hcju/isabelleps huggingface dataset:
infinite_part1: fixes V :: "'a set" assumes "infinite V" shows "infinite (part1 V)"
generalized_fw_join_2_nomatch: fixes bm :: "'a simple_match"   and p :: "('a, 'b) simple_packet_scheme"   and as :: "('a simple_match \<times> 'c) list"   and bd :: "'d"   and bs :: "('a simple_match \<times> 'd) list" assumes "\<not> simple_matches bm p" shows "generalized_sfw (generalized_fw_join as ((bm, bd) # bs)) p = generalized_sfw (generalized_fw_join as bs) p"
ceiling_zero: shows "\<lceil>0::'c\<rceil> = 0"
successive_successors: fixes xs :: "nat list"   and ys :: "nat list"   and a :: "nat"   and b :: "nat"   and c :: "nat"   and zs :: "nat list"   and P :: "nat \<times> nat \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" assumes "xs = ys @ a # b # c # zs"   and "successive P (arcs i j xs)" shows "\<not> P (a, b) \<or> \<not> P (b, c)"
has_start_end_points_path_iff: fixes x :: "'a" shows "has_start_end_points_path x = (has_start_points_path x \<and> has_end_points_path x)"
ground_fmla_aux_simps_7: fixes x :: "name"   and A :: "fmla"   and S :: "atom set" shows "ground_fmla_aux (exi x A) S = ground_fmla_aux A (S \<union> {atom x})"
f8: shows "s \<otimes> s' \<otimes> s'' \<otimes> (s' \<otimes> s'' \<otimes> (r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (r' \<otimes> r'')) = s \<otimes> s' \<otimes> s'' \<otimes> (s' \<otimes> s'' \<otimes> (r \<otimes> r'')) \<oplus> s \<otimes> s' \<otimes> s'' \<otimes> (s \<otimes> s'' \<otimes> (r' \<otimes> r''))"
iT_Minus_Plus_commute: fixes n :: "nat"   and k :: "nat"   and I :: "nat set" assumes "n \<le> k" shows "k \<ominus> (I \<oplus> n) = k - n \<ominus> I"
meth_C: shows "P \<turnstile> C sees M: Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"
cs_select_is_cs_2: fixes \<pi> :: "nat \<Rightarrow> 'n"   and xs :: "'n list"   and k :: "nat" assumes "is_path \<pi>"   and "xs \<noteq> []"   and "xs \<prec> cs\<^bsup>\<pi>\<^esup> k" shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> (\<pi> \<exclamdown> xs)"
rel_prod2_def: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'e" shows "rel_prod2 R a = (\<lambda>(c, b). R a b)"
last_mem_excluded: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "d_gchain G (Suc n) f"   and "f n \<noteq> f (Suc n)" shows "f (Suc n) \<notin> f ` {i. i \<le> n}"
assms_3: fixes x :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "h \<turnstile> ok g x"
finite_measure_axioms_def: fixes M :: "'b measure" shows "finite_measure_axioms M \<equiv> emeasure M (space M) \<noteq> top"
flatten_monom_mult: fixes c :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'b" shows "flatten (punit.monom_mult c t p) = punit.monom_mult (1::'b) t (c * flatten p)"
lookup_update: fixes ks :: "'d list"   and v :: "'c"   and t :: "('d, 'c) trie"   and ks' :: "'d list" shows "lookup_trie (update_trie ks v t) ks' = (if ks = ks' then Some v else lookup_trie t ks')"
trace_parallel_plan_strips_not_nil: fixes I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" shows "trace_parallel_plan_strips I \<pi> \<noteq> []"
add_ac_2: fixes a :: "'a"   and b :: "'a" shows "a + b = b + a"
Ord_def: fixes k :: "hf" shows "Ord k \<equiv> Transset k \<and> HBall k Transset"
inj_freshenLC: fixes i :: "nat" shows "inj (freshenLC i)"
left_edge_explicit: shows "(\<lambda>y. twoC (0, y)) = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))"
