hcju/isabelleps huggingface dataset:
set_spmf_parametric: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_fun (rel_spmf A) (rel_set A) set_spmf set_spmf"
rtrancl_while_def: fixes p :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a list"   and x :: "'a" shows "rtrancl_while p f x = while_option (While_Combinator.rtrancl_while_test p) (While_Combinator.rtrancl_while_step f) ([x], {x})"
subst_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "subst (imp \<phi> \<chi>) t x = imp (subst \<phi> t x) (subst \<chi> t x)"
SystemDiffAssignSubst_def: shows "SystemDiffAssignSubst = \<lparr>SFunctions = Map.empty, SPredicates = \<lambda>p. if p = vid1 then Some (Geq ($f (Inr vid1) local.empty) (Const 0)) else None, SContexts = Map.empty, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
signed_take_bit_Suc: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not (Suc n) a = plus (modulo a (numeral.numeral one plus (num.Bit0 num.One))) (times (numeral.numeral one plus (num.Bit0 num.One)) (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n (divide a (numeral.numeral one plus (num.Bit0 num.One)))))"
interior_cbox: fixes a :: "'b"   and b :: "'b" shows "interior (cbox a b) = box a b"
induct: fixes P :: "poly \<Rightarrow> bool"   and poly :: "poly" assumes "\<And>x. P (C x)"   and "\<And>x. P (Bound x)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Add x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Sub x1a x2a)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (Mul x1a x2a)"   and "\<And>x. P x \<Longrightarrow> P (Neg x)"   and "\<And>x1a x2a. P x1a \<Longrightarrow> P (Pw x1a x2a)"   and "\<And>x1a x2a x3. \<lbrakk>P x1a; P x3\<rbrakk> \<Longrightarrow> P (CN x1a x2a x3)" shows "P poly"
DisjointPaths_extend: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P :: "'c list" assumes "DisjointPaths G v0 v1 paths"   and "v0 \<leadsto>P\<leadsto>\<^bsub>G\<^esub> v1"   and "\<And>xs v. \<lbrakk>xs \<in> paths; xs \<noteq> P; v \<in> set xs; v \<in> set P\<rbrakk> \<Longrightarrow> v = v0 \<or> v = v1" shows "DisjointPaths G v0 v1 (insert P paths)"
h_lmeas: fixes n :: "nat"   and y :: "real" shows "{t. h n (g t) = y} \<inter> S \<in> sets lebesgue"
analz_image_pan: fixes evs :: "event list"   and KK :: "nat set"   and P :: "nat" assumes "evs \<in> set_cr"   and "KK \<subseteq> - invKey ` pubEK ` range CA" shows "(Pan P \<in> analz (Key ` KK \<union> knows Spy evs)) = (Pan P \<in> analz (knows Spy evs))"
delete_simps_2: fixes k :: "'a"   and p :: "'a \<times> 'b"   and ps :: "('a \<times> 'b) list" shows "delete k (p # ps) = (if fst p = k then delete k ps else p # delete k ps)"
mult_scalar_distrib_left: fixes r :: "'a \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "r \<odot> (p + q) = r \<odot> p + r \<odot> q"
outgoing_transitions_def: fixes e :: "(('a \<times> 'b) \<times> 'c) fset"   and s :: "'a" shows "outgoing_transitions e s = ffilter (\<lambda>((o, uu_), uu_). o = s) e"
surjective: fixes r :: "('b, 'c, 'd, 'e) spr_simWorld_scheme" shows "r = \<lparr>sprFst = sprFst r, sprLst = sprLst r, sprCRel = sprCRel r, \<dots> = spr_simWorld.more r\<rparr>"
Spy_analz_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> yahalom" shows "(Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
empty_suffix_exteq: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" assumes "emptyseq s" shows "(s |\<^sub>s n) m = s m"
