hcju/isabelleps huggingface dataset:
rep_inj: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat"   and "rep_nat x = rep_nat y" shows "x = y"
aencrypt_alt: fixes \<alpha> :: "'grp"   and msg :: "'grp" shows "aencrypt \<alpha> msg = map_spmf (\<lambda>y. (\<^bold>g [^] y, \<alpha> [^] y \<otimes> msg)) (sample_uniform (order \<G>))"
dense_repr_set_fst: fixes p :: "'a list"   and K :: "'a set" assumes "set p \<subseteq> K" shows "fst ` set (dense_repr p) \<subseteq> K - {\<zero>}"
ncrit_specD: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) set"   and m :: "nat"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'e"   and q_in_bs :: "bool"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" assumes "ncrit_spec crit"   and "dickson_grading d"   and "set gs \<union> set bs \<union> set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "snd ` set ps \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "q_in_bs \<Longrightarrow> q \<in> set gs \<union> set bs"   and "\<And>p' q'. \<lbrakk>(p', q') \<in>\<^sub>p snd ` set ps; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "\<And>p' q'. \<lbrakk>p' \<in> set gs \<union> set bs; q' \<in> set gs \<union> set bs; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "p \<in> set hs"   and "q \<in> set gs \<union> set bs \<union> set hs"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "crit data gs bs hs q_in_bs ps p q" shows "crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q)"
vsubset_vsingleton_right: fixes A :: "V"   and a :: "V" shows "(A \<subseteq>\<^sub>\<circ> set {a}) = (A = set {a} \<or> A = 0)"
FVT_finite: fixes t :: "trm" shows "finite (FVT t)"
steps_lsteps: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat"   and l :: "nat"   and u :: "nat"   and v :: "nat" shows "steps A f l u v = lsteps (rev (sorted_list_of_set A)) f l u v"
weighted_RP_measure_decreasing_P: fixes St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and C :: "'a literal multiset"   and i :: "nat" assumes "St \<leadsto>\<^sub>w St'"   and "(C, i) \<in># wP_of_wstate St" shows "(RP_combined_measure (weight (C, i)) St', RP_combined_measure (weight (C, i)) St) \<in> natLess <*lex*> (natLess <*lex*> natLess <*lex*> natLess) <*lex*> natLess <*lex*> natLess <*lex*> natLess"
step: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and c :: "nat \<Rightarrow> ('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat"   and k :: "nat"   and l :: "nat" assumes "program_trace program_step c n k"   and "l < k" shows "computation.concurrent_step direct_memop_step empty_storebuffer_step program_step (\<lambda>p p' is sb. sb) (c (n + l)) (c (n + Suc l))"
transpose_uminus: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "(- A)\<^sup>T = - A\<^sup>T"
assms_1: shows "chaos \<sqsubseteq> (i::'a::refinement_lattice)"
is_sum2sq_nat_def: fixes n :: "nat" shows "is_sum2sq_nat n \<equiv> \<exists>a b. n = sum2sq_nat a b"
nlt__le: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Lt Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Le Bet Cong C D A B"
case_2: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
f6: shows "2 ^ LENGTH('b) + - 1 = - 1 + 2 ^ LENGTH('b)"
nzip_def: fixes xs :: "'c \<Rightarrow> 'a"   and ys :: "'c \<Rightarrow> 'b"   and i :: "'c" shows "(xs || ys) i = (xs i, ys i)"
project_enc: fixes w :: "'a list"   and x :: "nat + nat set"   and I :: "(nat + nat set) list" shows "map \<pi> (enc (w, x # I)) = enc (w, I)"
a_coset_add_inv1: fixes G :: "('a, 'b) ring_scheme"   and M :: "'a set"   and x :: "'a"   and y :: "'a" assumes "abelian_group G"   and "M +>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> y = M"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "M \<subseteq> carrier G" shows "M +>\<^bsub>G\<^esub> x = M +>\<^bsub>G\<^esub> y"
