hcju/isabelleps huggingface dataset:
Decompose: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S"   and "ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t \<cdot> \<I>"   and "Ana t = (K, M)"   and "\<And>k. k \<in> set K \<Longrightarrow> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c k \<cdot> \<I>" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> (S @ [Decomp t])"
ortho: fixes u :: "'a vec" assumes "u \<in> S" shows "w \<bullet> u = (0::'a)"
intro: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d" assumes "finite X"   and "\<forall>c\<in>X. 0 < v c"   and "\<forall>c. c \<notin> X \<longrightarrow> n < v c"   and "bij_betw v X {1..n}"   and "X \<noteq> {}"   and "not_in_X \<notin> X" shows "Regions' X v n not_in_X"
less_imp_not_less: fixes x :: "'b DBMEntry"   and y :: "'b DBMEntry" assumes "x \<prec> y" shows "(\<not> y \<prec> x) = True"
tendsto_obtains_N: fixes f :: "nat \<Rightarrow> 'b"   and f0 :: "'b"   and S :: "'b set" assumes "f \<longlonglongrightarrow> f0"   and "open S"   and "f0 \<in> S" obtains N :: "nat" where "\<forall>n\<ge>N. f n \<in> S"
undual_dual: fixes x :: "'d" shows "undual (dual x) = x"
M_3: shows "\<forall>l. Discl \<subseteq> M l"
relcompI: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e"   and s :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'f" assumes "r a b"   and "s b c" shows "(r OO s) a c"
nat_filtration_def: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat" assumes "infinite_coin_toss_space p M" shows "infinite_coin_toss_space.nat_filtration M n = fct_gen_subalgebra M M (pseudo_proj_True n)"
order_finite_set_obtain_foundation: fixes M :: "'b set"   and t :: "'b" assumes "finite M"   and "t \<in> M" obtains s :: "'b" where "s \<in> M" and "s \<le> t" and "\<forall>u\<in>M. \<not> u < s"
proj_tuple_in_join_mask_idle: fixes n :: "nat"   and X :: "nat set"   and as :: "'b option list"   and pos :: "bool"   and t :: "'b option list set" assumes "wf_tuple n X as" shows "proj_tuple_in_join pos (join_mask n X) as t = join_cond pos t as"
spectrum_imp_cxt_quotient_ring: fixes \<pp> :: "('a \<times> 'a) set set" assumes "\<pp> \<in> qr.spectrum" shows "Comm_Ring.quotient_ring (qr.carrier_quotient_ring\<setminus>\<pp>) qr.carrier_quotient_ring qr.add_rel qr.mult_rel qr.zero_rel qr.one_rel"
lower_on_ground: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list"   and V :: "'a set" assumes "lower_on \<sigma> \<eta> V"   and "ground_on V \<eta>" shows "ground_on V \<sigma>"
delete_twist: fixes x :: "'a"   and y :: "'a"   and al :: "('a \<times> 'b) list" shows "AList.delete x (AList.delete y al) = AList.delete y (AList.delete x al)"
set_cinsert: fixes x :: "'f"   and A :: "'f cset" assumes "cin x A" obtains B :: "'f cset" where "A = cinsert x B" and "\<not> cin x B"
invertible_iff_is_unit: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "invertible A = (Determinants.det A dvd (1::'b))"
coeff_reciprocal_less: fixes p :: "nat"   and i :: "nat"   and P :: "'b poly" assumes "p < i"   and "degree P \<le> p" shows "coeff (reciprocal_poly p P) i = (0::'b)"
lappend_assoc: fixes xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" shows "lappend (lappend xs ys) zs = lappend xs (lappend ys zs)"
bind_advantage: fixes \<A> :: "'a \<Rightarrow> ('a \<times> nat \<times> nat \<times> nat \<times> nat) spmf" shows "Schnorr_\<Sigma>_commit.abstract_com.bind_advantage \<A> \<le> Schnorr_\<Sigma>_commit.rel_advantage (Schnorr_\<Sigma>_commit.adversary \<A>)"