eq_\<I>_gpv_map_gpv': fixes f :: "'h \<Rightarrow> 'l"   and f' :: "'i \<Rightarrow> 'm"   and A :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and g :: "'j \<Rightarrow> 'n"   and h :: "'o \<Rightarrow> 'k"   and \<I> :: "('n, 'o) \<I>"   and gpv1 :: "('h, 'j, 'k) gpv"   and gpv2 :: "('i, 'j, 'k) gpv" assumes "eq_\<I>_gpv (BNF_Def.vimage2p f f' A) (map_\<I> g h \<I>) gpv1 gpv2" shows "eq_\<I>_gpv A \<I> (map_gpv' f g h gpv1) (map_gpv' f' g h gpv2)"
is_strict_monoidal_functor: shows "strict_monoidal_functor (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C \<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D E.map"
soundness: fixes p :: "('a, 'b) fm"   and E :: "nat \<Rightarrow> 'c"   and F :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c list \<Rightarrow> bool" assumes "\<turnstile> p" shows "\<lbrakk>E, F, G\<rbrakk> p"
Step: shows "cfg2' \<turnstile> msg \<mapsto> cfg3"
add_block_fin: fixes b :: "'a set" assumes "finite b" shows "finite_incidence_system (\<V> \<union> b) (sys1.add_block b)"
simps: fixes M1 :: "('d, 'e, 'f) FSM"   and TS :: "'d list set"   and \<Omega> :: "('d, 'e) ATC set"   and M2 :: "('d, 'e, 'g) FSM" shows "M1 \<preceq>\<lbrakk>TS.\<Omega>\<rbrakk> M2 = (\<forall>iseq\<in>TS. atc_io_reduction_on M1 M2 iseq \<Omega>)"
diff_eq_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a - b = c) = (a = c + b)"
finite_imp_null_set_lborel: fixes A :: "'b set" assumes "finite A" shows "A \<in> null_sets lborel"
product_category_axioms: shows "product_category (\<cdot>) VxV.comp"
test_bit_lenD: fixes x :: "'b word"   and n :: "nat" assumes "x !! n" shows "n < LENGTH('b) \<and> x !! n"
countable_subset: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B"   and "countable B" shows "countable A"
adjuster_in_span: fixes w :: "complex vec"   and us :: "complex vec list" assumes "w \<in> carrier_vec d"   and "set us \<subseteq> carrier_vec d"   and "distinct us" shows "adjuster d w us \<in> local.span (set us)"
that: fixes a :: "'a" assumes "S \<subseteq> cbox (- a) a" shows "thesis"
sinks_aux_single_event: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and U :: "'e set"   and x :: "'f" shows "sinks_aux I D U [x] = (if \<exists>v\<in>U. (v, D x) \<in> I then insert (D x) U else U)"
change_respecting_doesnt_modify: fixes cms :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val"   and X :: "'Var set"   and x :: "'Var" assumes "change_respecting (cms, mem) (cms', mem') X"   and "(cms, mem) \<leadsto> (cms', mem')"   and "x \<in> X \<union> vars_\<C> X" shows "mem x = mem' x"
assms: shows "D f \<mapsto> (\<lambda>h. h *\<^sub>R f' x) at x within T"
coboundedI1: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "(c::'b) \<le> (a::'b)" shows "(c::'b) \<le> sup (a::'b) (b::'b)"
subst_def: fixes x :: "'a"   and t :: "('b, 'a) Term.term" shows "subst x t = Var(x := t)"
map_pre_F_def: shows "map_pre_F \<equiv> \<lambda>f1 f2. BNF_Composition.id_bnf \<circ> map_F0 f1 f2 \<circ> BNF_Composition.id_bnf"
sQc: shows "sound (\<lambda>s. bound_of Q - Q s)"
het_gt_0I: fixes a :: "real"   and xs :: "real list"   and b :: "real" assumes "a \<in> set xs"   and "b \<in> set xs"   and "a \<noteq> b" shows "0 < het xs"
prems: fixes x :: "'c" assumes "x \<in> insert x__ F" shows "f x \<in>\<^sub>i g x"