indices_append: fixes xs :: "('c \<times> ('c, 'd) operation) event list"   and ys :: "('c \<times> ('c, 'd) operation) event list" shows "indices (xs @ ys) = indices xs @ indices ys"
stable_imp_take_bit_eq: fixes a :: "'a"   and n :: "nat" assumes "a div (2::'a) = a" shows "take_bit n a = (if even a then 0::'a else (2::'a) ^ n - (1::'a))"
share_reconstruct_xor: fixes x :: "bool"   and y :: "bool" shows "share x \<bind> (\<lambda>(a1, a2). share y \<bind> (\<lambda>(b1, b2). xor_protocol (a1, b1) (a2, b2) \<bind> (\<lambda>(a, b). reconstruct (a, b)))) = xor_evaluate x y"
stoch_proc_subalg_nat_filt: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'c"   and n :: "nat" assumes "borel_adapt_stoch_proc nat_filtration X" shows "subalgebra (nat_filtration n) (stoch_proc_filt M X borel n)"
TUren: fixes I :: "('sf, 'a, 'sz) interp"   and \<theta> :: "('sf, 'sz) trm"   and x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "dsafe \<theta>" shows "dterm_sem I (TUrename x y \<theta>) \<nu> = dterm_sem I \<theta> (Radj x y \<nu>)"
that: fixes S :: "'b set" assumes "S \<in> lmeasurable"   and "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> S"   and "Sigma_Algebra.measure lebesgue S \<le> 2 * e * (2 * m) ^ (DIM('b) - 1)" shows "thesis"
runit'_simps_3: fixes f :: "'b" assumes "D.ide f" shows "trg\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[f] = trg\<^sub>D f"
assms_6: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and c :: "'a" assumes "\<And>c. P c scrut"   and "\<And>c. P c e1"   and "\<And>c. P c e2" shows "P c (scrut ? e1 : e2)"
merge_union: fixes A :: "'s"   and B :: "'s"   and f :: "'a \<Rightarrow> 'm" shows "merge (A \<^bold>\<union> B) f = bind (merge A f) (\<lambda>A'. bind (merge B f) (\<lambda>B'. return (A' \<^bold>\<union> B')))"
simps: fixes R :: "('d, 'e) ring_scheme"   and Ps :: "('f multiset \<Rightarrow> 'd) list"   and i :: "'f" assumes "ring R" shows "ring.indexed_eval R Ps i = ring.indexed_eval_aux R (rev Ps) i"
trans: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c'' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "c \<Rightarrow>\<^sup>* c'"   and "c' \<Rightarrow>\<^sup>* c''"   and "P c c'"   and "P c' c''" shows "P c c''"
bij_complete: fixes C :: "'c set"   and A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'c" assumes "finite C"   and "A \<subseteq> C"   and "B \<subseteq> C"   and "bij_betw f A B" obtains f' :: "'c \<Rightarrow> 'c" where "bij_betw f' C C" and "\<And>x. x \<in> A \<Longrightarrow> f' x = f x" and "\<And>x. x \<in> C - A \<Longrightarrow> f' x \<in> C - B"
scaleR_add_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "real"   and x :: "'b"   and y :: "'b" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (plus x y) = plus (scaleR a x) (scaleR a y)"
"3_4": fixes x :: "'a" assumes "x \<in> set (x__ # xs)"   and "P x" shows "False"
lemma_order_pderiv1: fixes a :: "'b"   and n :: "nat"   and q :: "'b poly" shows "pderiv ([:- a, 1::'b:] ^ Suc n * q) = [:- a, 1::'b:] ^ Suc n * pderiv q + smult (of_nat (Suc n)) (q * [:- a, 1::'b:] ^ n)"
nat_sum: fixes a :: "int"   and b :: "int" assumes "0 \<le> a"   and "0 \<le> b" shows "nat (a + b) = nat a + nat b"
