hcju/isabelleps huggingface dataset:
IIA: fixes P :: "indi \<Rightarrow> alt \<Rightarrow> real"   and a :: "alt"   and b :: "alt"   and P' :: "indi \<Rightarrow> alt \<Rightarrow> real" assumes "\<And>i. (P i a < P i b) = (P' i a < P' i b)" shows "(F P a < F P b) = (F P' a < F P' b)"
implmt1D: fixes C :: "qtname"   and I :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<leadsto>1I" shows "C \<noteq> Object \<and> (? c:class G C: I \<in> set (superIfs c))"
p: shows "atom ` (p \<bullet> Vs) \<sharp>* Vs"
is_Zp_cauchy_def: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "is_Zp_cauchy s = (s \<in> carrier Zp\<^bsup>\<omega>\<^esup> \<and> (\<forall>n. \<exists>N. \<forall>m k. N < m \<and> N < k \<longrightarrow> eint n < val_Zp_dist (s m) (s k)))"
inA: fixes x :: "nat"   and y :: "nat" assumes "P x y" shows "x \<in> A"
simps_2: fixes l :: "('b \<times> nat) tree"   and uu :: "'b \<times> nat"   and r :: "('b \<times> nat) tree" shows "T_del_min \<langle>l, uu, r\<rangle> = T_merge l r + 1"
bl_sbin_sign_aux: fixes n :: "nat"   and w :: "int"   and bs :: "bool list" shows "hd (bin_to_bl_aux (Suc n) w bs) = (bin_sign (signed_take_bit n w) = - 1)"
slice_edges_Nil_split: fixes S :: "'node SDG_node set"   and cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list" assumes "slice_edges S cs (as @ as') = []" shows "slice_edges S cs as = [] \<and> slice_edges S (upd_cs cs as) as' = []"
absorb2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "semilattice_order f less_eq less"   and "less_eq b a" shows "f a b = b"
eexp_ereal: fixes x :: "real" shows "eexp (ereal x) = ereal (exp x)"
effect_lengthI: fixes h' :: "heap"   and h :: "heap"   and r :: "nat"   and a :: "'a array" assumes "h' = h"   and "r = Array.length h a" shows "effect (Array.len a) h h' r"
insert: fixes A :: "'k set"   and x :: "'k"   and g :: "'k \<Rightarrow> 'l" assumes "finite A"   and "x \<notin> A" shows "sum g (insert x A) = g x + sum g A"
Lp_Lq_duality_2: fixes f :: "'b \<Rightarrow> real"   and p :: "ennreal"   and M :: "'b measure"   and q :: "ennreal" assumes "f \<in> space\<^sub>N (\<LL> p M)"   and "1 / p + 1 / q = 1"   and "p = \<infinity> \<Longrightarrow> sigma_finite_measure M" shows "Norm (\<LL> p M) f = (\<Squnion>g\<in>{g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1}. LINT x|M. f x * g x)"
tllist_all2_TCons1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'a"   and ts :: "('a, 'c) tllist"   and ts' :: "('b, 'd) tllist" shows "tllist_all2 P Q (TCons x ts) ts' = (\<exists>x' ts''. ts' = TCons x' ts'' \<and> P x x' \<and> tllist_all2 P Q ts ts'')"
replicate_mset_def: fixes n :: "nat"   and x :: "'b" shows "replicate_mset n x = (add_mset x ^^ n) {#}"
num_reachable_zero: fixes G :: "'a graph"   and v :: "'a" assumes "wf_graph G"   and "num_reachable G v = 0" shows "succ_tran G v = {}"
wt_exprs_eq: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and t :: "expr list"   and U :: "tys" shows "(E,dt\<Turnstile>In3 t\<Colon>U) = (\<exists>Ts. U = Inr Ts \<and> E,dt\<Turnstile>t\<Colon>\<doteq>Ts)"
antimono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'c" shows "decseq f = (\<forall>n. f (Suc n) \<le> f n)"
dexn2: shows "TypeExn (mk_id mn_ cn_) \<in> defined_types s_"
space_exponential: fixes l :: "real" shows "space (exponential l) = UNIV"
simps_9: fixes bs :: "real list"   and p :: "fm" shows "Ifm bs (NOT p) = (\<not> Ifm bs p)"
lit_semantics_cases: fixes x1ca :: "'b \<Rightarrow> bool"   and xa :: "'b literal" shows "lit_semantics x1ca xa = (case xa of x2ba\<^sup>+ \<Rightarrow> x1ca x2ba | x2ca\<inverse> \<Rightarrow> \<not> x1ca x2ca)"