fstate_def: fixes tr :: "'b \<times> 'a" shows "fstate tr \<equiv> snd tr"
hd_filter_prop: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "filter P xs \<noteq> []" shows "P (hd (filter P xs))"
ArrayClear': fixes s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and x :: "char list"   and \<pi> :: "char list \<Rightarrow> com option" assumes "s' = s(x := \<lambda>_. 0)" shows "\<pi>: (CLEAR x[], s) \<Rightarrow> s'"
T_on_rand'_as_sum: fixes A :: "('state \<Rightarrow> 'b pmf) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> ('answer \<times> 'b) pmf)"   and s0 :: "('state \<times> 'b) pmf"   and rs :: "'request list" shows "T_on_rand' A s0 rs = sum (T_on_rand'_n A s0 rs) {..<length rs}"
ground\<^sub>t\<^sub>s_vars\<^sub>t\<^sub>s: fixes ts :: "fterm list" assumes "ground\<^sub>t\<^sub>s ts" shows "vars\<^sub>t\<^sub>s ts = {}"
hyps_4: assumes "Keys (set (snd (sym_preproc_addnew gs vs' fs v))) = set (ks @ [v]) \<union> set (fst (sym_preproc_addnew gs vs' fs v))" shows "set (fst (sym_preproc_aux gs (ks @ [v]) (sym_preproc_addnew gs vs' fs v))) = Keys (set (snd (sym_preproc_aux gs (ks @ [v]) (sym_preproc_addnew gs vs' fs v))))"
form_of_cnf_def: fixes F :: "'b literal list list" shows "form_of_cnf F \<equiv> \<^bold>\<And>map disj_of_clause F"
invariant_1_2_def: fixes rai :: "int poly \<times> rat \<times> rat" shows "invariant_1_2 rai \<equiv> invariant_1 rai \<and> 1 < degree (poly_real_alg_1 rai)"
horner_sum_bit_eq_take_bit: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) (map (bit a) [0..<n]) = take_bit n a"
strict_implies_order: fixes b :: "'f"   and a :: "'f" assumes "b < a" shows "b \<le> a"
one_vec_def: shows "1 \<equiv> \<chi>i. (1::'a)"
map_rexp_ident: shows "map_rexp (\<lambda>x. x) = (\<lambda>r. r)"
walk_in_V: fixes xs :: "Vertices list" assumes "walk xs" shows "set xs \<subseteq> V\<^bsub>T2\<^esub>"
interval_tendsto_def: fixes X :: "nat \<Rightarrow> 'a interval"   and x :: "'a" shows "X \<longlonglongrightarrow>\<^sub>i x \<equiv> (interval.upper \<circ> X) \<longlonglongrightarrow> x \<and> (interval.lower \<circ> X) \<longlonglongrightarrow> x"
many_reducible_to_via_trans: fixes A :: "nat set"   and B :: "nat set"   and f :: "nat \<Rightarrow> nat"   and C :: "nat set"   and g :: "nat \<Rightarrow> nat" assumes "many_reducible_to_via A B f"   and "many_reducible_to_via B C g" shows "many_reducible_to_via A C (g \<circ> f)"
convergent_Cauchy: fixes "open" :: "'b set \<Rightarrow> bool"   and uniformity :: "('b \<times> 'b) filter"   and X :: "nat \<Rightarrow> 'b" assumes "class.uniform_space open uniformity"   and "topological_space.convergent open X" shows "uniform_space.Cauchy uniformity X"
induce_subgraph_verts: fixes G :: "('c, 'd) pre_digraph"   and vs :: "'c set" shows "verts (G \<restriction> vs) = vs"
sublist_step: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "i + length xs < length ys"   and "sublist_at xs ys i"   and "ys ! (i + length xs) = x" shows "sublist_at (xs @ [x]) ys i"
\<omega>_limit_set_positively_invariant: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.positively_invariant f X (auto_ll_on_open.\<omega>_limit_set f X x)"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (y \<le> x \<and> x \<le> y)"