empty_not_insert: fixes a :: "'a"   and A :: "'a set" shows "{} \<noteq> insert a A"
lerch_lemma: fixes h :: "real \<Rightarrow> 'a"   and u :: "real" assumes "continuous_on {0..1} h"   and "\<And>n. ((\<lambda>u. u ^ n *\<^sub>R h u) has_integral (0::'a)) {0..1}"   and "0 \<le> u"   and "u \<le> 1" shows "h u = (0::'a)"
digit_encode_take: fixes n :: "nat"   and ds :: "nat list"   and a :: "nat" shows "take n (digit_encode ds a) = digit_encode (take n ds) a"
inj_eq: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "inj f" shows "(f x = f y) = (x = y)"
Tstruct_def: fixes intT :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Tstruct intT \<equiv> \<forall>\<sigma>. \<exists>a. intT \<sigma> a"
eq': shows "\<phi> (2, 1) [^]\<^bsub>S\<^esub> 2 ^ p = \<one>\<^bsub>S\<^esub>"
overrides_Package_old: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides old"   and "accmodi new = Package"   and "wf_prog G" shows "accmodi old = Package"
strong_law_of_large_numbers_iid: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "indep_vars (\<lambda>_. borel) X UNIV"   and "\<And>i. distr M borel (X i) = distr M borel (X 0)"   and "integrable M (X 0)" shows "AE x in M. (\<lambda>n. (\<Sum>i<n. X i x) / real n) \<longlonglongrightarrow> expectation (X 0)"
hb_consistent_prefix: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat" assumes "xs prefix of i" shows "hb.hb_consistent (node_deliver_messages xs)"
path_def: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme" shows "path A \<equiv> transition_system.path (\<lambda>a p. snd a) (\<lambda>a p. snd a \<in> succ A (fst a) p)"
Basis_zero: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "zero \<notin> Basis"
assms_2: shows "wave (bipartite_web_of \<Gamma>) f"
dtor_rel_coinduct: fixes JR :: "'c coF \<Rightarrow> 'd coF \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>ya xa. JR ya xa \<Longrightarrow> rel_pre_coF R JR (dtor_coF ya) (dtor_coF xa)" shows "JR \<le> rel_coF R"
poly2_const: fixes a :: "'c"   and x :: "'c"   and y :: "'c" shows "poly2 [:[:a:]:] x y = a"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<lbrace>t \<^bold>\<star> u\<rbrace> = \<lbrace>t\<rbrace> \<star> \<lbrace>u\<rbrace>"
trans: fixes b :: "'a option \<Rightarrow>\<^sub>0 nat"   and a :: "'a option \<Rightarrow>\<^sub>0 nat"   and c :: "'a option \<Rightarrow>\<^sub>0 nat" assumes "extended_ord b a"   and "extended_ord c b" shows "extended_ord c a"
r\<^sub>0s\<^sub>1_is_cospan: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>1 h) (tab\<^sub>0 g \<star> Tfg.p\<^sub>0)"
perp_foot_incident_2: fixes p :: "proj2"   and l :: "proj2_line" shows "proj2_incident (perp_foot p l) (drop_perp p l)"
plus_uint16_def: shows "(+) \<equiv> map_fun Rep_uint16 (map_fun Rep_uint16 Abs_uint16) (+)"
superclasses_def: fixes G :: "prog"   and C :: "qtname" shows "superclasses G C = class_rec G C {} (\<lambda>C c superclss. if C = Object then {} else insert (super c) superclss)"
TSdet_Suc: fixes n :: "nat"   and qs :: "nat list"   and initH :: "nat list"   and init :: "nat list" assumes "Suc n \<le> length qs" shows "TSdet init initH qs (Suc n) = Partial_Cost_Model.Step (rTS initH) (TSdet init initH qs n) (qs ! n)"
zero_right: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a" assumes "bounded_bilinear prod" shows "prod a (0::'b) = (0::'c)"
igSwapAbsIPresIGWlsAbs_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSwapAbsIPresIGWlsAbs MOD \<equiv> \<forall>zs z1 z2 us s A. isInBar (us, s) \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSwapAbs MOD zs z1 z2 A)"