lift_mult_disjoint: fixes x :: "V"   and y :: "V"   and a :: "V" assumes "x \<noteq> y" shows "lift (a * x) a \<sqinter> lift (a * y) a = 0"
ideal_apow_n_val: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)" shows "Vr K v \<diamondsuit>\<^sub>p x = vp K v\<^bsup> Vr K v n_val K v x\<^esup>"
scalar_matrix_mult_iarrays_def: fixes c :: "'c"   and A :: "'c iarray iarray" shows "c *ssi A = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>i j. c * A !! i !! j)"
2: shows "regular (f \<sqinter> - (top * choose_arc h * f\<^sup>T\<^sup>\<star>) \<squnion> (f \<sqinter> top * choose_arc h * f\<^sup>T\<^sup>\<star>)\<^sup>T \<squnion> choose_arc h) \<and> regular (forest_components f) \<and> regular (choose_arc h\<^sup>T)"
add: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "additive f" shows "f (plus x y) = plus (f x) (f y)"
suminf_minus_initial_segment: fixes k :: "nat" assumes "summable f" shows "(\<Sum>n. f (n + k)) = (\<Sum>n. f n) - sum f {..<k}"
assms_2: shows "(i, j) \<in> set (identify_blocks A k)"
assms: shows "\<forall>A. square_mat A \<and> isDiagonal_mat A \<longrightarrow> (\<exists>P Q. P \<in> carrier_mat (dim_row A) (dim_row A) \<and> Q \<in> carrier_mat (dim_row A) (dim_row A) \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q))"
cindex_poly_inverse_add_cross: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < b"   and "poly (p * q) a \<noteq> 0"   and "poly (p * q) b \<noteq> 0" shows "cindex_poly a b q p + cindex_poly a b p q = cross (p * q) a b"
one_final_but_makeFaceFinal: fixes g :: "graph"   and E :: "(nat \<times> nat) set"   and f :: "face" assumes "minGraphProps g"   and "one_final_but g E"   and "E \<subseteq> \<E> f"   and "f \<in> \<F> g"   and "\<not> final f" shows "one_final (makeFaceFinal f g)"
extensional'_def: fixes A :: "'a set" shows "extensional' A \<equiv> {f. \<forall>x. x \<notin> A \<longrightarrow> f x = unity}"
act_conj_act_3: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<A>\<phi> \<^bold>& \<^bold>\<A>\<psi> \<^bold>\<rightarrow> \<^bold>\<A>(\<phi> \<^bold>& \<psi>) in v]"
assms_3: shows "\<D>\<^sub>\<circ> r = \<D>\<^sub>\<circ> s"
fold_if_in_set: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "fold (\<lambda>x m. if P x m then x else m) xs x \<in> set (x # xs)"
not_add_less_right: fixes y :: "V"   and x :: "V" assumes "Ord y" shows "\<not> x + y < x"
vblcase_def: fixes a :: "'a"   and f :: "vbl \<Rightarrow> 'a"   and n :: "vbl" shows "(case n of zeroX \<Rightarrow> a | nextX x \<Rightarrow> f x) = (SOME z. (n = zeroX \<longrightarrow> z = a) \<and> (\<forall>x. n = nextX x \<longrightarrow> z = f x))"
one_in_unit_circle_set: shows "1\<^sub>h \<in> circline_set unit_circle"
has_field_derivative_cosh: fixes g :: "'a \<Rightarrow> 'a"   and Db :: "'a"   and x :: "'a"   and s :: "'a set" assumes "(g has_field_derivative Db) (at x within s)" shows "((\<lambda>x. cosh (g x)) has_field_derivative sinh (g x) * Db) (at x within s)"
Subspace_is_findim: fixes U :: "'b set" assumes "Subspace U" shows "\<exists>ms. set ms \<subseteq> U \<and> RSpan ms = U"
hyps: shows "map ($ R) (map ($ S) xl) = map ($ (\<lambda>x. $ R (S x))) xl"
mult_is_mul: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "mult r = mul r"
padd_correct: fixes a :: "int"   and b :: "int"   and p\<^sub>1 :: "int \<times> int \<times> int"   and p\<^sub>2 :: "int \<times> int \<times> int" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curvep a b p\<^sub>1"   and "on_curvep a b p\<^sub>2" shows "make_affine (padd a p\<^sub>1 p\<^sub>2) = local.add a (make_affine p\<^sub>1) (make_affine p\<^sub>2)"