simps: fixes l :: "'a tree"   and x :: "'a"   and r :: "'a tree" shows "mrbst_push_down l x r = pmf_of_set {0..size l + size r} \<bind> (\<lambda>k. if k < size l then case l of \<langle>ll, y, lr\<rangle> \<Rightarrow> map_pmf (Node ll y) (mrbst_push_down lr x r) else if k < size l + size r then case r of \<langle>rl, y, rr\<rangle> \<Rightarrow> map_pmf (\<lambda>l'. \<langle>l', y, rr\<rangle>) (mrbst_push_down l x rl) else return_pmf \<langle>l, x, r\<rangle>)"
openin_subtopology_Int_subset: fixes u :: "'b set"   and S :: "'b set"   and v :: "'b set" assumes "openin (top_of_set u) (u \<inter> S)"   and "v \<subseteq> u" shows "openin (top_of_set v) (v \<inter> S)"
pmc_vacuity: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "PartialMeetContraction Cn selection"   and "K = Cn A"   and "\<phi> \<notin> Cn K" shows "Tarskian_logic.meet_contraction K selection \<phi> = K"
sum0: shows "(\<Sum>x\<in>X. of_rat (\<beta> x) * exp x) = 0"
\<alpha>0pow: fixes x :: "nat" shows "\<alpha> i ^ x \<noteq> 0"
cut_less_in_imp: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I \<down>< t" shows "x \<in> I"
x1neqx2: shows "flow0 x t1 \<noteq> flow0 x t2"
card_Union_le_sum_card: fixes U :: "'a set set" assumes "\<forall>u\<in>U. finite u" shows "card (\<Union> U) \<le> sum card U"
lossless_gpv_conv_finite: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" shows "lossless_gpv \<I> gpv = (finite_gpv \<I> gpv \<and> colossless_gpv \<I> gpv)"
map_scale_eq_0_iff: fixes c :: "'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "(c \<cdot> t = 0) = (c = (0::'b) \<or> t = 0)"
type_substitutivity: fixes x :: "name"   and T' :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and e :: "trm"   and T :: "ty"   and e' :: "trm" assumes "(x, T') # \<Gamma> \<turnstile> e : T"   and "\<Gamma> \<turnstile> e' : T'" shows "\<Gamma> \<turnstile> e[x::=e'] : T"
analytic_at_two: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and g :: "complex \<Rightarrow> complex" shows "(f analytic_on {z} \<and> g analytic_on {z}) = (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
llast_conv_lnth: fixes xs :: "'a llist"   and n :: "nat" assumes "llength xs = eSuc (enat n)" shows "llast xs = lnth xs n"
trie_update_def: fixes k :: "'c list"   and v :: "'d"   and t :: "('c, 'd) trie" shows "trie_update k v t = trie_update_with' k t v (\<lambda>v'. v)"
blocksSeen_def: fixes s :: "state"   and p :: "Proc" shows "blocksSeen s p = allBlocksRead s p \<union> {dblock s p}"
sqrt_int_floor_def: fixes x :: "int" shows "sqrt_int_floor x = root_int_floor 2 x"
Ch_unilateral_substitutes: shows "\<forall>h. TruePref.unilateral_substitutes_on UNIV (Ch h)"
eqC: fixes r :: "real" shows "B r \<union> S r = C r"
transition: shows "P \<rightarrow> \<langle>\<alpha>,P'\<rangle>"
square_free_prod_list_distinct: fixes us :: "'b poly list" assumes "square_free (prod_list us)"   and "\<And>u. u \<in> set us \<Longrightarrow> 0 < degree u" shows "distinct us"
content_smult_int: fixes a :: "int"   and p :: "int poly" shows "content (Polynomial.smult a p) = \<bar>a\<bar> * content p"
Suc_le_length_iff: fixes n :: "nat"   and xs :: "'b list" shows "order.greater_eq (length xs) (Suc n) = (\<exists>x ys. xs = x # ys \<and> order.greater_eq (length ys) n)"
seq'_def: shows "seq' = (\<lambda>i. (pp_of_term (seq i), component_of_term (seq i)))"