dX_range': fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> Pdds'.dX X d" shows "x \<in> X \<and> Xd x = d"
min_lvar_not_in_bounds_lvars: fixes min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and s :: "('a, 'b) state"   and x\<^sub>i :: "nat" assumes "MinLVarNotInBounds min_lvar_not_in_bounds" shows "min_lvar_not_in_bounds s = Some x\<^sub>i \<longrightarrow> x\<^sub>i \<in> lvars (\<T> s)"
size_4: fixes x21 :: "'a"   and x22 :: "'a list" shows "\<parallel>x21 \<Colon> x22\<parallel> = \<parallel>x22\<parallel> + Suc 0"
orthogonal_transformation_isometry: fixes f :: "'a \<Rightarrow> 'a" shows "orthogonal_transformation f = (f (0::'a) = (0::'a) \<and> (\<forall>x y. dist (f x) (f y) = dist x y))"
valid_valid_in: fixes p :: "fm"   and U :: "nat set" assumes "valid p" shows "valid_in U p"
zip_simps_4: fixes ys :: "['c]" shows "zip\<cdot>[]\<cdot>ys = []"
hm_it_next_def: fixes it :: "nat \<times> ('c \<times> 'd) list \<times> ('c, 'd) hashtable" shows "hm_it_next it \<equiv> case it of (0, [a], ht) \<Rightarrow> return (a, 0, [], ht) | (0, a # b # l, ht) \<Rightarrow> return (a, 0, b # l, ht) | (Suc nat, [a], ht) \<Rightarrow> hm_it_adjust nat ht \<bind> (\<lambda>i. Array.nth (the_array ht) i \<bind> (\<lambda>l. return (a, i, rev l, ht))) | (Suc nat, a # b # l, ht) \<Rightarrow> return (a, Suc nat, b # l, ht)"
restrict_singleton_product: fixes k :: "'d"   and f :: "'d \<times> 'd \<Rightarrow> 'c"   and l :: "'d"   and m :: "'d"   and g :: "'d \<times> 'd \<Rightarrow> 'c"   and n :: "'d"   and i :: "'d"   and j :: "'d" shows "([k] \<langle>f\<rangle> [l] \<odot> [m] \<langle>g\<rangle> [n]) (i, j) = (if i = k \<and> l = m \<and> j = n then f (i, l) * g (m, j) else bot)"
sel_1: fixes x1 :: "'b word"   and x2 :: "nat" shows "pfxm_prefix (PrefixMatch x1 x2) = x1"
lc_minus_higher: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "punit.lc (p - punit.higher p v) = lookup p v"
simps_4: fixes x2 :: "\<Pi>\<^sub>1 set"   and x1 :: "\<omega>" shows "\<alpha>\<nu> x2 \<noteq> \<omega>\<nu> x1"
bind_is_OK_pure_I: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog" assumes "pure f h"   and "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> h \<turnstile> ok g x" shows "h \<turnstile> ok Heap_Error_Monad.bind f g"
that: fixes a :: "V"   and b :: "V" assumes "x = [a, b]\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> A"   and "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "thesis"
problem_plan_bound_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" shows "problem_plan_bound PROB \<equiv> Sup (MPLS PROB)"
same_image_on_spanset_imp_same_hom: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and S :: "'b \<Rightarrow> 'c"   and ms :: "'b list" assumes "VectorSpaceHom smult V smult' T"   and "VectorSpaceHom smult V smult' S"   and "set ms \<subseteq> V"   and "V = R_scalar_mult.RSpan UNIV smult ms"   and "\<forall>m\<in>set ms. S m = T m" shows "S = T"
simps_2: fixes n :: "nat"   and A :: "'a list set" shows "A ^^ Suc n = A @@ A ^^ n"
less_le_not_le: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_prod x y = (less_eq_prod x y \<and> \<not> less_eq_prod y x)"
elims: fixes x :: "'b \<Rightarrow> ('a, 'b) Term.term"   and y :: "('a, 'b) Term.term set" assumes "subst_range x = y" obtains \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" where "x = \<sigma>" and "y = \<sigma> ` subst_domain \<sigma>"