map_in_trees: fixes t :: "'d tree"   and f :: "'d \<Rightarrow> 'e"   and S :: "'e set" assumes "\<And>x. x \<in> set_tree t \<Longrightarrow> f x \<in> S" shows "map_tree f t \<in> trees S"
qSwapped_qSkel: fixes X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and Y :: "('d, 'e, 'f, 'g, 'h) qTerm" assumes "(X, Y) \<in> qSwapped" shows "qSkel Y = qSkel X"
map_id: fixes t :: "'a pmf" shows "map_pmf id t = t"
irreducible\<^sub>d_smultI: fixes c :: "'b"   and p :: "'b poly" assumes "c \<noteq> (0::'b)"   and "irreducible\<^sub>d (smult c p)" shows "irreducible\<^sub>d p"
choice: shows "dh < dg \<or> H = 0 \<and> F \<noteq> 0 \<and> G \<noteq> 0"
mult_1_right: fixes a :: "'a" shows "a + (0::'a) = a"
cong_trans: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d"   and d :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo c d a" shows "unique_euclidean_semiring.cong modulo b d a"
concI_if_Nil1: fixes A :: "'b list set"   and xs :: "'b list"   and B :: "'b list set" assumes "[] \<in> A"   and "xs \<in> B" shows "xs \<in> A @@ B"
gc_complete_Liminf: fixes Ns :: "('f \<times> 'l) set llist"   and B :: "'f" assumes "chain (\<leadsto>GC) Ns"   and "active_subset (lhd Ns) = {}"   and "passive_subset (Liminf_llist Ns) = {}"   and "B \<in> Bot_F"   and "fst ` lhd Ns \<Turnstile>\<inter>\<G> {B}" shows "\<exists>BL\<in>Bot_FL. BL \<in> Liminf_llist Ns"
llast_simps_2: fixes r :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "r \<in> A\<^sup>\<star>" shows "LList2.llast (a ## r) = (if r = LNil then a else LList2.llast r)"
retractionI: fixes g :: "'a"   and f :: "'a" assumes "ide (g \<cdot> f)" shows "retraction g"
compute_monom_mult_poly_mapping: fixes c :: "'b"   and t :: "'a"   and xs :: "('a, 'b) fmap" shows "punit.monom_mult c t (Pm_fmap xs) = Pm_fmap (if c = (0::'b) then fmempty else punit.shift_map_keys t ((*) c) xs)"
map_sel: fixes a :: "'b option"   and f :: "'b \<Rightarrow> 'c" assumes "a \<noteq> None" shows "the (map_option f a) = f (the a)"
not_eq_extremum: fixes a :: "'b word" shows "(a \<noteq> 0) = (0 < a)"
setcompr_eq_image: fixes f :: "'g \<Rightarrow> 'f"   and P :: "'g \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
monomial_eq_itself: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "is_monomial p" shows "monomial (lcf p) (lpp p) = p"
inject_3: fixes x4 :: "ty"   and y4 :: "ty" shows "(ArrayT x4 = ArrayT y4) = (x4 = y4)"
irreducible_imp_rsquarefree_of_int_poly: fixes p :: "int poly" assumes "irreducible p"   and "0 < Polynomial.degree p" shows "rsquarefree (Ring_Hom_Poly.of_int_poly p)"
Inf_nat_def: fixes X :: "nat set" shows "Inf X = (LEAST n. n \<in> X)"
not_less_iff_gr_or_eq: fixes x :: "'b word"   and y :: "'b word" shows "(\<not> x <s y) = (y <s x \<or> x = y)"
cbb: shows "cball \<xi> d \<subseteq> ball \<xi> r"
minus_add_distrib: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ab_group_add plus zero minus uminus" shows "uminus (plus a b) = plus (uminus a) (uminus b)"
measurable_on_add: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f measurable_on S"   and "g measurable_on S" shows "(\<lambda>x. f x + g x) measurable_on S"
W_consts_3: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n false\<^sub>n W\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<phi>"
Inf_pred_def: fixes A :: "'a pred set" shows "\<Sqinter> A = Pred (\<Sqinter> (eval ` A))"
itrm_sym: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and x :: "'a itrm"   and y :: "'a itrm" assumes "itrm_cong R x y" shows "itrm_cong R y x"
