hcju/isabelleps huggingface dataset:
conj_assoc: fixes P :: "'d upred"   and Q :: "'d upred"   and S :: "'d upred" shows "((P \<and> Q) \<and> S) = (P \<and> Q \<and> S)"
gfim: shows "g ` f ` S \<subseteq> S"
in_prime_factors_imp_dvd: fixes p :: "'a"   and x :: "'a" assumes "p \<in> prime_factors x" shows "p dvd x"
pmf_embed_pmf: fixes f :: "'c \<Rightarrow> real"   and x :: "'c" assumes "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1" shows "pmf (embed_pmf f) x = f x"
compact_def: fixes S :: "'b set" shows "compact S = (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
ileq_interleave: fixes w :: "'a list"   and w1 :: "'a list"   and w2 :: "'a list" assumes "w \<in> w1 \<otimes> w2" shows "w1 \<preceq> w \<and> w2 \<preceq> w"
wave_A_in_RF: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "wave \<Gamma> f"   and "x \<in> A \<Gamma>" shows "x \<in> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"
conf_cong: fixes G :: "prog"   and l :: "lname \<Rightarrow> val option"   and s :: "st"   and v :: "val"   and T :: "ty" shows "G,set_locals l s\<turnstile>v\<Colon>\<preceq>T = G,s\<turnstile>v\<Colon>\<preceq>T"
EndPhase2_HInv3_R: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase2 s s' t"   and "HInv3_R s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_R s' p q d"
d_def: fixes n :: "nat"   and fs :: "int vec list"   and k :: "nat" shows "fs_int.d n fs k = gram_schmidt.Gramian_determinant n fs k"
simps_1: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cLengthF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cLengthF"
symp_conversep: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "symp R = (R\<inverse>\<inverse> \<le> R)"
iptables_goto_bigstepD_10: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and chain :: "char list" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m (Goto chain)]"   and "s = Undecided"   and "matches \<gamma> m p" shows "\<exists>rs. \<Gamma> chain = Some rs \<and> \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"
exhaust: fixes y :: "'e trm" obtains x1 :: "Preliminaries.var" where "y = Var x1"   | x21 :: "'e" and x22 :: "'e trm list" where "y = Fn x21 x22"
simps_15: fixes sh :: "sheap"   and b :: "'a exp"   and e\<^sub>1 :: "'a exp"   and e\<^sub>2 :: "'a exp" shows "iconf sh (if (b) e\<^sub>1 else e\<^sub>2) = (iconf sh b \<and> \<not> sub_RI e\<^sub>1 \<and> \<not> sub_RI e\<^sub>2)"
cod_def: fixes f :: "Arr" shows "cod f = (if codomains f \<noteq> {} then SOME b. b \<in> codomains f else null)"
d_fixpoint: fixes x :: "'a" shows "(\<exists>y. x = ar ar y) = (x = ar ar x)"
entails_bot: shows "N \<Turnstile>\<inter>\<G> {B}"
prime_factorization_radical: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "prime_factorization (radical x) = mset_set (prime_factors x)"
pair_pmf_def: fixes A :: "'c pmf"   and B :: "'d pmf" shows "pair_pmf A B = A \<bind> (\<lambda>x. B \<bind> (\<lambda>y. return_pmf (x, y)))"
that: fixes ws :: "nat list" assumes "0 < length ws"   and "last ws \<noteq> 0"   and "f = ws \<odot> 0\<^sup>\<infinity>" shows "thesis"
sel_1: fixes x1 :: "'c" shows "projl (Inl x1) = x1"
nsq: shows "\<not> ps (S t j) q \<noteq> None"
zs_def: shows "zs = takeWhile ((=) (0::'a)) (coeffs f)"
simps_197: fixes x111 :: "char list"   and x112 :: "ty"   and x113 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "{x111:x112; x113} \<noteq> Call x101 x102 x103 x104"
order_iff: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (a = inf a b)"
map_pure_tree: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "map_tree f (pure x) = pure (f x)"
proj_k_le_one: fixes k :: "nat" shows "proj_k k \<le>\<^sub>L 1\<^sub>m K"
special_soundness: shows "local.\<Sigma>_protocols_base.special_soundness"
30: shows "minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T"
