hcju/isabelleps huggingface dataset:
proj5_addpreRT: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and npre :: "nat set"   and ip :: "nat" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>5 (the (the (addpreRT rt dip npre) ip)) = \<pi>\<^sub>5 (the (rt ip))"
ran_map_of_list: fixes xs :: "'c list" shows "ran (map_of_list xs) = set xs"
spmf_rel_conversep: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_spmf R\<inverse>\<inverse> = (rel_spmf R)\<inverse>\<inverse>"
B_def: shows "B = (\<lambda>n. pmf (binomial_pmf n q))"
inject_2: fixes x3 :: "nat"   and y3 :: "nat" shows "(User x3 = User y3) = (x3 = y3)"
PLS_works: fixes x :: "nat"   and s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" assumes "x \<in> PLS s as" shows "\<exists>as'. exec_plan s as = exec_plan s as' \<and> length as' = x \<and> subseq as' as"
equiv'_D: fixes a :: "'a"   and b :: "'b" assumes "A_B.equiv' a b" shows "a \<sim> b"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "Left.seq g f" shows "Left.comp g f = g \<cdot> f"
integral_nonneg_AE: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "AE x in M. 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
size_image_mset: fixes f :: "'l \<Rightarrow> 'k"   and M :: "'l multiset" shows "size (image_mset f M) = size M"
app_size: fixes u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" shows "size (app u\<^sub>1 u\<^sub>2) = size u\<^sub>1 + size u\<^sub>2 + 1"
ring_irreducibleE_4: fixes r :: "'a list" assumes "r \<in> carrier (K [X])"   and "pirreducible K r" shows "r \<notin> Units (K [X])"
COMB_pairwise: shows "List_Factoring.pairwise (COMB [])"
simps_1: fixes i :: "nat"   and n :: "nat"   and u :: "Core.term" shows "subst_bv2 (Bv i) n u = (if i < n then Bv i else if i = n then u else Bv (i - 1))"
prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "g (p ^ k) = (if k = 2 then - 1 else 0)"
FL_Inf_def: shows "FL_Inf = \<Union> (FL_Infer_of ` F_Inf)"
fps_cutoff_code: fixes n :: "nat"   and x :: "'b ratfps" shows "fps_cutoff n (fps_of_ratfps x) = fps_of_poly (ratfps_cutoff n x)"
le_multiset_empty_right: fixes M :: "'a multiset" shows "\<not> M < {#}"
monoid_axioms: shows "monoid H (\<cdot>) \<one>"
simps: fixes B :: "ty"   and C :: "ty"   and X :: "ntrm set" shows "ANDLEFT1 (B AND C) X = {y:AndL1 x.M y |x y M. fin (AndL1 x.M y) y \<and> x:M \<in> X}"
strict_implies_order: fixes b :: "'c"   and a :: "'c" assumes "b < a" shows "b \<le> a"
ht_at_phl_export1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "\<lbrace>\<alpha> x \<cdot> \<tau> y\<rbrace>z\<lbrace>w\<rbrace>" shows "\<lbrace>\<alpha> x\<rbrace>\<tau> y \<cdot> z\<lbrace>w\<rbrace>"
bind_alt1: fixes m :: "'m"   and m' :: "'m"   and f :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (alt m m') f = alt (bind m f) (bind m' f)"
sinv: shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) Q"
1: shows "e \<cdot> induced_arrow (C.dom e') (D.mkCone e') = e'"
assms: shows "(RETURN x, RETURN y) \<in> \<langle>A\<rangle>nres_rel"
RN_deriv_unique_sigma_finite: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and N :: "'d measure" assumes "f \<in> borel_measurable M"   and "density M f = N"   and "sigma_finite_measure N" shows "AE x in M. f x = RN_deriv M N x"