N_def: shows "\<forall>i n. N \<le> n \<longrightarrow> \<parallel>\<c>\<o>\<l> i (X n - to_mtx (Finite_Cartesian_Product.transpose L))\<parallel> < \<epsilon> / real CARD('a)"
FinFunc_def: fixes r :: "('h \<times> 'h) set"   and s :: "('g \<times> 'g) set" shows "FinFunc r s = Func (Field s) (Field r) \<inter> fin_support (zero r) (Field s)"
inv_allI: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat" assumes "\<And>i j. \<lbrakk>i < n; j < n\<rbrakk> \<Longrightarrow> p A i j" shows "local.inv_all p A"
blocking_on_def: fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X' :: "'a set" shows "Singleton_for_d.blocking_on ds X h X' = (X' \<noteq> Ch h X \<and> X' = Ch h (X \<union> X') \<and> X' \<subseteq> Singleton_for_d.CD_on ds (X \<union> X'))"
equalityI: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
rev_pair_relator_list: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "class.group_add minus plus zero uminus" shows "rev (alternating_list (2 * monoid_add.add_order plus zero (plus s t)) s t) = alternating_list (2 * monoid_add.add_order plus zero (plus t s)) t s"
drest_proj_succ_eq_drest_succ: fixes h :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap"   and vs :: "'d set" assumes "fst h \<subseteq>\<^sub>f s"   and "fmdom' (snd h) \<subseteq> fmdom' s" shows "fmrestrict_set vs (state_succ s (action_proj h vs)) = fmrestrict_set vs (state_succ s h)"
sep_disj_prod_commute_1: fixes ps :: "'a"   and n :: "'b" shows "(ps, 0::'b) ## (0::'a, n)"
evalsub_def: fixes E :: "char list \<Rightarrow> 'b"   and F :: "char list \<Rightarrow> 'b list \<Rightarrow> 'b"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "evalsub E F \<sigma> = eval\<^sub>t E F \<circ> \<sigma>"
dehomogenize_times: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "dehomogenize x (p * q) = dehomogenize x p * dehomogenize x q"
homeomorphic_maps_def: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "homeomorphic_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y)"
assms: shows "wffp (sl1 @ [State st1 u1 c1] @ [State st2 u2 c2] @ sl2)"
apply_fun_def: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'c" shows "(A .. B) = (\<lambda>x. A x (B x))"
Dube_aux_ge_d: fixes j :: "nat" shows "d \<le> local.Dube_aux j"
m_s_h: fixes X :: "char list set"   and S :: "char list \<Rightarrow> 'av" assumes "finite X" shows "m\<^sub>s S X \<le> h * card X"
exp_diff: fixes x :: "'d"   and y :: "'d" shows "exp (x - y) = exp x / exp y"
design_support_def: shows "des.design_support \<equiv> set_mset (str_del_point_blocks x)"
term_poly_list_rel_same_rightD: fixes a :: "char list list"   and aa :: "char list multiset"   and ab :: "char list multiset" assumes "(a, aa) \<in> term_poly_list_rel"   and "(a, ab) \<in> term_poly_list_rel" shows "aa = ab"
\<phi>_in_terms_of_\<eta>: fixes y :: "'c"   and f :: "'b"   and x :: "'b" assumes "C.ide y"   and "\<guillemotleft>f : F' y \<rightarrow>\<^sub>B x\<guillemotright>" shows "\<phi>' y f = G' f \<cdot>\<^sub>C F'G'.\<eta> y"
IH: fixes ys :: "'a list" shows "cnt y (remove_all_cycles xs ys) \<le> max 1 (cnt y ys)"
elemI: fixes a' :: "'a"   and A :: "'a set"   and R :: "('a, 'b) eq_object_scheme"   and a :: "'a" assumes "a' \<in> A"   and "a .=\<^bsub>R\<^esub> a'" shows "a .\<in>\<^bsub>R\<^esub> A"
B: shows "F \<r>\<^sub>C[g] = (F g \<star>\<^sub>D D.inv (unit (src\<^sub>C g))) \<cdot>\<^sub>D D.inv (\<Phi> (g, src\<^sub>C g))"