assms_1: shows "a_consistent (ae, a, as) (\<Gamma>, Terms.Let \<Delta> e, S)"
callee_invariant_on_run_resource: fixes \<I> :: "('a, 'b) \<I>" shows "callee_invariant_on run_resource (WT_resource \<I>) \<I>"
fdomain_iff: fixes a :: "V"   and r :: "V" shows "(a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r) = (\<exists>y. [a, y]\<^sub>\<circ> \<in>\<^sub>\<circ> r)"
flj: fixes k :: "nat" assumes "k < j" shows "f' k = f k"
map_poly_0: fixes f :: "'c \<Rightarrow> 'b" shows "map_poly f 0 = 0"
sigma_finite_measure_count_space: fixes A :: "'b set" shows "sigma_finite_measure (count_space A)"
ordered_set_of: fixes t :: "dag"   and var :: "ref \<Rightarrow> nat"   and x :: "ref" assumes "ordered t var"   and "x \<in> set_of t" shows "var x \<le> var (root t)"
lastxx_is_4cases: fixes x :: "nat"   and y :: "nat" shows "lang (L_4cases x y) = lang (L_lasthasxx x y)"
iso_lunit: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "category.iso C (monoidal_category.lunit C T \<alpha> \<iota> a)"
zero_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
tndFalse: fixes A :: "('b set set \<times> 'b set set) set"   and B :: "('b set set \<times> 'b set set) set"   and D :: "'b"   and F :: "'b" assumes "\<forall>a b c d. (a, b) \<in> A \<and> (c, d) \<in> B \<longrightarrow> twoNetsDistinct a b c d"   and "\<exists>(a, b)\<in>A. a \<in> subnetsOfAdr D \<and> b \<in> subnetsOfAdr F"   and "\<exists>(a, b)\<in>B. a \<in> subnetsOfAdr D \<and> b \<in> subnetsOfAdr F" shows "False"
oracle_decrypt_def: fixes decrypt :: "'c \<Rightarrow> 'b \<Rightarrow> 'a option"   and k :: "'c"   and L :: "'b set"   and c :: "'b" shows "ind_cca.oracle_decrypt decrypt k L c = return_spmf (if c \<in> L then None else decrypt k c, L)"
f_2: fixes x :: "complex" assumes "x \<in> A" shows "f x \<noteq> 0"
total_var: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.total x = matrix_stone_relation_algebra.total_var x"
max_step: fixes j :: "nat" assumes "t j \<le> t1" shows "max_stepsize j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (t1 - t 0) + 1) - 1))"
arg_inv_not_pi: fixes z :: "complex" assumes "z \<noteq> 0"   and "arg z \<noteq> pi" shows "arg (1 / z) = - arg z"
list_all2_append2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and zs :: "'c list" shows "list_all2 P xs (ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> list_all2 P us ys \<and> list_all2 P vs zs)"
weight_w0_unary_1: fixes t :: "('f, 'b) Term.term"   and f :: "'f"   and ts :: "('f, 'b) Term.term list"   and t1 :: "('f, 'b) Term.term"   and ts' :: "('f, 'b) Term.term list" assumes "weight t = w0"   and "t = Fun f ts"   and "ts = t1 # ts'" shows "ts' = []"
LxxE: fixes qs :: "nat list"   and x :: "nat"   and y :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> bool" assumes "qs \<in> Lxx x y"   and "qs \<in> lang (seq [Atom x, Atom x]) \<Longrightarrow> P x y qs"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y]) \<Longrightarrow> P x y qs"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x]) \<Longrightarrow> P x y qs"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y]) \<Longrightarrow> P x y qs" shows "P x y qs"
negative_cycle_dest: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "less (linordered_ab_monoid_add.len plus m i j xs) (linordered_ab_monoid_add.len plus m i j (rem_cycles i j xs))" shows "\<exists>i' ys. less (linordered_ab_monoid_add.len plus m i' i' ys) neutral \<and> set ys \<subseteq> set xs \<and> i' \<in> set (i # j # xs)"
that: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"   and "(\<Psi>, P', Q') \<in> Rel" shows "thesis"
that: fixes \<I>\<^sub>\<tau> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)"   and "\<lbrakk>{}; {}; \<A>\<rbrakk>\<^sub>s \<I>\<^sub>\<tau>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>)" shows "thesis"