simps: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "rpo_optim ground_heads_var gt_sym extf arity_sym arity_var" shows "rpo_optim.gt ground_heads_var gt_sym extf a1 a2 = ((\<exists>ti t s. a1 = t \<and> a2 = s \<and> ti \<in> set (args t) \<and> (rpo_optim.gt ground_heads_var gt_sym extf ti s \<or> ti = s)) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> ground_heads.gt_hd gt_sym ground_heads_var (head t) (head s) \<and> rpo_basis.chkvar t s \<and> rpo_optim.chkargs (rpo_optim.gt ground_heads_var gt_sym extf) t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> rpo_optim.chkargs (rpo_optim.gt ground_heads_var gt_sym extf) t s \<and> (\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (rpo_optim.gt ground_heads_var gt_sym extf) (args t) (args s))))"
double_of_integer_def: shows "double_of_integer \<equiv> map_fun int_of_integer id double_of_int"
uw\<theta>: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" assumes "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>" shows "partial_magma.ide V w \<and> partial_magma.in_hom V \<theta> (H f w) u"
the_Transaction_inverse: fixes x :: "transaction" shows "transaction_merkle.Transaction' (transaction_merkle.the_Transaction x) = x"
is_red_adds_iff: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and p :: "'t \<Rightarrow>\<^sub>0 'd" shows "is_red F p = (\<exists>f\<in>F. \<exists>v\<in>keys p. f \<noteq> 0 \<and> lt f adds\<^sub>t v)"
one_fun_var2: fixes x :: "'c" assumes "x \<notin> E" shows "1 x = \<bottom>"
hcomp_conc_unit1: fixes P :: "conc_interface" shows "\<epsilon>\<^sub>c \<otimes>\<^sub>c P \<simeq> P"
cod_in_codomains: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "codomains f \<noteq> {}" shows "cod f \<in> codomains f"
part_overlap_eq: fixes x :: "i"   and y :: "i" assumes "EM" shows "P x y = (\<forall>z. O z x \<longrightarrow> O z y)"
span_minimal: fixes T :: "'b set"   and S :: "'b set" assumes "T \<subseteq> U\<^sub>V\<^sub>S"   and "S \<subseteq> T"   and "local.subspace T" shows "local.span S \<subseteq> T"
wf_until_aux_Cons: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and pos :: "bool"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula"   and a :: "nat \<times> event_data option list set \<times> event_data option list set"   and aux :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and ne :: "nat" assumes "wf_until_auxlist \<sigma> V n R pos \<phi> I \<psi> (a # aux) ne" shows "wf_until_auxlist \<sigma> V n R pos \<phi> I \<psi> aux (Suc ne)"
Abs_nd_fun_inject: fixes x :: "'b \<Rightarrow> 'b set"   and y :: "'b \<Rightarrow> 'b set" assumes "s2p {f. s2p UNIV f} x"   and "s2p {f. s2p UNIV f} y" shows "(Abs_nd_fun x = Abs_nd_fun y) = (x = y)"
insert_subset: fixes x :: "'g"   and A :: "'g set"   and B :: "'g set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
eq1: shows "lookup_pair (sort_oalist xs) = lookup_dflt xs"
Dube_ge_d: fixes d :: "nat"   and n :: "nat" shows "d \<le> Dube n d"
floor_correct: fixes x :: "'b" shows "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)"
copy_first_matrix_dim_2: fixes nr :: "nat"   and nc :: "nat" shows "dim_col (copy_first_matrix nr nc) = nc"
amult_0_l: fixes z :: "int" shows "0 * ant z = 0"
C_SC_equiv: fixes a :: "'p"   and b :: "'p"   and c :: "'p"   and d :: "'p" shows "(a b \<equiv> c d) = SC {a, b} {c, d}"
nodes_semi_def: fixes G :: "nat graph_rec" assumes "cfg_doms G" shows "cfg_doms.nodes_semi G \<equiv> ((rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G)), nodes_le, nodes_sup)"