exhaust: fixes y :: "natural" obtains "y = 0"   | natural :: "natural" where "y = Code_Numeral.Suc natural"
mk_rtrancl_sound: fixes init :: "'b" shows "mk_rtrancl init \<subseteq> R\<^sup>* `` set_of init"
r_univ_recfn: shows "recfn 2 r_univ"
l11_22a: fixes B :: "'p"   and P :: "'p"   and A :: "'p"   and C :: "'p"   and B' :: "'p"   and P' :: "'p"   and A' :: "'p"   and C' :: "'p" assumes "B P TS A C"   and "B' P' TS A' C'"   and "A B P CongA A' B' P'"   and "P B C CongA P' B' C'" shows "A B C CongA A' B' C'"
iso3: fixes x :: "'a" shows "d (d x \<cdot> U) = d x"
d_adj: shows "n = max_degree (adjugate (\<^bold>X - \<^bold>C A))"
real_euclid_th5_1: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec"   and d :: "(real, 'a) vec" assumes "a \<noteq> b"   and "B\<^sub>\<real> a b c"   and "B\<^sub>\<real> a b d" shows "B\<^sub>\<real> a c d \<or> B\<^sub>\<real> a d c"
preserves_comp: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>C.seq g f" shows "\<F>C.D (g \<cdot> f) = \<F>C.D g \<cdot>\<^sub>S \<F>C.D f"
inject: fixes x1 :: "'b literal set"   and x2 :: "'b trm set"   and y1 :: "'b literal set"   and y2 :: "'b trm set" shows "(Ecl x1 x2 = Ecl y1 y2) = (x1 = y1 \<and> x2 = y2)"
id1: shows "pderiv f = g * pderiv k + k * pderiv g"
cross_components_3: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "cross3 x y $ 3 = x $ 1 * y $ 2 - y $ 1 * x $ 2"
val_of_nonzero_2: fixes f :: "nat \<Rightarrow> int"   and p :: "int" assumes "f \<in> padic_set p"   and "f \<noteq> padic_zero p"   and "prime p" shows "f (nat (padic_val p f)) = \<zero>\<^bsub>residue_ring (p ^ nat (padic_val p f))\<^esub>"
ind: shows "{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci (take n gs)) ft {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
splitting_lemma_right_gen: fixes C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and g' :: "'b \<Rightarrow> 'b" assumes "short_exact_sequence C B A g f"   and "g' \<in> hom C B"   and "g \<circ> g' \<in> Group.iso C C" obtains H :: "'b set" and K :: "'b set" where "H \<lhd> B" and "subgroup K B" and "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}" and "H <#>\<^bsub>B\<^esub> K = carrier B" and "f \<in> Group.iso A (subgroup_generated B H)" and "g \<in> Group.iso (subgroup_generated B K) C"
cobounded2: fixes b :: "'k multiset"   and a :: "'k multiset" shows "b \<subseteq># a \<union># b"
ereal_minus_less_minus: fixes a :: "ereal"   and b :: "ereal" shows "(- a < - b) = (b < a)"
braking_movement_axioms: shows "braking_movement a\<^sub>o v\<^sub>o"
Df_def: fixes a :: "'a"   and i :: "nat"   and h :: "'a" shows "Df a i h = (if i = 0 then f a else f' a (i - 1) h)"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "butlast (x # xs) = (if xs = \<epsilon> then \<epsilon> else x # butlast xs)"
IH2_1: assumes "M \<Turnstile> G" shows "M(p := True) \<Turnstile> G"
omax_maxim: fixes R :: "('c \<times> 'c) set set"   and r :: "('c \<times> 'c) set" assumes "finite R"   and "\<forall>r\<in>R. Well_order r"   and "r \<in> R" shows "r \<le>o omax R"