hyps: fixes x2a :: "('a, 'b) Term.term" assumes "x2a \<in> set ts" shows "x2a \<cdot> Var = x2a"
replacefacesAt2_length: fixes nvs :: "nat list"   and f' :: "face"   and f'' :: "face list"   and vs :: "face list list" shows "|replacefacesAt2 nvs f' f'' vs| = |vs|"
rel_simps_47: shows "(1::'a) \<le> (1::'a)"
abs_eq: fixes x :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and xa :: "perm" assumes "eq_onp hereditarily_fs x x" shows "xa \<bullet> Abs_formula x = Abs_formula (xa \<bullet> x)"
distinct_41: fixes x61 :: "Arraytype"   and x62 :: "nat" shows "arrV x61 x62 \<noteq> nullV"
mem_monad_SReal_HFinite: fixes x :: "real star"   and a :: "real" assumes "x \<in> monad (hypreal_of_real a)" shows "x \<in> HFinite"
r1_trans: fixes s :: "trm"   and s' :: "trm"   and t :: "trm" assumes " s \<mapsto>\<^sup>* s' " shows " App s t \<mapsto>\<^sup>* App s' t "
not_summable_harmonic: shows "\<not> summable (\<lambda>n. inverse (of_nat n))"
T: fixes N1 :: "'a quasinorm"   and N2 :: "'a quasinorm"   and N3 :: "'a quasinorm" assumes "N1 \<le> N2"   and "N2 \<le> N3" shows "N1 \<le> N3"
wt_t_ge_s: shows "wt t \<ge>\<^sub>p wt s"
suffix_appendD: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "suffix (zs @ xs) ys" shows "suffix xs ys"
a_strong: fixes x :: "real" assumes "0 \<le> x" shows "x * A x - T x \<in> {0..c1 * x}"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_is_shadow_root_ptr: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr"
ring_irreducibleE_1: fixes r :: "'a" assumes "r \<in> carrier R"   and "ring_irreducible r" shows "r \<noteq> \<zero>"
that: fixes \<Gamma>' :: "'a formula list" assumes "set \<Gamma>' \<subseteq> \<Gamma>"   and "\<Turnstile> \<^bold>\<And>\<Gamma>' \<^bold>\<rightarrow> F" shows "thesis"
whisker_right: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and f :: "'a" assumes "seq \<nu> \<mu>"   and "ide f" shows "\<nu> \<cdot> \<mu> \<star> f = (\<nu> \<star> f) \<cdot> (\<mu> \<star> f)"
succsr_def: shows "succsr \<equiv> {(x, y). y \<in> set (succs x)}"
assms_1: shows "\<forall>S. open S \<longrightarrow> open {x. f x \<in> S}"
coeff_diff: fixes p :: "'b poly"   and q :: "'b poly"   and n :: "nat" shows "coeff (p - q) n = coeff p n - coeff q n"
RETURN_SPEC_refine: fixes x :: "'e"   and R :: "('e \<times> 'd) set"   and \<Phi> :: "'d \<Rightarrow> bool" assumes "\<exists>x'. (x, x') \<in> R \<and> \<Phi> x'" shows "RETURN x \<le> \<Down> R (SPEC \<Phi>)"
prod_pOp_commute: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and a :: "'c \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'd" assumes "\<forall>k\<in>I. aGroup (A k)"   and "a \<in> carr_prodag I A"   and "b \<in> carr_prodag I A" shows "prod_pOp I A a b = prod_pOp I A b a"
last_prog_p\<^sub>2: shows "last_prog p\<^sub>2 (sb\<^sub>1 @ sb') = p"
convex_scaling: fixes S :: "'c set"   and c :: "real" assumes "convex S" shows "convex ((*\<^sub>R) c ` S)"
simps_2: fixes l :: "'a tree"   and x :: "'a"   and r :: "'a tree" shows "wbalanced \<langle>l, x, r\<rangle> = (\<bar>int (size l) - int (size r)\<bar> \<le> 1 \<and> wbalanced l \<and> wbalanced r)"
GREATEST_FP_def: fixes L :: "('c, 'd) gorder_scheme"   and f :: "'c \<Rightarrow> 'c" shows "GFP\<^bsub>L\<^esub> f = \<Squnion>\<^bsub>L\<^esub>{u \<in> carrier L. u \<sqsubseteq>\<^bsub>L\<^esub> f u}"
simps_2: fixes n :: "nat"   and m :: "nat" shows "cut_bound n m \<bottom> = n"
simps: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a stream" shows "holds P xs = P (shd xs)"
simps_2: shows "\<lbrace>\<^bold>\<I>\<rbrace> = \<I>"
arrE: fixes f :: "'a" assumes "Right.arr f" shows "right (src f) f"
is_full: fixes a :: "'d"   and a' :: "'d"   and g :: "'c" assumes "D.ide a"   and "D.ide a'"   and "\<guillemotleft>g : G a' \<rightarrow>\<^sub>C G a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow>\<^sub>D a\<guillemotright> \<and> G f = g"