front_tickFree_append: fixes s :: "'b event list"   and t :: "'b event list" assumes "tickFree s"   and "front_tickFree t" shows "front_tickFree (s @ t)"
simps_2: fixes P :: "'state"   and t :: "('b, 'pred, 'act) Tree" shows "valid_Tree P (tNot t) = (\<not> valid_Tree P t)"
Gromov_boundary_rep_converging: fixes x :: "'c Gromov_completion" assumes "x \<in> Gromov_boundary" shows "Gromov_converging_at_boundary (rep_Gromov_completion x)"
induct: fixes P :: "('a simple_match \<Rightarrow> 'a word \<times> nat) \<Rightarrow> 'a simple_rule list \<Rightarrow> bool"   and a0 :: "'a simple_match \<Rightarrow> 'a word \<times> nat"   and a1 :: "'a simple_rule list" assumes "\<And>uu_. P uu_ []"   and "\<And>sel m uv_ ss. P sel ss \<Longrightarrow> P sel (SimpleRule m uv_ # ss)" shows "P a0 a1"
space_PiM: fixes I :: "'a set"   and M :: "'a \<Rightarrow> 'b measure" shows "space (Pi\<^sub>M I M) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
prems_2: shows "k \<in> wordinterval_to_set (snd e)"
sinvar_no_offending: fixes G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar G nP" shows "set_offending_flows G nP = {}"
While: fixes r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and P :: "'d set"   and b :: "'d set"   and c :: "('d, 'e, 'f) com"   and A :: "'d set" assumes "wf r"   and "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<sigma>} \<inter> P \<inter> b) c ({t. (t, \<sigma>) \<in> r} \<inter> P),A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P While b c (P \<inter> - b),A"
cring_simprules_5: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
simps: fixes a :: "'a"   and hts :: "'a hterm list" shows "term_of_hterm (HApp a hts) = App a (terms_of_hterms hts)"
intros_5: fixes P :: "pi"   and Rs :: "residual"   and a :: "name"   and b :: "name" assumes "P \<longmapsto> Rs"   and "a \<noteq> b" shows "[a\<noteq>b]P \<longmapsto> Rs"
wp_conseq: fixes c :: "com"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "local.wp c P s"   and "\<And>s. P s \<Longrightarrow> Q s" shows "local.wp c Q s"
assms: shows "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
ereal_of_enat_less_iff: fixes m :: "enat"   and n :: "enat" shows "(ereal_of_enat m < ereal_of_enat n) = (m < n)"
\<tau>move2_pc_length_compE2: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk e pc xcp" shows "pc < length (compE2 e)"
green_typeI_chain_axioms_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" shows "green_typeI_chain_axioms i j F two_chain s \<equiv> valid_typeI_division s two_chain \<and> (\<forall>twoC\<in>two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j)"
enc_eq_welldef: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and w' :: "'b list"   and I' :: "(nat + nat set) list"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.enc \<Sigma> (w, I) = formula.enc \<Sigma> (w', I')"   and "formula.wf_formula \<Sigma> (length I) \<phi>"   and "formula.wf_interp_for_formula \<Sigma> (w, I) \<phi>"   and "formula.wf_interp_for_formula \<Sigma> (w', I') \<phi>" shows "formula.satisfies \<Sigma> (w, I) \<phi> = formula.satisfies \<Sigma> (w', I') \<phi>"
H3_1: shows "u3 \<in> borel_measurable M"
baEANow: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and n :: "nat"   and \<phi> :: "'cmp \<Rightarrow> bool" assumes "eval c t t' n [\<phi>]\<^sub>b"   and "\<parallel>c\<parallel>\<^bsub>t n\<^esub>" shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t n)"