step_z_dbm_preserves_int: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and v :: "'e \<Rightarrow> nat"   and n :: "nat"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and X :: "'e set"   and k :: "'e \<Rightarrow> 'f" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X k"   and "dbm_int D n" shows "dbm_int D' n"
assms: fixes r :: "'a state \<times> (v, v) result" shows "evaluate_match True env1 st1 v1 pes1 err_v1 r = evaluate_match True env2 st2 v2 pes2 err_v2 r"
IH_4: assumes "insert\<langle>t,s\<rangle> \<in> set (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<sigma>)" shows "\<exists>t' s'. t = t' \<cdot> \<sigma> \<and> s = s' \<cdot> \<sigma> \<and> insert\<langle>t',s'\<rangle> \<in> set S"
Re_interval_times: fixes A :: "complex_interval"   and B :: "complex_interval" shows "Re_interval (A * B) = Re_interval A * Re_interval B - Im_interval A * Im_interval B"
bind_Let_refine2: fixes m' :: "'a nres"   and R' :: "('a \<times> 'b) set"   and x :: "'b"   and f' :: "'a \<Rightarrow> 'c nres"   and R :: "('c \<times> 'd) set"   and f :: "'b \<Rightarrow> 'd nres" assumes "m' \<le> \<Down> R' (RETURN x)"   and "\<And>x'. \<lbrakk>inres m' x'; (x', x) \<in> R'\<rbrakk> \<Longrightarrow> f' x' \<le> \<Down> R (f x)" shows "m' \<bind> f' \<le> \<Down> R (Let x f)"
inst_lift_pcpo: shows "\<bottom> = Abs_lift \<bottom>"
ordinal_plus_0: fixes x :: "ordinal" shows "x + 0 = x"
map_member_plus: fixes x :: "'c"   and y :: "'d"   and f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "((x, y) \<in>\<^sub>m f ++ g) = (x \<notin> dom g \<and> (x, y) \<in>\<^sub>m f \<or> (x, y) \<in>\<^sub>m g)"
mult_eq_0_iff: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k" assumes "class.semiring_no_zero_divisors plus zero times" shows "(times a b = zero) = (a = zero \<or> b = zero)"
const_in_iff: fixes c :: "'b"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "((\<lambda>_. c) \<in> o[F](f)) = ((\<lambda>_. 1::'b) \<in> o[F](f))"
dma_\<C>_vars: fixes y :: "'Var"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<forall>x\<in>\<C>_vars y. mem\<^sub>1 x = mem\<^sub>2 x" shows "dma mem\<^sub>1 y = dma mem\<^sub>2 y"
genby_in_sum_list_lists: fixes w :: "'b"   and S :: "'b set" assumes "w \<in> \<langle>S\<rangle>" shows "w \<in> (\<Union>ss\<in>lists (S \<union> uminus ` S). {sum_list ss})"
map_transfer: fixes Rb :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and Sd :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (rel_fun Rb Sd) (rel_fun (rel_pmf Rb) (rel_pmf Sd)) map_pmf map_pmf"
globalize_local_consts: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and is' :: "nat list"   and "is" :: "nat list"   and f :: "'a \<Rightarrow> 'a"   and t :: "('b, 'd, 'c, 'a) itnode rose_tree" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "is' \<in> it_paths (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent is f t)" shows "subst_lconsts (iSubst (tree_at (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent is f t) is')) \<subseteq> Abstract_Task.fresh_at_path freshenLC anyP antecedent (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent is f t) is' \<union> range f"
signed_div_arith: fixes a :: "'b word"   and b :: "'b word" shows "sint (a sdiv b) = signed_take_bit (LENGTH('b) - 1) (sint a sdiv sint b)"
subtopology_eq_discrete_topology_empty: fixes X :: "'b topology" shows "(X = discrete_topology {}) = (topspace X = {})"