fix_add: fixes ap :: "(action \<times> nat) list"   and x :: "nat"   and n :: "nat"   and b :: "cell" shows "fetch ap (x + 2 * n) b = fetch ap (2 * n + x) b"
enabled_def: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and r :: "'d"   and s :: "'e" shows "RuleSystem_Defs.enabled eff r s \<equiv> \<exists>sl. eff r s sl"
state_measure_integral_split: fixes x :: "nat"   and A :: "nat set"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "x \<in> A"   and "finite A"   and "f \<in> borel_measurable (state_measure A \<Gamma>)" shows "integral\<^sup>N (state_measure A \<Gamma>) f = \<integral>\<^sup>+ y. \<integral>\<^sup>+ \<sigma>. f (\<sigma>(x := y)) \<partial>state_measure (A - {x}) \<Gamma> \<partial>stock_measure (\<Gamma> x)"
Cong_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D \<or> Cong A B D C \<or> Cong B A C D \<or> Cong B A D C \<or> Cong C D A B \<or> Cong C D B A \<or> Cong D C A B \<or> Cong D C B A" shows "Cong A B C D"
lfinite_induct: fixes xs :: "'b llist"   and P :: "'b llist \<Rightarrow> bool" assumes "lfinite xs"   and "\<And>xs. lnull xs \<Longrightarrow> P xs"   and "\<And>xs. \<lbrakk>lfinite xs; \<not> lnull xs; P (ltl xs)\<rbrakk> \<Longrightarrow> P xs" shows "P xs"
Posix1_2: fixes s :: "'b list"   and r :: "'b rexp"   and v :: "'b val" assumes "s \<in> r \<rightarrow> v" shows "flat v = s"
RedSeq: fixes E :: "char list \<Rightarrow> ty option"   and v :: "val"   and e\<^sub>2 :: "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" shows "P,E \<turnstile> \<langle>Val v;; e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e\<^sub>2,s\<rangle>"
codomains_comp: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "B.codomains (g \<cdot>\<^sub>B f) = B.codomains g"
SD_pref_profileD_2: fixes i :: "'agent"   and q :: "'alt pmf"   and p :: "'alt pmf" assumes "i \<in> agents"   and "SD (R i) q p" shows "q \<in> lotteries"
nth_last_index: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" shows "xs ! last_index xs x = x"
assms_1: shows "boa \<le> vimage2p ha ha (=)"
bounded_linear_imp_differentiable_on: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "bounded_linear f" shows "f differentiable_on S"
assms_1: shows "ins n (d + 1) x l = Unbal l'"
ArrCod_def: shows "ArrCod = 2\<^sub>\<nat>"
nat_of_bool_mod_eq: fixes n :: "nat" shows "of_bool (n mod 2 = 1) = n mod 2"
Horn_def: fixes C :: "'at Literal set" shows "Horn C = (card (positive_part C) \<le> 1)"
a1: shows "local.adverse 1 i j (Suc xmin) \<down>= 1"
append_T_imp_tickFree: fixes t :: "'b event list"   and s :: "'b event list"   and P :: "'b process" assumes "t @ s \<in> T P"   and "s \<noteq> []" shows "tickFree t"
mod_smult_right: fixes a :: "'b"   and x :: "'b poly"   and y :: "'b poly" assumes "a \<noteq> (0::'b)" shows "x mod smult a y = x mod y"
simps: fixes prec :: "nat"   and f :: "floatarith"   and I :: "float interval list" shows "compute_bound_fa prec f I = approx prec f (map Some I)"
intros_3: fixes c :: "nat"   and u :: "'d \<Rightarrow> real"   and x :: "'d" assumes "real c < u x" shows "intv_elem x u (Greater c)"
smult_0_left: fixes p :: "'d poly" shows "smult (0::'d) p = 0"
dual_chainI: fixes C :: "'g set" assumes "is_chain C" shows "is_dual_chain C"