process_rule_6_futures: fixes ys :: "'e list"   and P :: "'e process"   and xs :: "'e list"   and Y :: "'e set" assumes "ys \<in> future_divergences P xs" shows "(ys, Y) \<in> futures P xs"
keys_bind_pmf_aux: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf"   and A :: "'d set" shows "Mapping.keys (bind_pmf_aux p f A) = (\<Union>x\<in>A. set_pmf (f x))"
base_correctness: fixes V :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and R :: "'a option list set"   and z :: "'a option list" assumes "card V = 1"   and "rwf_query n V Q Qn"   and "R = genericJoin V Q Qn" shows "(z \<in> genericJoin V Q Qn) = (wf_tuple n V z \<and> (\<forall>(A, X)\<in>Q. restrict A z \<in> X) \<and> (\<forall>(A, X)\<in>Qn. restrict A z \<notin> X))"
interaction_bounded_by_exec_gpv_bad_count: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('d, 'a, 'b) gpv"   and n :: "enat"   and s :: "'s"   and bad :: "'s \<Rightarrow> bool"   and count :: "'s \<Rightarrow> nat"   and k :: "real" assumes "interaction_bounded_by consider gpv n"   and "I s"   and "\<not> bad s"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> Suc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; \<not> consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<And>s' x. \<lbrakk>I s'; \<not> bad s'; enat (count s') < n + enat (count s); consider x; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> spmf (map_spmf (bad \<circ> snd) (callee s' x)) True \<le> k"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; \<not> bad s; bad s'; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> consider x"   and "0 \<le> k"   and "\<I> \<turnstile>g gpv \<surd>" shows "ennreal (spmf (map_spmf (bad \<circ> snd) (exec_gpv callee gpv s)) True) \<le> ennreal k * ennreal_of_enat n"
continuous_apply: fixes f :: "'a set \<Rightarrow> 'b set"   and C :: "nat \<Rightarrow> 'a set" assumes "continuous f"   and "chain C" shows "f (natUnion C) = natUnion (f \<circ> C)"
rep_eq: fixes x :: "int" shows "Rep_uint32 (uint32_of_int x) = word_of_int x"
inject: fixes x1 :: "'f"   and y1 :: "'f" shows "(Node x1 = Node y1) = (x1 = y1)"
stable_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "TruePref_tax.stable_on ds X = (TruePref_tax.individually_rational_on ds X \<and> TruePref_tax.stable_no_blocking_on ds X)"
regset_spec: fixes xs :: "'b list"   and d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "(xs \<in> regset d i j k) = ((\<forall>n\<in>set (butlast (trace d i xs)). n < k) \<and> deltas d xs i = j)"
mds\<^sub>s_def: shows "mds\<^sub>s \<equiv> \<lambda>m. {}"
dropWhile_Cons: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and ys :: "'b list" assumes "dropWhile P xs = x # ys" shows "\<not> P x"
space_as_set_inverse: fixes x :: "'d ccsubspace" shows "Abs_clinear_space (space_as_set x) = x"
bij_perm_swap_iff: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'c" shows "bij (perm_swap a b f) = bij f"
drop_set_of_apprD: fixes xs :: "real list"   and XS :: "'b list"   and n :: "nat" assumes "xs \<in> set_of_appr XS" shows "drop n xs \<in> set_of_appr (drop n XS)"
ask_option_def: fixes ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> ('a, 'b) optionT" shows "ask_option ask f = OptionT (ask (\<lambda>r. run_option (f r)))"
from_nat_inj: fixes i :: "nat"   and j :: "nat" assumes "i < CARD('b)"   and "j < CARD('b)"   and "Bij_Nat.from_nat i = Bij_Nat.from_nat j" shows "i = j"
Y_ne_X: shows "mset (y # ys) \<noteq> mset (x # xs)"
main_step_aux1_R: fixes qs :: "real poly list" assumes "check_all_const_deg qs = True" shows "set (find_consistent_signs_R qs) = consistent_sign_vectors_R qs UNIV"