nn_integral_sum: fixes P :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>i. i \<in> P \<Longrightarrow> f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M = (\<Sum>i\<in>P. integral\<^sup>N M (f i))"
bisimp_bisim_eq: shows "bisimp = (\<sim>)"
ex0_def: shows "ex0 \<equiv> \<lambda>F. F dj"
alpha_def_4: shows "val_Zp (a \<ominus> \<alpha>) = val_Zp (local.divide (f \<bullet> a) (pderiv f \<bullet> a))"
samequot_soundness: fixes as :: "'a"   and r :: "'b rexp"   and b :: "bool"   and rs :: "'b rexp list"   and n :: "nat" assumes "while_option fst (rderiv_and_add as) (True, [\<guillemotleft>r\<guillemotright>]) = Some (b, rs)" shows "lang n (PLUS rs) = \<Union> (lang n ` {\<guillemotleft>rderivs (replicate m as) r\<guillemotright> |m. True})"
Let_def: fixes s :: "'d"   and f :: "'d \<Rightarrow> 'c" shows "HOL.Let s f \<equiv> f s"
rad_of_def: fixes w :: "real" shows "rad_of w = w * pi / 180"
squarefree_part_1: shows "squarefree_part (1::'a) = (1::'a)"
left_mult_one_mat': fixes A :: "'a mat"   and n :: "nat" assumes "dim_row A = n" shows "1\<^sub>m n * A = A"
IH_1: shows "KD_Tree.size_kdt l \<le> 2 ^ height l"
Chain_def: fixes D :: "('c, 'd) Order_scheme"   and C :: "'c set" shows "Chain D C = (C \<subseteq> carrier D \<and> Torder (Iod D C))"
simps: fixes a1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and a2 :: "'d"   and a3 :: "('c \<Rightarrow> real) set"   and a4 :: "'d"   and a5 :: "('c \<Rightarrow> real) set" shows "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto>\<^sub>\<alpha> \<langle>a4, a5\<rangle> = (\<exists>A l Z l' Z'. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l' \<and> a5 = Closure\<^sub>\<alpha> Z' \<and> A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>)"
f_dim: fixes i :: "nat" assumes "i < length fs" shows "dim_vec (fs ! i) = n"
UNIV_4: fixes P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>UNIV. P x) = (\<exists>x. P x)"
count_mset_set_1: fixes A :: "'k set"   and x :: "'k" assumes "finite A"   and "x \<in> A" shows "count (mset_set A) x = 1"
brec_det_algo_def: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and Qi :: "('a, unit) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist" shows "brec_det_algo rqrm Qi \<delta> \<equiv> \<lparr>dwa_cond = brec_cond, dwa_step = brec_step rqrm Qi, dwa_initial = brec_initial Qi \<delta>, dwa_invar = brec_invar (hs.\<alpha> Qi) (ls.\<alpha> \<delta>)\<rparr>"
monofun_eq_f: fixes N :: "('g, 'b, 'h) Module_scheme"   and h :: "'a"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "h \<in> H"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N" shows "monofun R M N f H h h = f h"
vec_hom_zero_iff: fixes x :: "'a vec"   and n :: "nat" shows "(vec\<^sub>h x = 0\<^sub>v n) = (x = 0\<^sub>v n)"
assertion_isor: fixes p :: "'a"   and y :: "'a"   and x :: "'a" assumes "test p" shows "(y \<le> x \<cdot> p) = (y \<cdot> p\<^sup>o \<le> x)"
ipset_from_cidr_eq_ip_cidr_set: fixes base :: "'b word"   and len :: "nat" shows "ipset_from_cidr base len = ip_cidr_set base len"
cases: fixes x :: "tm" obtains v :: "name" and p :: "tm" and sx :: "name" and y :: "name" and ax :: "name" and xa :: "name" where "atom v \<sharp> (p, sx, y, ax, xa)" and "atom xa \<sharp> (p, sx, y, ax)" and "atom ax \<sharp> (p, sx, y)" and "atom y \<sharp> (p, sx)" and "atom sx \<sharp> p" and "x = p"