are_all_operators_applicable_set: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list" shows "are_all_operators_applicable s ops = (\<forall>op\<in>set ops. \<forall>v\<in>set (precondition_of op). s v = Some True)"
two_reduced_heads_imp_reduced_alt_step: fixes s :: "'w"   and t :: "'w"   and w :: "'w"   and bs :: "'w list"   and n :: "nat"   and cs :: "'w list" assumes "s \<noteq> t"   and "S_reduced_for w (t # bs)"   and "order.greater (add_order (s + t)) n \<or> add_order (s + t) = 0"   and "S_reduced_for w (alternating_list n s t @ cs)" shows "\<exists>ds. S_reduced_for w (alternating_list (Suc n) t s @ ds)"
dist_fs_triangle: fixes A :: "complex_homo"   and B :: "complex_homo"   and C :: "complex_homo" shows "dist_fs A B \<le> dist_fs A C + dist_fs C B"
rbt_min_def: fixes t :: "('b, unit) RBT_Impl.rbt" shows "rbt_min t = rbt_fold1_keys min t"
fls_deriv_lr_inverse_2: fixes x :: "'a"   and f :: "'a fls"   and y :: "'a" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * y = (1::'a)" shows "fls_deriv (fls_right_inverse f y) = - fls_right_inverse f y * fls_deriv f * fls_right_inverse f y"
mHom_tOp_commute: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and g :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N"   and "g \<in> mHom R M N" shows "tOp_mHom R M N f g = tOp_mHom R M N g f"
subst_bv_def: fixes u :: "Core.term"   and t :: "Core.term" shows "subst_bv u t = subst_bv2 t 0 u"
reduced_row_echelon_form_upt_k_Gauss_Jordan_column_k: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "reduced_row_echelon_form_upt_k (snd (Gauss_Jordan_column_k (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1, A) k)) (Suc k)"
mag_divide: fixes x :: "('a, 'b, 'c) Quantity_scheme"   and y :: "('a, 'b, 'c) Quantity_scheme" shows "mag (x / y) = mag x / mag y"
strict_implies_order: fixes a :: "'d list"   and b :: "'d list" assumes "strict_prefix a b" shows "prefix a b"
elims: fixes x :: "'c tree"   and y :: "'c tree set" assumes "subtrees x = y" obtains "x = \<langle>\<rangle>" and "y = {\<langle>\<rangle>}"   | l :: "'c tree" and a :: "'c" and r :: "'c tree" where "x = \<langle>l, a, r\<rangle>" and "y = {\<langle>l, a, r\<rangle>} \<union> subtrees l \<union> subtrees r"
assms_2: fixes x' :: "'b"   and x :: "'a" assumes "REMOVE_INTERNAL c' x'"   and "(x, x') \<in> R"   and "I x'" shows "(f x, f' $ x') \<in> \<langle>R\<rangle>nres_rel"
poly_root_induct: fixes Q :: "'b poly \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and p :: "'b poly" assumes "Q 0"   and "\<And>p. (\<And>a. P a \<Longrightarrow> poly p a \<noteq> (0::'b)) \<Longrightarrow> Q p"   and "\<And>a p. \<lbrakk>P a; Q p\<rbrakk> \<Longrightarrow> Q ([:a, - (1::'b):] * p)" shows "Q p"
forward_backward: fixes s :: "'b"   and os :: "'a list" assumes "s \<in> \<S>" shows "(\<Sum>t\<in>\<S>. forward s t os) = backward s os"
lm016: fixes P :: "('a \<times> 'b) set"   and X :: "'a set"   and Y :: "'a set" shows "P || X \<subseteq> P || (X \<union> Y) \<and> Outside' X P \<subseteq> Outside' (X \<inter> Y) P"
assms_5: shows "\<not> is_fun_call (body fd ! pc)"
distinct_5: fixes x1 :: "nat" shows "VNat x1 \<noteq> Wrong"
strictly_preserves_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g" shows "F (\<F>C.tensor f g) = F f \<otimes>\<^sub>D F g"
ra_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "composition (inf x (composition y top)) z = inf (composition x z) (composition y top)"
universal_denial_def: fixes A :: "'a set"   and B :: "'a set" shows "A E B \<equiv> \<forall>b\<in>B. b \<notin> A"
less_add_same_cancel1: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus a b) = less zero b"
le_filter_def: fixes F :: "'a filter"   and F' :: "'a filter" shows "(F \<le> F') = (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