correct: fixes m1 :: "bool \<times> bool"   and m2 :: "bool" shows "etp.OT_12.correctness m1 m2"
right_ideal_minus: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "right_ideal I"   and "a \<in> I"   and "b \<in> I" shows "a - b \<in> I"
step_invariant_weakenE: fixes A :: "('e, 'f) automaton"   and PI :: "'f \<Rightarrow> bool"   and P :: "'e \<times> 'f \<times> 'e \<Rightarrow> bool"   and Q :: "'e \<times> 'f \<times> 'e \<Rightarrow> bool"   and QI :: "'f \<Rightarrow> bool" assumes "A \<TTurnstile>\<^sub>A (PI \<rightarrow>) P"   and "\<And>t. P t \<Longrightarrow> Q t"   and "\<And>a. QI a \<Longrightarrow> PI a" shows "A \<TTurnstile>\<^sub>A (QI \<rightarrow>) Q"
divide_p32_def: fixes p :: "uint32"   and x :: "uint32"   and y :: "uint32" shows "divide_p32 p x y = mult_p32 p x (inverse_p32 p y)"
\<Sigma>_protocol: fixes \<eta> :: "nat" shows "chaum_ped_sigma.\<Sigma>_protocol \<eta>"
CollectD: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "a \<in> {x. P x}" shows "P a"
le_less_trans: fixes x :: "nat list"   and y :: "nat list"   and z :: "nat list" assumes "x \<le>\<^sub>v y"   and "y <\<^sub>v z" shows "x <\<^sub>v z"
y: shows "\<epsilon>(r' \<mapsto> (\<epsilon>, \<epsilon>, e)) \<leadsto>\<^sup>* s'"
fail_state_def: shows "local.fail_state = StateT (\<lambda>s. fail)"
zero_bcontfun_def: shows "0 \<equiv> Bcontfun (\<lambda>_. 0::'d)"
d_delta_lnexp_cf3_nonneg: fixes x :: "real" assumes "0 < numer_cf3 x"   and "0 < numer_cf3 (- x)" shows "0 \<le> diff_delta_lnexp_cf3 x"
disc_12: fixes x2 :: "'a" shows "is_Prim \<^bold>\<langle>x2\<^bold>\<rangle>"
is_empty_def: fixes m :: "('a, 'b) mapping" shows "Mapping.is_empty m = (Mapping.keys m = {})"
hyps_3: assumes "v \<in> set l"   and "takeWhile ((\<noteq>) v) (rev l) = c" shows "path_entry E (rev c) v"
flatten_card_sf_induct: fixes X :: "'b ltln set"   and N :: "'b ltln set"   and M :: "'b ltln set" assumes "finite X"   and "\<And>x. x \<in> X \<Longrightarrow> sf x \<subseteq> X" shows "card (\<Union>\<phi>\<in>X. sf (\<phi>[N]\<^sub>\<Sigma>\<^sub>1)) \<le> card X \<and> card (\<Union>\<phi>\<in>X. sf (\<phi>[M]\<^sub>\<Pi>\<^sub>1)) \<le> card X \<and> card (\<Union>\<phi>\<in>X. sf \<phi>[M]\<^sub>\<Sigma>\<^sub>2 \<union> sf (\<phi>[M]\<^sub>\<Pi>\<^sub>1)) \<le> 3 * card X"
induct: fixes P :: "'b tree \<Rightarrow> 'b \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b tree"   and a1 :: "'b"   and a2 :: "'b tree" assumes "\<And>a uu_. P \<langle>\<rangle> a uu_"   and "\<And>x uv_ a. P \<langle>\<langle>\<rangle>, x, uv_\<rangle> a \<langle>\<rangle>"   and "\<And>l1 x1 r1 a l2 x2 r2. \<lbrakk>\<lbrakk>\<not> (a \<le> x1 \<and> a \<le> x2); x1 \<le> x2\<rbrakk> \<Longrightarrow> P l1 a r1; \<lbrakk>\<not> (a \<le> x1 \<and> a \<le> x2); \<not> x1 \<le> x2\<rbrakk> \<Longrightarrow> P l2 a r2\<rbrakk> \<Longrightarrow> P \<langle>l1, x1, r1\<rangle> a \<langle>l2, x2, r2\<rangle>"   and "\<And>vc vd ve va vb b. P \<langle>\<langle>vc, vd, ve\<rangle>, va, vb\<rangle> b \<langle>\<rangle>" shows "P a0 a1 a2"
catch_return: fixes x :: "'a \<times> 's"   and m :: "'m" shows "catch (return x) m = return x"