ad3: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) \<le> - (x \<sqinter> - - y)"
insert_DiffM: fixes x :: "'d"   and M :: "'d multiset" assumes "x \<in># M" shows "add_mset x (M - {#x#}) = M"
boundedI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> b \<cdot> c"
icard_Diff_singleton_if: fixes A :: "'a set"   and x :: "'a" shows "icard (A - {x}) = (if x \<in> A then icard A - 1 else icard A)"
bdd_below_Ici: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.preorder less_eq less" shows "preorder.bdd_below less_eq (ord.atLeast less_eq a)"
P_D2_H2_d_linear: fixes d :: "D2" shows "Linear_order (P_D2_H2_d d)"
F_one: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "abel_semigroup_set f z" shows "f (abel_semigroup_set.F f z g A) z = abel_semigroup_set.F f z g A"
prems_1: shows "sharing_consistent \<S> \<O> (x # sb)"
poincare_map_step_flow: fixes P :: "'a set"   and x :: "'a"   and t :: "real" assumes "returns_to P x"   and "closed P"   and "\<And>s. \<lbrakk>0 < s; s \<le> t\<rbrakk> \<Longrightarrow> flow0 x s \<notin> P"   and "0 < t"   and "t \<in> existence_ivl0 x" shows "poincare_map P (flow0 x t) = poincare_map P x"
False: shows "\<not> \<alpha> \<unlhd> \<beta>"
grounding_\<rho>_def: fixes x :: "'v" shows "grounding_\<rho> x = (let s = Hd (Sym (SOME f. f \<in> ground_heads_var x)) in apps s (replicate (the_enat (arity s - arity_var x)) s))"
rev_drop: fixes i :: "nat"   and xs :: "'c list" shows "rev (drop i xs) = take (length xs - i) (rev xs)"
ps_equal_to_snapshot_def: fixes c :: "('d, 'e, 'f, 'g) configuration_scheme"   and c' :: "('d, 'e, 'h, 'i) configuration_scheme" shows "ps_equal_to_snapshot c c' \<equiv> \<forall>p. Some (states c p) = ps c' p"
assms_20: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
map_rec: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "map f xs = rec_list [] (\<lambda>x _. (#) (f x)) xs"
ip_cidr_set_def: fixes i :: "'a word"   and r :: "nat" shows "ip_cidr_set i r \<equiv> {j. i && ~~ mask (LENGTH('a) - r) = j && ~~ mask (LENGTH('a) - r)}"
oprod_antisym: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "antisym r"   and "antisym r'" shows "antisym (r *o r')"
prems: fixes i :: "'a" assumes "i \<in> insert i__ I" shows "poly_fun (f i)"
inj_map: fixes f :: "'b \<Rightarrow> 'c" shows "inj (map f) = inj f"
set_restrict_spmf: fixes p :: "'h spmf"   and A :: "'h set" shows "set_spmf (p \<upharpoonleft> A) = set_spmf p \<inter> A"
raw_match_extend_map_preserve': fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs'' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'"   and "subs' \<subseteq>\<^sub>m subs''" shows "tsubst t (convert_subs subs') = tsubst t (convert_subs subs'')"
simps_2: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "pre (x ::= a) Q = (\<lambda>s. eSuc (Q (s[a/x])))"
Unit_def: fixes R :: "('h, 'i) Ring_scheme"   and a :: "'h" shows "Unit R a = (a \<in> carrier R \<and> (\<exists>b\<in>carrier R. a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = 1\<^sub>r\<^bsub>R\<^esub>))"
simps: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t1 :: "('b \<times> 'c) tree"   and t2 :: "('b \<times> 'c) tree" assumes "Set2_Join join inv" shows "Set2_Join.inter join t1 t2 = (if t1 = \<langle>\<rangle> then \<langle>\<rangle> else if t2 = \<langle>\<rangle> then \<langle>\<rangle> else case t1 of \<langle>l1, (a, x), r1\<rangle> \<Rightarrow> let (l2, b, r2) = Set2_Join.split join t2 a; l' = Set2_Join.inter join l1 l2; r' = Set2_Join.inter join r1 r2 in if b then join l' a r' else Set2_Join.join2 join l' r')"