is_word_Suc_j_b: assumes "n \<noteq> length D" shows "is_word (take (Suc j) b)"
assms_2: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "plossless_gpv \<I>' (callee s x)"
intg: shows "\<forall>a b. g integrable_on cbox a b"
reachable_not_node: fixes r :: "'b"   and g :: "'b ugraph" assumes "r \<notin> nodes g" shows "reachable_nodes g r = {r}"
Int_Un_eq_1: fixes S :: "'d set"   and T :: "'d set" shows "S \<inter> T \<union> S = S"
observable3_def: fixes ob :: "'e \<Rightarrow> 'f"   and P :: "'e set" shows "observable3 ob P \<equiv> ob -` ob ` P \<subseteq> P"
run_to_execute: fixes c :: "'b Heap"   and h :: "heap"   and \<sigma>' :: "heap option"   and r :: "'b" assumes "run c (Some h) \<sigma>' r" shows "if \<sigma>' = None then execute c h = None else execute c h = Some (r, the \<sigma>')"
equal_multiset_def: fixes A :: "'a multiset"   and B :: "'a multiset" shows "equal_class.equal A B = (A = B)"
emeasure_P: fixes J :: "'i set"   and K :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "J \<subseteq> K"   and "finite K"   and "K \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emeasure (P K) (emb K J X) = emeasure (P J) X"
simps_1: fixes xs :: "nat set" shows "omit xs [] = []"
gram_schmidt_fs_lin_indpt_axioms: shows "gram_schmidt_fs_lin_indpt n fs"
simps_1: shows "complex_roots_to_real_factorization [] = []"
countableI: fixes f :: "'c \<Rightarrow> nat"   and S :: "'c set" assumes "inj_on f S" shows "countable S"
prSubstAbs_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'c"   and SEM :: "('d, 'e, 'f, 'sort, 'g, 'b, 'h) semDom_scheme" shows "prSubstAbs g gA SEM \<equiv> \<forall>ys Y y us s A val. wls (asSort ys) Y \<and> wlsAbs (us, s) A \<and> sWlsVal SEM val \<longrightarrow> gA (A $[Y / y]_ys) val = gA A (val (y := g Y val)_ys)"
local_state_of_finite_restricted_global_state_is_finite_1: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r' :: "'b"   and ls :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and r :: "'b" assumes "s r' = Some ls"   and "finite (RID\<^sub>G (s(r := None)))"   and "r \<noteq> r'" shows "finite (RID\<^sub>L ls)"
clock_repMC_cec: fixes t :: "'s Trace"   and aec :: "'s odlist"   and a :: "'a"   and cec :: "'s odlist"   and x :: "'s"   and y :: "'s" assumes "t \<in> Clock.jkbpC"   and "toSet aec = agent_abs a t"   and "toSet cec = common_abs t"   and "x \<in> toSet aec"   and "y \<in> toSet aec" shows "(x, y) \<in> relations (clock_repMC (toSet cec)) a"
le_zero_eq: fixes n :: "'f multiset" shows "(n \<subseteq># {#}) = (n = {#})"
rotate_conv_mod: fixes n :: "nat"   and xs :: "'b list" shows "rotate n xs = rotate (n mod length xs) xs"
SourcesA22_L1: shows "Sources level1 sA22 = {sA11, sA22, sA23, sA31, sA32, sA41}"
lift_eligible_literal: fixes L :: "'a literal"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and \<theta> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "eligible_literal L C \<sigma>"   and "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>" shows "eligible_literal L C \<theta>"
ordinal_number_mem_ODnums: fixes D :: "'c Order" assumes "Worder D" shows "ordinal_number D \<in> ODnums"
ta_hb_consistent_lappendI: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) list"   and E' :: "('b \<times> ('c, 'b) obs_event action) llist"   and E'' :: "('b \<times> ('c, 'b) obs_event action) llist" assumes "ta_hb_consistent P E E'"   and "ta_hb_consistent P (E @ list_of E') E''"   and "lfinite E'" shows "ta_hb_consistent P E (lappend E' E'')"