poly_pCons: fixes a :: "'c"   and p :: "'c poly"   and x :: "'c" shows "poly (pCons a p) x = a + x * poly p x"
mono_sup: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a"   and B :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "order.mono less_eq f" shows "f A \<squnion> f B \<le> f (sup A B)"
less_imp_not_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "less x y" shows "(x = y) = False"
ltln_expand_Release: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = \<xi> \<Turnstile>\<^sub>n \<psi> and\<^sub>n \<phi> or\<^sub>n X\<^sub>n (\<phi> R\<^sub>n \<psi>)"
scan_const: fixes xs :: "'c list"   and a :: "'c" shows "scan (\<lambda>x _. x) xs a = xs"
f_A_def: fixes n :: "nat" shows "f_A n = int (min (mtf_A ! n) (index (swaps (sw_A ! n) (s_A n)) (rs ! n)))"
individually_rational_on_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.individually_rational_on Pd Ch ds X = (Contracts.CD_on Pd ds X = X \<and> Contracts.CH Ch X = X)"
f_def: shows "f \<equiv> \<lambda>X. M \<union> (\<Union>x\<in>X. \<Union>(a, b)\<in>TI. set \<langle>a --\<guillemotright> b\<rangle>\<langle>x\<rangle>)"
matrix_to_iarray_Gauss_Jordan_in_ij: fixes i :: "'b"   and j :: "'c"   and A :: "(('a, 'c) vec, 'b) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))" shows "matrix_to_iarray (Gauss_Jordan_in_ij A i j) = Gauss_Jordan_in_ij_iarrays (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j)"
iUntil_True_left: fixes I :: "nat set"   and Q :: "nat \<Rightarrow> bool" shows "(True. t' \<U> t I. Q t) = (\<diamond> t I. Q t)"
iso_char: fixes f :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" shows "S.iso f = (S.arr f \<and> bij_betw (S.Fun f) (S.Dom f) (S.Cod f))"
units_of_pow: fixes x :: "'a"   and n :: "nat" assumes "x \<in> Units R" shows "x [^]\<^bsub>units_of R\<^esub> n = x [^] n"
eval_def: fixes return :: "int \<Rightarrow> 'f"   and bind :: "'f \<Rightarrow> (int \<Rightarrow> 'f) \<Rightarrow> 'f"   and fail :: "'f"   and E :: "'e \<Rightarrow> 'f" shows "exp_base.eval return bind fail E \<equiv> rec_exp E return (\<lambda>e1 e2 e1a e2a. bind e1a (\<lambda>i. bind e2a (\<lambda>j. return (i + j)))) (\<lambda>e1 e2 e1a e2a. bind e1a (\<lambda>i. bind e2a (\<lambda>j. if j = 0 then fail else return (i div j))))"
bijec_def: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g" shows "bijec\<^bsub>F,G\<^esub> f = (injec\<^bsub>F,G\<^esub> f \<and> surjec\<^bsub>F,G\<^esub> f)"
1: fixes t :: "real" assumes "t \<noteq> 0" shows "cmod ((iexp (- (t * a)) - iexp (- (t * b))) / (\<i> * complex_of_real t) - complex_of_real (b - a)) \<le> a\<^sup>2 / 2 * \<bar>t\<bar> + b\<^sup>2 / 2 * \<bar>t\<bar>"
set_times_def: fixes A :: "'b set"   and B :: "'b set" shows "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
generate_pow_card: fixes a :: "'a" assumes "a \<in> carrier (mult_of R)" shows "G.ord a = card (generate (mult_of R) {a})"
hom_0: fixes x :: "'a poly" assumes "poly_x_mult_y x = 0" shows "x = 0"
ounit_circle_def: shows "ounit_circle \<equiv> abs_ocircline unit_circle_clmat"
image_add_0: fixes S :: "'b set" shows "(+) (0::'b) ` S = S"
