ys)"
sign_swap_id: fixes a :: "'d"   and b :: "'d" shows "sign (Fun.swap a b id) = (if a = b then 1 else - 1)"
invertible_right_inverse: fixes A :: "(('b, 'c) vec, 'c) vec" shows "invertible A = (\<exists>B. A ** B = mat (1::'b))"
DirProds_comm_group_imp_comm_groups: fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set"   and i :: "'e" assumes "comm_group (DirProds G I)"   and "i \<in> I" shows "comm_group (G i)"
cfg_on_valid: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> cfg_on_div s" shows "cfg \<in> MDP.valid_cfg"
coeffs_smult: fixes a :: "'a"   and p :: "'a poly" shows "coeffs (smult a p) = (if a = (0::'a) then [] else map ((*) a) (coeffs p))"
VUnion_iff: fixes A :: "V"   and C :: "V" shows "(A \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> C) = (\<exists>x\<in>elts C. A \<in>\<^sub>\<circ> x)"
indoc_Nil: fixes a :: "'item" shows "indoc a [] = False"
that: fixes e :: "real" assumes "0 < e" shows "\<exists>x0. \<forall>x\<ge>x0. dist (f x) l < e"
mono_implies_arr: fixes f :: "'a \<times> 'a" assumes "CC.mono f" shows "CC.arr f"
Not\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Not\<^sub>\<alpha> t\<^sub>\<alpha>)"
index_mult_mat_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and B :: "'a mat" assumes "i < dim_row A"   and "j < dim_col B" shows "(A * B) $$ (i, j) = Matrix.row A i \<bullet> col B j"
init_ri_same_loc: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and M :: "char list"   and a :: "nat" assumes "P \<turnstile> \<langle>e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>"   and "e = INIT C (Cs,b) \<leftarrow> unit \<or> e = C\<bullet>\<^sub>sM([]) \<or> e = RI (C,Throw a) ; Cs \<leftarrow> unit \<or> e = RI (C,C\<bullet>\<^sub>sclinit([])) ; Cs \<leftarrow> unit" shows "l = l'"
X0: shows "x (m + k) \<in> carrier (residue_ring (p ^ (m + k)))"
runiq_paste2: fixes Q :: "('c \<times> 'd) set"   and P :: "('c \<times> 'd) set" assumes "runiq Q"   and "runiq P" shows "runiq (P +< Q)"
pid_gstep_correct: fixes pgc :: "('c, 'ls, 'gs) pid_global_config" shows "gstep_succ (pidgc_\<alpha> pgc) = pidgc_\<alpha> ` pid_gstep_succ pgc"
eq_imp_def: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "eq_imp f g \<equiv> \<forall>s s'. (\<forall>x. f x s = f x s') \<longrightarrow> g s = g s'"
bra_def: fixes v :: "complex mat" shows "\<langle>v| \<equiv> mat 1 (dim_row v) (\<lambda>(i, j). cnj (v $$ (j, i)))"
ladder_i_def: fixes L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" shows "ladder_i L index = (if index = 0 then deriv_i (hd L) else ladder_j L (index - 1))"
harm_expand_2: shows "harm (Suc 0) = (1::'b)"
SUP_empty: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` {}) = bot"
isDERIV_is_analytic: fixes i :: "nat"   and f :: "floatarith"   and xs :: "real list"   and n :: "nat" assumes "isDERIV i f xs" shows "isDERIV i (local.deriv i f n) xs"
assms_2: shows "x \<in> set (map fst op_list)"
prems_2: shows "ax # asx \<noteq> [] \<longrightarrow> n' = last (targetnodes (ax # asx))"
Sabs_3: shows "not_shadows_consts_env \<Gamma>"
L: fixes T :: "'a \<Rightarrow> 'a stream \<Rightarrow> enat"   and N :: "'a measure"   and f :: "'a \<Rightarrow> 'a" assumes "(\<lambda>(x, \<omega>). T x \<omega>) \<in> N \<Otimes>\<^sub>M stream_space M \<rightarrow>\<^sub>M count_space UNIV"   and "f \<in> N \<rightarrow>\<^sub>M M" shows "(\<lambda>x. L (T x) (f x)) \<in> N \<rightarrow>\<^sub>M prob_algebra (stream_space M)"
D_flds: shows "P \<turnstile> D has_fields pre @ FDTs"
rank_None_before: fixes n :: "nat"   and x :: "nat" assumes "n < x" shows "\<UU>.rank x n = None"
statically_complete_calculus_axioms: shows "statically_complete_calculus Bot Inf (\<Turnstile>) Red_I Red_F"
rel_spmf_mono: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d spmf"   and g :: "'e spmf"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_spmf B f g"
path_reversepath: fixes g :: "real \<Rightarrow> 'c" shows "path (reversepath g) = path g"
intro: fixes I :: "'a \<Rightarrow> bool"   and subsumed :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and qle :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and key :: "'a \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>I x; I y\<rbrakk> \<Longrightarrow> subsumed x y = qle x y"   and "key = key" shows "set_mod_maps_axioms qle subsumed I key"