diff_fun_on_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'd" shows "diff_fun_on A f = (\<exists>W. A \<subseteq> W \<and> W \<subseteq> carrier \<and> open W \<and> (\<exists>f'. diff_fun k (charts_submanifold W) f' \<and> (\<forall>x\<in>A. f x = f' x)))"
snth_siterate: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" shows "siterate f x !! n = (f ^^ n) x"
less: fixes a :: "nat \<Rightarrow> nat"   and b :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "a \<in> s"   and "b \<in> s"   and "a i < b i" shows "a < b"
set_of_graph_rulesD_2: fixes Rs :: "(('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) set"   and R :: "('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph" assumes "set_of_graph_rules Rs"   and "R \<in> Rs" shows "finite_graph (snd R)"
hom_char: fixes a :: "'a"   and b :: "'a" shows "R.hom a b = hom a b \<inter> Collect (right f)"
DERIV: shows "\<forall>m t. m < n \<and> 0 \<le> t \<and> t \<le> h \<longrightarrow> (diff m has_real_derivative diff (Suc m) t) (at t)"
path_append_def: fixes \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'a" shows "IFC_def.path_append \<pi> m \<pi>' = (\<lambda>n. if n \<le> m then \<pi> n else \<pi>' (n - m))"
assms_2: shows "Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle>"
tmod1: fixes mn :: "char list"   and s1 :: "'b state"   and ds :: "dec list"   and ck :: "bool"   and env :: "v sem_env"   and s2 :: "'b state"   and new_env :: "v sem_env"   and specs :: "spec list option" assumes "[mn] \<notin> defined_mods s1 \<and> no_dup_types ds \<and> evaluate_decs ck [mn] env s1 ds (s2, Rval new_env)" shows "evaluate_top ck env s1 (Tmod mn specs ds) (update_defined_mods (\<lambda>_. {[mn]} \<union> defined_mods s2) s2, Rval (make_sem_env (nsLift mn (sem_env.v new_env)) (nsLift mn (c new_env))))"
le_imp_less_or_eq: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le>s y" shows "x <s y \<or> x = y"
borderD_spref: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "x <p w"
indRelR_satisfies_pred_impl_enc_satisfies_pred: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>(P, Q)\<in>indRelR. Pred (P, Q)" shows "\<forall>S. Pred (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>))"
log_nat_power: fixes x :: "real"   and b :: "real"   and n :: "nat" assumes "0 < x" shows "log b (x ^ n) = real n * log b x"
list_all2_iff: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). P x y))"
lunit_coherence2: fixes f :: "'a" assumes "C.ide f" shows "lF f = F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f)"
rep_of_min: fixes l :: "nat list"   and i :: "nat" assumes "ufa_invar l"   and "i < length l" shows "l ! rep_of l i = rep_of l i"
conversepD: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and b :: "'e"   and a :: "'d" assumes "r\<inverse>\<inverse> b a" shows "r a b"
flow_has_derivative: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "(flow t0 x0 has_derivative (\<lambda>i. i *\<^sub>R f t (flow t0 x0 t))) (at t)"
cross_ratio_S_S_hyp2_hyp2_positive: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and b :: "proj2" assumes "are_endpoints_in_S p q a b" shows "0 < cross_ratio p q a b"
in_fold_lQuot: fixes v :: "'c list"   and w :: "'c list"   and L :: "'c list set" shows "(v \<in> fold lQuot w L) = (w @ v \<in> L)"
pseudonatural_transformation_whisker_right_axioms: shows "pseudonatural_transformation_whisker_right (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1"
conf_widen: fixes P :: "'b prog"   and h :: "heap"   and v :: "val"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> v :\<le> T"   and "P \<turnstile> T \<le> T'" shows "P,h \<turnstile> v :\<le> T'"