vint_struct_def: shows "\<SS>\<^sub>\<int> = [\<int>\<^sub>\<circ>, vint_plus, vint_mult]\<^sub>\<circ>"
upper_arctan_float_interval: fixes p :: "nat"   and x :: "float interval" shows "upper (arctan_float_interval p x) = ub_arctan p (upper x)"
rep_eq: fixes x :: "('a, 'b) lifted_boolean_algebra"   and xa :: "('a, 'b) lifted_boolean_algebra" shows "Rep_lifted_boolean_algebra (x \<sqinter> xa) = (\<lambda>f. Rep_lifted_boolean_algebra x f \<sqinter> Rep_lifted_boolean_algebra xa f)"
Compl_set_less_eq_aux_fusion_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and proper_interval :: "'c option \<Rightarrow> 'c option \<Rightarrow> bool"   and g1 :: "('c, 'd) generator"   and g2 :: "('c, 'e) generator"   and ao :: "'c option"   and s1 :: "'d"   and s2 :: "'e" shows "proper_intrvl.Compl_set_less_eq_aux_fusion less proper_interval g1 g2 ao s1 s2 = proper_intrvl.Compl_set_less_eq_aux less proper_interval ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
encRRel: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel"
poly_mult_one'_1: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "poly_mult [\<one>] p = local.normalize p"
negative_mono: fixes m :: "'b SecurityInvariant_configured"   and N :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "\<not> c_sinvar m \<lparr>nodes = N, edges = E'\<rparr>" shows "\<not> c_sinvar m \<lparr>nodes = N, edges = E\<rparr>"
valid_wgraph_repr_def: fixes l :: "(('a \<times> 'a) \<times> 'b) list" shows "valid_wgraph_repr l = (valid_graph_repr (map fst l) \<and> valid_weight_repr l)"
BIT_x: fixes x :: "'a"   and y :: "'a"   and init :: "'a list"   and qs :: "'a list"   and r :: "'a list" assumes "x \<noteq> y"   and "init \<in> {[x, y], [y, x]}"   and "qs \<in> lang (question (Atom x))" shows "T\<^sub>p_on_rand' BIT (type0 init x y) (qs @ r) = T\<^sub>p_on_rand' BIT (type0 init x y) r \<and> Partial_Cost_Model.config'_rand BIT (type0 init x y) qs = type0 init x y"
ordinal_number_eq: fixes D :: "'c Order"   and E :: "'c Order" assumes "Worder D"   and "Worder E" shows "ord_equiv D E = (ordinal_number D = ordinal_number E)"
assms_1: shows "m1 \<in> carrier_mat d1 d1"
lprefix_down_linear: fixes xs :: "'a llist"   and zs :: "'a llist"   and ys :: "'a llist" assumes "lprefix xs zs"   and "lprefix ys zs" shows "lprefix xs ys \<or> lprefix ys xs"
leq_antisym: fixes x :: "'c nres"   and y :: "'c nres" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
star_of_1_le: fixes y :: "'a" shows "(1 \<le> star_of y) = ((1::'a) \<le> y)"
tangent_space_eq: fixes p :: "'a" shows "tangent_space p = {X. is_derivation X p} \<inter> {X. extensional0 diff_fun_space X}"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and err :: "'b err" shows "P (case err of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = ((err = Err \<longrightarrow> P f1) \<and> (\<forall>x2. err = OK x2 \<longrightarrow> P (f2 x2)))"
less_supI2: fixes x :: "'a multiset"   and b :: "'a multiset"   and a :: "'a multiset" assumes "x \<subset># b" shows "x \<subset># a \<union># b"
Sem_def: fixes s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c :: "OBJ"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" shows "s , c \<Down> t  = (\<exists>n.  s , c \<rightarrow>\<^sub>n  t )"