app_1: fixes \<Gamma>2 :: "name list"   and s1 :: "name"   and s2 :: "name" assumes "length \<Gamma>1 = length \<Gamma>2"   and "distinct \<Gamma>1"   and "distinct \<Gamma>2"   and "wellformed' (length \<Gamma>1) t\<^sub>1"   and "fBall (frees t\<^sub>1 |\<union>| fset_of_list \<Gamma>1) (\<lambda>y. y \<le> s1)"   and "fdisjnt (fset_of_list \<Gamma>1) (frees t\<^sub>1)"   and "fBall (frees t\<^sub>1 |\<union>| fset_of_list \<Gamma>2) (\<lambda>y. y \<le> s2)"   and "fdisjnt (fset_of_list \<Gamma>2) (frees t\<^sub>1)" shows "alpha_equiv (fmap_of_list (zip \<Gamma>1 \<Gamma>2)) (fst (run_state (term_to_nterm \<Gamma>1 t\<^sub>1) s1)) (fst (run_state (term_to_nterm \<Gamma>2 t\<^sub>1) s2))"
nres_relD: fixes c :: "'b nres"   and a :: "'c nres"   and R :: "('b \<times> 'c) set" assumes "(c, a) \<in> \<langle>R\<rangle>nres_rel" shows "c \<le> \<Down> R a"
case_a2c_case_caseR: fixes f :: "'a"   and g :: "nat discr Box \<rightarrow> nat discr Box llist \<rightarrow> 'a"   and w :: "R" shows "(case a2c\<cdot>w of lnil \<Rightarrow> f | x :@ xs \<Rightarrow> g\<cdot>x\<cdot>xs) = (case nextR\<cdot>w of Nothing \<Rightarrow> f | Just\<cdot>(x, r) \<Rightarrow> g\<cdot>x\<cdot>(a2c\<cdot>r))"
pair_map_spmf2: fixes p :: "'e spmf"   and f :: "'g \<Rightarrow> 'f"   and q :: "'g spmf" shows "pair_spmf p (map_spmf f q) = map_spmf (apsnd f) (pair_spmf p q)"
P_false: assumes "\<exists>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> = false \<tau>" shows "oid_of (x \<tau>) \<in> oid_of ` \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>"
random_member_is_member: fixes ss :: "'a fset"   and s :: "'a" assumes "random_member ss = Some s" shows "s |\<in>| ss"
differentiable_mult: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f differentiable at x within s"   and "g differentiable at x within s" shows "(\<lambda>x. f x * g x) differentiable at x within s"
assms_1: shows "x \<in> env_dom (e\<lparr>a:X\<rparr>\<lparr>b:Y\<rparr>)"
seq_INF_distrib: fixes X :: "'c::type set"   and c :: "'a::refinement_lattice"   and d :: "'c::type \<Rightarrow> 'a::refinement_lattice" assumes "(X::'c::type set) \<noteq> {}" shows "(c::'a::refinement_lattice) ; \<Sqinter> ((d::'c::type \<Rightarrow> 'a::refinement_lattice) ` (X::'c::type set)) = (\<Sqinter>x::'c::type\<in>X. c ; d x)"
id_left: fixes CC :: "('d, 'e, 'f) category_scheme"   and f :: "'e" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>" shows "Id\<^bsub>CC\<^esub> (Cod\<^bsub>CC\<^esub> f) \<bullet>\<^bsub>CC\<^esub> f = f"
apply_bcontfun: fixes x :: "'c \<Rightarrow>\<^sub>C 'd" shows "apply_bcontfun x \<in> bcontfun"
simps: fixes it :: "('a, 'b, 'c, 'd) itnode rose_tree" shows "iAnnot it = iAnnot' (rose_tree.root it)"
RES_SPEC_eq: fixes \<Phi> :: "'a set" shows "RES \<Phi> = SPEC (\<lambda>P. P \<in> \<Phi>)"
trivial: fixes t :: "('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" shows "UNIF1 Var (add_mset (t, t) E) E"
append_eq_append_conv: fixes xs :: "'b list"   and ys :: "'b list"   and us :: "'b list"   and vs :: "'b list" assumes "\<^bold>|xs\<^bold>| = \<^bold>|ys\<^bold>| \<or> \<^bold>|us\<^bold>| = \<^bold>|vs\<^bold>|" shows "(xs \<cdot> us = ys \<cdot> vs) = (xs = ys \<and> us = vs)"
ccApprox_paths: fixes t :: "Vars.var ttree" shows "ccApprox t = lub (ccFromList ` paths t)"
eql: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "eql t1 t2 \<in> fmla"
cut_and_updateD: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option"   and ttas :: "('thread_id \<times> ('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action) list"   and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and x :: "'x"   and ta :: "('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "cut_and_update s vs"   and "s -\<triangleright>ttas\<rightarrow>* s'"   and "ta_seq_consist P vs (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))"   and "thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s'\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "actions_ok s' t ta" shows "\<exists>ta' x'' m''. t \<turnstile> \<langle>x, shr s'\<rangle> -ta'\<rightarrow> \<langle>x'', m''\<rangle> \<and> actions_ok s' t ta' \<and> ta_seq_consist P (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>) \<and> eq_upto_seq_inconsist P \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))"