simps_3: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and xs :: "'b list" shows "successively P (x # y # xs) = (P x y \<and> successively P (y # xs))"
assms_2: fixes \<beta> :: "V" assumes "\<beta> \<in> elts \<alpha>" shows "\<beta> + \<alpha> = \<alpha>"
lOfWax_def: fixes \<sigma> :: "'e" shows "lOfWax \<sigma> = Fn (Tag \<sigma>) [rOfWax \<sigma>]"
prems: shows "is_subobj P (D, D # rev (C' # Cs'))"
length_two_unique: assumes "length \<GG> = 2" shows "\<GG> = [{\<one>\<^bsub>G\<^esub>}, carrier G]"
distinct_53: fixes x3 :: "nat \<times> bool"   and x71 :: "'b"   and x72 :: "nat" shows "Val x3 \<noteq> PubConstAtom x71 x72"
AbstrLevels_S12_A81: fixes i :: "AbstrLevelsID" assumes "sS12 \<in> AbstrLevel i" shows "sA81 \<notin> AbstrLevel i"
t1: fixes t :: "'a euclidean_space" assumes "\<And>x. rep x \<in> B" shows "Abs_euclidean_space (Rep_euclidean_space t) = t"
CompFun_ChiRel: fixes S1 :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and SA :: "('a, 'b, 'c) seqauto"   and S2 :: "'a" assumes "S1 \<in> HAStates A"   and "SA \<in> the (CompFun A S1)"   and "S2 \<in> States SA" shows "(S1, S2) \<in> ChiRel A"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and sum :: "'c + 'd" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((\<forall>x1. sum = Inl x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum = Inr x2 \<longrightarrow> P (f2 x2)))"
naturality: fixes f :: "'a" assumes "Adj.C.arr f" shows "Adj'.\<epsilon> (Adj.C.cod f) \<cdot>\<^sub>C Adj'.FG.map f = Adj.C.map f \<cdot>\<^sub>C Adj'.\<epsilon> (Adj.C.dom f)"
nstri_mul_ext_map: fixes ss :: "'a list"   and ts :: "'a list"   and order :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and order' :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s t. \<lbrakk>s \<in> set ss; t \<in> set ts; fst (order s t)\<rbrakk> \<Longrightarrow> fst (order' (f s) (f t))"   and "\<And>s t. \<lbrakk>s \<in> set ss; t \<in> set ts; snd (order s t)\<rbrakk> \<Longrightarrow> snd (order' (f s) (f t))"   and "snd (mul_ext order ss ts)" shows "snd (mul_ext order' (map f ss) (map f ts))"
n1i1: shows "n1 = pickFT cl \<and> i1 = 0"
upper_exp_float_interval: fixes p :: "nat"   and x :: "float interval" shows "upper (exp_float_interval p x) = ub_exp p (upper x)"
expand\<^sub>T_def: fixes n_ns :: "'b node \<times> 'b node set" shows "expand\<^sub>T n_ns \<equiv> REC\<^sub>T (\<lambda>expand (n, ns). if new n = {} then if \<exists>n'\<in>ns. old n' = old n \<and> next n' = next n then RETURN (name n, upd_incoming n ns) else expand (\<lparr>name = expand_new_name (name n), incoming = {name n}, new = next n, old = {}, next = {}\<rparr>, {n} \<union> ns) else SPEC (\<lambda>x. x \<in> new n) \<bind> (\<lambda>\<phi>. let n = n\<lparr>new := new n - {\<phi>}\<rparr> in if \<exists>q. \<phi> = prop\<^sub>r(q) \<or> \<phi> = nprop\<^sub>r(q) then if not\<^sub>r \<phi> \<in> old n then RETURN (name n, ns) else expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = true\<^sub>r then expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = false\<^sub>r then RETURN (name n, ns) else if \<exists>\<nu> \<mu>. \<phi> = \<nu> and\<^sub>r \<mu> \<or> \<phi> = X\<^sub>r \<nu> then expand (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) else expand (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) \<bind> (\<lambda>(nm, nds). expand (n\<lparr>name := nm, new := new2 \<phi> \<union> new n, old := {\<phi>} \<union> old n\<rparr>, nds)))) n_ns"