M_FullModel_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "M_FullModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP \<equiv> FullModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"
v2: fixes x :: "'a"   and r :: "complex"   and b :: "'b" shows "h x (r *\<^sub>C b) = r *\<^sub>C h x b"
set_spmf_of_set: fixes A :: "'c set" shows "set_spmf (spmf_of_set A) = (if finite A then A else {})"
real_of_ereal_image_greaterThanLessThan_iff: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal"   and d :: "ereal" shows "(real_of_ereal ` {a<..<b} = real_of_ereal ` {c<..<d}) = (b \<le> a \<and> d \<le> c \<or> a = c \<and> b = d)"
map_G_id: fixes x :: "('n, 'o, 'p, 'q, 'r, 's, 't) G" shows "map_G id id id id id id x = x"
IncrEnvironment_axioms: shows "IncrEnvironment jkbp clock_jview envObs clock_jviewInit clock_jviewIncr"
exhaust: fixes y :: "ty" obtains x1 :: "prim_ty" where "y = PrimT x1"   | x2 :: "ref_ty" where "y = RefT x2"
sgn1_zero_impl_arg_zero: fixes x :: "nat" assumes "sgn1 x = 0" shows "x = 0"
induct: fixes x1 :: "pi"   and x2 :: "pi"   and P :: "pi \<Rightarrow> pi \<Rightarrow> bool" assumes "x1 \<equiv>\<^sub>s x2"   and "\<And>P. P P P"   and "\<And>P Q. \<lbrakk>P \<equiv>\<^sub>s Q; P P Q\<rbrakk> \<Longrightarrow> P Q P"   and "\<And>P Q R. \<lbrakk>P \<equiv>\<^sub>s Q; P P Q; Q \<equiv>\<^sub>s R; P Q R\<rbrakk> \<Longrightarrow> P P R"   and "\<And>P Q. P (P \<oplus> Q) (Q \<oplus> P)"   and "\<And>P Q R. P ((P \<oplus> Q) \<oplus> R) (P \<oplus> Q \<oplus> R)"   and "\<And>P. P (P \<oplus> \<zero>) P"   and "\<And>P Q. P (P \<parallel> Q) (Q \<parallel> P)"   and "\<And>P Q R. P ((P \<parallel> Q) \<parallel> R) (P \<parallel> Q \<parallel> R)"   and "\<And>P. P (P \<parallel> \<zero>) P"   and "\<And>a P. P ([a\<frown>a]P) P"   and "\<And>x. P (<\<nu>x>\<zero>) \<zero>"   and "\<And>x y P. P (<\<nu>x><\<nu>y>P) (<\<nu>y><\<nu>x>P)"   and "\<And>x P Q. P (<\<nu>x>(P \<oplus> Q)) (<\<nu>x>P \<oplus> <\<nu>x>Q)"   and "\<And>x P Q. x \<sharp> P \<Longrightarrow> P (<\<nu>x>(P \<parallel> Q)) (P \<parallel> <\<nu>x>Q)"   and "\<And>x a y P. \<lbrakk>x \<noteq> a; x \<noteq> y\<rbrakk> \<Longrightarrow> P (<\<nu>x>a<y>.P) (a<y>.(<\<nu>x>P))"   and "\<And>x a b P. \<lbrakk>x \<noteq> a; x \<noteq> b\<rbrakk> \<Longrightarrow> P (<\<nu>x>a{b}.P) (a{b}.(<\<nu>x>P))"   and "\<And>x P. P (<\<nu>x>\<tau>.P) (\<tau>.(<\<nu>x>P))"   and "\<And>P. P (!P) (P \<parallel> !P)" shows "P x1 x2"
nonzero: shows "\<forall>z\<in>B. z \<noteq> 0"
wfP_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "wfP r"   and "\<And>x. \<forall>y. r y x \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
chamber_singleton: fixes w :: "'w" assumes "w \<in> W" shows "chamber (dual_order.poset_simplex_map \<P> {w})"
cases: fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and \<I> :: "('j, 'k) \<I>"   and D :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and a1 :: "('h, 'j, 'k \<Rightarrow> 'l) generat"   and a2 :: "('i, 'j, 'k \<Rightarrow> 'm) generat" assumes "eq_\<I>_generat A \<I> D a1 a2" obtains x :: "'h" and y :: "'i" where "a1 = Pure x" and "a2 = Pure y" and "A x y"   | out :: "'j" and c :: "'k \<Rightarrow> 'l" and c' :: "'k \<Rightarrow> 'm" where "a1 = IO out c" and "a2 = IO out c'" and "out \<in> outs_\<I> \<I>" and "\<And>input. input \<in> responses_\<I> \<I> out \<Longrightarrow> D (c input) (c' input)"
gap_def: fixes i :: "nat"   and t :: "nat" shows "gap i t \<equiv> pdec2 (the (eval r_state [t, i]))"