p_subid_idem: fixes x :: "'a" shows "x \<cdot> (0::'a) \<parallel> (x \<cdot> (0::'a)) = x \<cdot> (0::'a)"
evaln_elim_cases_2: fixes G :: "prog"   and s :: "st"   and n :: "nat"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r Skip\<succ>\<midarrow>n\<rightarrow> (x, s')" obtains "x = \<diamondsuit>" and "s' = Norm s"
dagger_ext: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "x \<subseteq> t_star x"
canonical_proj_is_hom: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "canonical_proj I J \<in> ring_hom R (RDirProd (R Quot I) (R Quot J))"
error_free_abupd_absorb: fixes s :: "state"   and j :: "jump" assumes "error_free s" shows "error_free (abupd (absorb j) s)"
add_zero_left: fixes V :: "'c set"   and x :: "'c" assumes "vectorspace V"   and "x \<in> V" shows "(0::'c) + x = x"
subst_forget: fixes x :: "name"   and t :: "lam"   and u :: "lam" assumes "x \<sharp> t" shows "t[x::=u] = t"
f: shows "u = f * p \<and> \<bar>f\<bar> = 1"
induced_pow_fun_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "inj_on ((`) f) (Pow A)"
array_get_oo'_def: fixes x :: "'a"   and a :: "'a array" shows "array_get_oo' x a \<equiv> array_get_oo x a \<circ> nat_of_integer"
union_le_mono1: fixes B :: "'b multiset"   and D :: "'b multiset"   and C :: "'b multiset" assumes "B < D" shows "B + C < D + C"
inf_sup_aci_2: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.lattice inf less_eq less sup" shows "inf (inf x y) z = inf x (inf y z)"
that: fixes t :: "'a \<Rightarrow>\<^sub>0 'b" shows "ord t t"
cil_commute: fixes w :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and w' :: "'a list" shows "w \<otimes>\<^bsub>\<alpha>\<^esub> w' = w' \<otimes>\<^bsub>\<alpha>\<^esub> w"
\<alpha>edges_irrefl: fixes g :: "'m" assumes "invar g" shows "irrefl (\<alpha>edges_aux g)"
Max_wrt_among_subset_2: fixes A :: "'alt set" shows "Max_wrt_among (Pareto R) A \<subseteq> A"
norm1_0_iff: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "(norm1 v = 0) = (v = 0)"
map_1: fixes f :: "'d \<Rightarrow> 'c" shows "map_option f None = None"
dg_prod_2_Dom_app: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and g :: "V"   and f :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>" shows "(\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Dom\<rparr> \<lparr>g, f\<rparr>\<^sub>\<bullet> = [\<AA>\<lparr>Dom\<rparr>\<lparr>g\<rparr>, \<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr>]\<^sub>\<circ>"
elims: fixes x :: "nat"   and y :: "hf" assumes "ord_of x = y" obtains "x = 0" and "y = 0"   | k :: "nat" where "x = Suc k" and "y = succ (ord_of k)"
E10: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<diamond>F = \<diamond>F"
check_single_imp: fixes \<C> :: "t_context"   and e :: "b_e"   and ctn :: "checker_type"   and ctm :: "checker_type" assumes "check_single \<C> e ctn = ctm"   and "ctm \<noteq> Bot" shows "check_single \<C> e = id \<or> check_single \<C> e = type_update_select \<or> (\<exists>cons prods. check_single \<C> e = (\<lambda>ctn. type_update ctn cons prods))"
vrestriction_atD_2: fixes A :: "V"   and a :: "V"   and b :: "V" assumes "(r \<restriction>\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A"   and "a \<in>\<^sub>\<circ> A" shows "r\<lparr>a\<rparr> = b"
c: shows "C \<in> path_components_of (subtopology Y V)"