UP_subring_eval': fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "a \<in> S" shows "to_fun g a = to_function (R\<lparr>carrier := S\<rparr>) g a"
fimage_fdomain: fixes r :: "V" shows "r `\<^sub>\<bullet> \<D>\<^sub>\<bullet> r = \<R>\<^sub>\<bullet> r"
gcd_poly_pseudo_mod: fixes q :: "'a poly"   and p :: "'a poly" assumes "q \<noteq> 0"   and "content p = (1::'a)"   and "content q = (1::'a)" shows "gcd p q = gcd q (primitive_part (pseudo_mod p q))"
reaches_eq_imp: fixes x :: "'e"   and y :: "'e" shows "eq_imp (\<lambda>r' s. \<Union> (ran ` obj_fields ` set_option (sys_heap s r'))) (x reaches y)"
stake_cycle_le: fixes u :: "'c list"   and n :: "nat" assumes "u \<noteq> []"   and "n < length u" shows "stake n (cycle u) = take n u"
simps_2: fixes p :: "linear_poly"   and c :: "rat"   and q :: "linear_poly"   and d :: "QDelta" shows "compatible_cs (Le_Constraint Lt_Rel p c) (Le_Constraint Leq_Rel q d) = (q = p \<and> qdfst d = c)"
Sim: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "(\<Psi>', R, S) \<in> Rel" shows "\<Psi>' \<rhd> R \<leadsto>[Rel] S"
h6: shows "lincomb a A = \<zero>\<^bsub>V\<^esub>"
mem_sphere: fixes y :: "'d"   and x :: "'d"   and e :: "real" shows "(y \<in> sphere x e) = (dist x y = e)"
match_list_comb_list_comb_none_name: fixes name :: "name"   and name' :: "name"   and ps :: "term list"   and vs :: "'c list" assumes "name \<noteq> name'" shows "match (name $$ ps) (name' $$ vs) = None"
dvd_div_eq_iff: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c dvd a"   and "c dvd b" shows "(a div c = b div c) = (a = b)"
SUP_identity_eq: fixes Sup :: "'f set \<Rightarrow> 'f"   and A :: "'f set" shows "Sup ((\<lambda>x. x) ` A) = Sup A"
simps_1: fixes x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and y2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "(Assume x2 = Assume y2) = (x2 = y2)"
add_uminus_conv_diff: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus b) = minus a b"
HPT_target_source: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "t \<in> TS"   and "target t \<in> States A"   and "A \<in> SAs (HA ST)" shows "source t \<in> States A"
minus_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier K"   and "y \<in> carrier K" shows "x \<ominus>\<^bsub>K\<^esub> y \<in> carrier K"
DERIV_continuous_on: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 'a" assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative D x) (at x within s)" shows "continuous_on s f"
get_ins_def: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int" shows "get_ins M l = fst (snd M)\<down>l"
rtrancl_impl_def: fixes gen_succ :: "('c \<times> 'c) list \<Rightarrow> 'c list \<Rightarrow> 'c list"   and un :: "'c list \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and emp :: "'d"   and rel :: "('c \<times> 'c) list" shows "rtrancl_impl gen_succ un memb emp rel = (let succ = gen_succ rel; n = length rel in (\<lambda>as. relpow_impl succ un memb as emp n))"
binary_endofunctor_axioms: shows "binary_endofunctor (\<cdot>) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star> snd \<mu>\<nu>)"
ex_inf: fixes x :: "'a"   and y :: "'a" shows "\<exists>inf. is_inf x y inf"
flattening_lemma_ii: fixes \<pi> :: "'a strips_operator list list"   and \<Pi> :: "'a strips_problem"   and G :: "'a \<Rightarrow> bool option"   and I :: "'a \<Rightarrow> bool option" assumes "\<forall>ops\<in>set \<pi>. \<exists>op. ops = [op] \<and> is_valid_operator_strips \<Pi> op"   and "G \<subseteq>\<^sub>m execute_parallel_plan I \<pi>" shows "G \<subseteq>\<^sub>m execute_serial_plan I (concat \<pi>)"