discr_transC_indis: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "discr c"   and "(c, s) \<rightarrow>c (c', s')" shows "s \<approx> s'"
in_roofed_mono': fixes x :: "'a"   and G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and S' :: "'a set" assumes "x \<in> roofed_gen G B S"   and "S \<subseteq> roofed_gen G B S'" shows "x \<in> roofed_gen G B S'"
strategyproofness: shows "pmf (sds R5) c + pmf (sds R5) a \<le> pmf (sds R17) c + pmf (sds R17) a"
evContainsConj: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and A0 :: "formula"   and A1 :: "formula"   and gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "EV (contains f (i, FConj Pos A0 A1))"   and "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))" shows "EV (contains f (0, A0)) \<or> EV (contains f (0, A1))"
positive_x_axis_def: shows "positive_x_axis = {z \<in> circline_set x_axis. z \<noteq> \<infinity>\<^sub>h \<and> 0 < Re (to_complex z)}"
gc_hs_get_roots_get_workD: fixes s :: "(('f, 'g, 'h) response, char list, 'e process_name, 'e process_name \<times> ('f, 'e, 'g, 'h) request_op, ('f, 'e, 'g, 'h) Model.local_state) system_state" assumes "atS gc get_roots_UN_get_work_locs s"   and "handshake_invL s" shows "sys_ghost_hs_phase s\<down> = hp_IdleMarkSweep \<and> sys_hs_type s\<down> \<in> {ht_GetWork, ht_GetRoots}"
cos_monotone_minus_pi_0: fixes y :: "real"   and x :: "real" assumes "- pi \<le> y"   and "y < x"   and "x \<le> 0" shows "cos y < cos x"
f: shows "arr (mkArr (Cod g) (Dom g) (inv_into (Dom g) (Fun g)))"
simps: fixes x :: "'b ext"   and y :: "'b ext" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
vdw_imp_vdw_lemma: fixes l :: "nat"   and m :: "nat"   and k :: "nat" assumes "\<And>k'. 0 < k' \<Longrightarrow> \<exists>n_k'. vdw k' l n_k'"   and "2 \<le> l"   and "0 < m"   and "0 < k" shows "\<exists>N. vdw_lemma k m l N"
lex_prodp_def: fixes r1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "lex_prodp r1 r2 x y \<equiv> r1 (fst x) (fst y) \<or> fst x = fst y \<and> r2 (snd x) (snd y)"
lt_map_scale: fixes c :: "'b"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "c \<noteq> (0::'b)" shows "lpp (c \<cdot> p) = lpp p"
nat_pow_eone: fixes x :: "'a mod_ring poly" assumes "x \<in> carrier R" shows "x [^]\<^bsub>R\<^esub> 1 = x"
assms_1: shows "\<lfloor>U1, s1\<rfloor> = \<lfloor>U2, s2\<rfloor>"
apply_bcontfun_cases: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> bcontfun" obtains x :: "'c \<Rightarrow>\<^sub>C 'd" where "y = apply_bcontfun x"
catch_Norm: fixes G :: "prog"   and s :: "st"   and tn :: "qtname" shows "\<not> G,Norm s\<turnstile>catch tn"
cUNION_cinsert: fixes f :: "'d \<Rightarrow> 'c cset"   and x :: "'d"   and A :: "'d cset" shows "cUnion (cimage f (cinsert x A)) = cUn (f x) (cUnion (cimage f A))"
next_Plus1: fixes fsm1 :: "('b, 'c) fsm_scheme"   and fsm2 :: "('b, 'd) fsm_scheme"   and st1 :: "hf"   and x :: "'b"   and st' :: "hf" shows "next (Plus fsm1 fsm2) (HF.Inl st1) x st' = (\<exists>st1'. st' = HF.Inl st1' \<and> next fsm1 st1 x st1')"
assms: shows "abc \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Obj\<rparr>"