cr_envT_stateT_run: fixes s :: "'d"   and M :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "(cr_envT_stateT s M ===> eq_onp ((=) s) ===> M) run_env run_state"
mem_box_2: fixes x :: "'e"   and a :: "'e"   and b :: "'e" shows "(x \<in> cbox a b) = (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
simps: fixes N :: "('a literal multiset \<times> nat) multiset"   and P :: "('a literal multiset \<times> nat) multiset"   and Q :: "('a literal multiset \<times> nat) multiset"   and n :: "nat" shows "wQ_of_wstate (N, P, Q, n) = Q"
existence_ivl_trans: fixes s :: "real"   and t0 :: "real"   and x0 :: "'a"   and t :: "real" assumes "s - t0 \<in> existence_ivl0 x0"   and "t \<in> existence_ivl0 (flow0 x0 (s - t0))" shows "s + t - t0 \<in> existence_ivl0 x0"
nextElem_in: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" shows "nextElem xs x y \<in> set (x # xs)"
log_ceiling_def: fixes b :: "int"   and x :: "int" shows "log_ceiling b x = (case log_main (into_base b) x of (y, by) \<Rightarrow> if x = by then y else Suc y)"
ctr: fixes t :: "('sequent \<times> 'rule) tree"   and a :: "'a" shows "konigDtree t a = (tree.root t, a) ## (let s = fst (tree.root t); r = snd (tree.root t); (s', a') = SOME (s', a'). s' |\<in>| (fst \<circ> tree.root) |`| cont t \<and> pred (s, a) r (s', a') \<and> inv (s', a'); t' = SOME t'. t' |\<in>| cont t \<and> s' = fst (tree.root t') in konigDtree t' a')"
not_interval_UNIV_1: fixes a :: "'b"   and b :: "'b" shows "cbox a b \<noteq> UNIV"
rel_mset_Zero: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_mset R {#} {#}"
simps_3: fixes c\<^sub>1 :: "cmd"   and c\<^sub>2 :: "cmd" shows "#:c\<^sub>1;; c\<^sub>2 = #:c\<^sub>1 + #:c\<^sub>2"
snd_convol': fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and x :: "'d" shows "snd (BNF_Def.convol f g x) = g x"
mult_isol_equiv_subdistl: shows "(\<forall>x y z. x \<le> y \<longrightarrow> z ; x \<le> z ; y) = (\<forall>x y z. z ; x \<le> z ; (x + y))"
b: shows "Diag b \<and> b \<noteq> \<^bold>\<I>"
parallel_parametric: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and D :: "'g \<Rightarrow> 'i \<Rightarrow> bool" shows "(rel_dds A B ===> rel_dds C D ===> rel_dds (rel_sum A C) (rel_sum B D)) (\<parallel>) (\<parallel>)"
Afix_cong: fixes heap2 :: "(Vars.var \<times> exp) list"   and aexp1 :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and aexp2 :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and heap1 :: "(Vars.var \<times> exp) list" assumes "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> aexp1 e = aexp2 e"   and "heap1 = heap2" shows "ArityAnalysis.Afix aexp1 heap1 = ArityAnalysis.Afix aexp2 heap2"
pnet_reachable_transfer': fixes n :: "net_tree"   and s :: "'s net_state"   and U :: "(nat \<Rightarrow> 'g) \<Rightarrow> (nat \<Rightarrow> 'g) \<Rightarrow> bool" assumes "wf_net_tree n"   and "s \<in> reachable (closed (pnet np n)) TT" shows "netgmap sr s \<in> netmask (net_tree_ips n) ` oreachable (oclosed (opnet onp n)) (\<lambda>_ _ _. True) U"
not_int_cmp_0_3: fixes i :: "int" shows "(NOT i < 0) = (0 \<le> i)"
SLC_ifFalse: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e1 :: "('r, 'l, 'v) expr"   and e2 :: "('r, 'l, 'v) expr"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV F)) e1 e2])"   and "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [e2]))"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
LLL_bound_invariant_def: fixes outside :: "bool"   and upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "LLL_bound_invariant outside upw i fs = (LLL_invariant upw i fs \<and> f_bound outside i fs \<and> g_bound fs)"
finestPartUnion: fixes X :: "'b set" shows "X = \<Union> (finestpart X)"
chop_elem_not_Nil: fixes ys :: "'a list"   and n :: "nat"   and xs :: "'a list" assumes "ys \<in> set (chop n xs)" shows "ys \<noteq> []"
parts_crypt_num: fixes s :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "Crypt (SesK SK) (Num 0) \<in> parts (used s)" shows "(\<exists>n. (Asset n, Crypt (SesK SK) (Num 0)) \<in> s) \<or> SesKey SK \<in> spied s"
distinct_289: fixes x14 :: "Syntax.term"   and x16 :: "Syntax.term" shows "Unroll x14 \<noteq> Unauth x16"
f_def: shows "f \<equiv> \<lambda>(l, u). (l, [u]\<^sub>\<R>)"
uint_add_ge0: fixes xa :: "'a word"   and x :: "'b word" shows "0 \<le> uint xa + uint x"