zero_closed: shows "0\<^sub>o\<^sub>w \<in> U"
trms\<^sub>s\<^sub>t_cons: fixes x :: "('d, 'e) strand_step"   and S :: "('d, 'e) strand_step list" shows "trms\<^sub>s\<^sub>t (x # S) = trms\<^sub>s\<^sub>t\<^sub>p x \<union> trms\<^sub>s\<^sub>t S"
group_incl_imp_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "Group.group (G\<lparr>carrier := H\<rparr>)" shows "subgroup H G"
ll_on_open_real_def: fixes T :: "real set"   and f :: "real \<Rightarrow> real \<Rightarrow> real"   and X :: "real set" shows "ll_on_open_real T f X \<equiv> ll_on_open T f X"
int_to_bv_def: fixes n :: "int" shows "int_to_bv n = (if 0 \<le> n then norm_signed (\<zero> # nat_to_bv (nat n)) else norm_signed (bv_not (\<zero> # nat_to_bv (nat (- n - 1)))))"
up_eq: fixes x :: "'b"   and y :: "'b" shows "(up\<cdot>x = up\<cdot>y) = (x = y)"
not_symKeys_publicKey: fixes b :: "bool"   and A :: "agent" shows "publicKey b A \<notin> symKeys"
mult_fract: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "Fract a b * Fract c d = Fract (a * c) (b * d)"
language_state_for_input_alt_def: fixes M :: "('e, 'f, 'g) FSM"   and q :: "'g"   and xs :: "'e list" shows "language_state_for_input M q xs = LS\<^sub>i\<^sub>n M q {xs}"
set_subset_Univ: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<phi> :: "'c \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'd"   and b :: "'c"   and a :: "'c" assumes "hom_functor C S \<phi>"   and "partial_magma.ide C b"   and "partial_magma.ide C a" shows "hom_functor.set C \<phi> (b, a) \<subseteq> Collect (category.terminal S)"
padic_simps_1: fixes p :: "int"   and n :: "nat" shows "padic_zero p n = \<zero>\<^bsub>residue_ring (p ^ n)\<^esub>"
simps: fixes l :: "'a"   and u :: "'a" shows "- Interval l u = Interval (- u) (- l)"
larger_is_join2: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.cpo below"   and "below x y" shows "cpo.join below x y = y"
compl_sets: fixes a :: "'c set"   and M :: "'c measure" assumes "a \<in> sets M" shows "space M - a \<in> sets M"
axioms_2: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"
simps_2: fixes xs :: "'a list"   and p :: "pexpr"   and ps :: "pexpr list" shows "nonzero xs (p # ps) = (peval xs p \<noteq> \<zero> \<and> nonzero xs ps)"
Q_hit_pre_edges: shows "H.edges_of_walk (Q_hit_pre @ [y]) \<subseteq> \<Union> (H.edges_of_walk ` paths_with_new)"
bounded_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(x \<preceq> \<Sqinter> A) = (\<forall>a\<in>A. x \<preceq> a)"
tendsto_zero_divide_iff: fixes c :: "'a"   and a :: "nat \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "(\<lambda>n. a n / c) \<longlonglongrightarrow> (0::'a) = a \<longlonglongrightarrow> (0::'a)"
weakCongWeakPsiCong: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "P \<doteq>\<^sub>c Q" shows "\<Psi> \<rhd> P \<doteq> Q"
func: shows "f \<in> carrier \<rightarrow> carrier\<^bsub>T\<^esub>"
inf_sup_aci_2: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
ev_induct_strong: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and x :: "'a stream"   and P :: "'a stream \<Rightarrow> bool" assumes "ev \<phi> x"   and "\<And>xs. \<phi> xs \<Longrightarrow> P xs"   and "\<And>xs. \<lbrakk>ev \<phi> (stl xs); \<not> \<phi> xs; P (stl xs)\<rbrakk> \<Longrightarrow> P xs" shows "P x"
select_convs_2: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "user_reg \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = user_reg"
couSim: shows "weak_reduction_coupled_simulation (indRelRTPO TRel) (STCal Source Target)"
notfinal_all_before: fixes args :: "nat list"   and A :: "abc_inst list"   and x :: "nat"   and y :: "nat" assumes "abc_notfinal (abc_steps_l (0, args) A x) A"   and "y \<le> x" shows "abc_notfinal (abc_steps_l (0, args) A y) A"