wrapB_def: shows "wrapB \<equiv> \<Lambda> f x. unbox\<cdot>x >>= f >>= box"
exhaust_sel: fixes phantom :: "('a, 'b) phantom" obtains "phantom = phantom (of_phantom phantom)"
nonintSI_def: shows "nonintSI \<equiv> \<forall>\<pi> \<pi>'. wfp UNIV \<pi> \<and> wfp UNIV \<pi>' \<and> stateOf \<pi> = s0 \<and> stateOf \<pi>' = s0 \<longrightarrow> (\<forall>i. (\<forall>j\<le>i. f (fst (\<pi> !! j)) (fst (\<pi>' !! j))) \<longrightarrow> g (fst (\<pi> !! i)) (fst (\<pi>' !! i)))"
notin_set_remove1: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "x \<notin> set xs" shows "x \<notin> set (remove1 y xs)"
F_is_totient_N: shows "F N = of_nat (totient N)"
degree_not_exists_monom: fixes p :: "'a mpoly"   and var :: "nat" assumes "p \<noteq> 0" shows "\<not> (\<exists>m\<in>monomials p. MPoly_Type.degree p var < lookup m var)"
1: fixes b :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and x :: "String.literal"   and y :: "(String.literal \<times> String.literal set list) list" assumes "alist_conds tao"   and "exe_ars_conds b"   and "dom (map_of (map (apsnd map_of) b)) = dom (map_of tao)"   and "(x, y) \<in> set b" shows "x \<in> fst ` set tao"
H_kat_to_kad: fixes p :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set" shows "rel_antidomain_kleene_algebra.H p x q = (rdom p \<subseteq> wp x (rdom q))"
ordIso_ordLeq_trans: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "r =o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
lb_lower_bound: shows "\<forall>g. (lb, g) \<in> gseq \<longrightarrow> g \<notin> BAD P"
RedNewArrayFail: fixes i :: "32 word"   and h :: "'heap"   and T :: "ty"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "0 \<le>s i"   and "allocate h (Array_type T (nat (sint i))) = {}" shows "extTA,P,t \<turnstile> \<langle>newA T\<lfloor>Val (Intg i)\<rceil>,(h, l)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt OutOfMemory),(h, l)\<rangle>"
fh: assumes "curr_method (stk, loc, C, M, pc, ics) \<noteq> clinit" shows "P,\<Phi> |- find_handler P xcp h frs sh [ok]"
all_conj_distrib: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<forall>x. P x \<and> Q x) = ((\<forall>x. P x) \<and> (\<forall>x. Q x))"
wf_list_graph_impl_axioms_locale_props: fixes V :: "'a list"   and E :: "('a \<times> 'a) list" shows "wf_list_graph_impl V E = (fst ` set E \<subseteq> set V \<and> snd ` set E \<subseteq> set V)"
sum_eq: shows "W (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2)) = sum W (P\<^sub>2 \<union> wrap B\<^sub>2)"
intros_2: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> Suc 0 \<Down> s[a/x]"
unit_simps_2: shows "local.dom \<eta>\<^sub>0 = src e\<^sub>0"
cis_conv_exp: fixes b :: "real" shows "cis b = exp (\<i> * complex_of_real b)"
get_scdom_component_subset_get_dom_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and c :: "(_) object_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> Shadow_DOM_DOM_Components.get_dom_component ptr \<rightarrow>\<^sub>r c" shows "set c \<subseteq> set sc"
trans: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "subseq a b"   and "subseq b c" shows "subseq a c"
times_canc_left: fixes h :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "h * p = h * q"   and "h \<noteq> 0" shows "p = q"
max_degree_def: fixes A :: "('b poly, 'c) sq_matrix" shows "max_degree A = (MAX (i, j). degree (to_fun A i j))"
port_range: fixes s :: "'b"   and e :: "'c"   and port :: "16 word" shows "s = (0::'b) \<and> e = (65535::'c) \<longrightarrow> port \<le> 65535"