subst_apply_pairs_pair_image_subst: fixes F :: "(('fun, 'b) Term.term \<times> ('fun, 'b) Term.term) list"   and \<theta> :: "'b \<Rightarrow> ('fun, 'a) Term.term" shows "pair ` set (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) = pair ` set F \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
StateT_def: shows "StateT \<equiv> \<Lambda> cfun. stateT_abs\<cdot>cfun"
omega_induct_all: fixes n :: "V"   and P :: "V \<Rightarrow> bool" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "\<And>x. \<lbrakk>x \<in>\<^sub>\<circ> \<omega>; \<And>y. y \<in>\<^sub>\<circ> x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P n"
component_of_term_def: fixes v :: "'a" shows "punit.component_of_term v = snd (to_pair_unit v)"
one_divide_eq_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(divide one a = zero) = (a = zero)"
redT_updWs_PostWS_imp_PostWS: fixes t :: "'a"   and ws :: "'a \<Rightarrow> 'b wait_set_status option"   and was :: "('a, 'b) wait_set_action list"   and ws' :: "'a \<Rightarrow> 'b wait_set_status option"   and t'' :: "'a"   and w :: "wake_up_status" assumes "redT_updWs t ws was ws'"   and "t'' \<noteq> t"   and "ws t'' = \<lfloor>PostWS w\<rfloor>" shows "ws' t'' = \<lfloor>PostWS w\<rfloor>"
induct: fixes x1 :: "'c list"   and x2 :: "'d list"   and P :: "'c list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "same_length x1 x2"   and "P [] []"   and "\<And>xs ys x y. \<lbrakk>same_length xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
lin_split_def: shows "lin_split \<equiv> \<lambda>(a, n) p. heap_WHILET (\<lambda>i. if i < n then Array.nth a i \<bind> (\<lambda>(uu_, s). return (s < p)) else return False) (\<lambda>i. return (i + 1)) 0 \<bind> return"
a_star_redu_AndL2_elim: fixes x :: "name"   and M :: "trm"   and y :: "name"   and R :: "trm" assumes "AndL2 x.M y \<longrightarrow>\<^sub>a* R" shows "\<exists>M'. R = AndL2 x.M' y \<and> M \<longrightarrow>\<^sub>a* M'"
lexp_lex: fixes order :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "lexp order xs ys = ((xs, ys) \<in> lex {(x, y). order x y})"
scaleR_scaleC: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and r :: "real" assumes "class.scaleC scaleR scaleC" shows "scaleR r = scaleC (complex_of_real r)"
ord_of_nat_add: fixes m :: "nat"   and n :: "nat" shows "ord_of_nat (m + n) = ord_of_nat m + ord_of_nat n"
Inv_preserves_Can_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Can C (monoidal_language.Inv C t)"
prpl_of_lasso_simps_2: fixes L :: "('e, 'f) lasso_scheme" shows "snd (prpl_of_lasso L) = lasso_va L # lasso_cysfx L"
strong_consensus_cluster_def: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool"   and W :: "'b set"   and I :: "'b set" assumes "with_w quorum_of" shows "with_w.strong_consensus_cluster quorum_of W I \<equiv> I \<subseteq> W \<and> (\<forall>p\<in>I. \<exists>Q\<subseteq>I. quorum_of p Q) \<and> (\<forall>Q Q'. with_w.quorum_of_set quorum_of I Q \<and> with_w.quorum_of_set quorum_of I Q' \<longrightarrow> I \<inter> Q \<inter> Q' \<noteq> {})"
leg1_iso: shows "\<exists>\<nu>. \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu>"