hcju/isabelleps huggingface dataset:
prime_elem_multiplicity_power_distrib: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "x \<noteq> zero" shows "factorial_semiring.multiplicity times one p (power.power one times x n) = n * factorial_semiring.multiplicity times one p x"
hyps_3: shows "cancels_to_1\<^sup>*\<^sup>* (a @ b) (ba @ b')"
simps_25: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "'a" shows "(case a\<^sub>5 of a\<^sub>1 \<Rightarrow> f1 | a\<^sub>2 \<Rightarrow> f2 | a\<^sub>3 \<Rightarrow> f3 | a\<^sub>4 \<Rightarrow> f4 | a\<^sub>5 \<Rightarrow> f5) = f5"
summable_real_powr_iff: fixes s :: "real" shows "summable (\<lambda>n. real n powr s) = (s < - 1)"
prog_2: shows "is_multi_arith_prog_on (l + 1) k start steps a b"
cosh_real_nonneg_less_iff: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "(cosh x < cosh y) = (x < y)"
intros_3: fixes n' :: "'node SDG_node"   and m :: "'node"   and n :: "'node SDG_node" assumes "valid_SDG_node n'"   and "m = parent_node n'"   and "n = CFG_node m"   and "n \<noteq> n'" shows "n s\<longrightarrow>\<^bsub>cd\<^esub> n'"
not_less_Least: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "less k (ord.Least less_eq P)" shows "\<not> P k"
arr_iff_in_hom: fixes f :: "'b" shows "B.arr f = \<guillemotleft>f : B.dom f \<rightarrow>\<^sub>B B.cod f\<guillemotright>"
chains_def: fixes A :: "'c set set" shows "chains A = {C. C \<subseteq> A \<and> chain\<^sub>\<subseteq> C}"
bdd_above_insert: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.lattice inf less_eq less sup" shows "preorder.bdd_above less_eq (insert a A) = preorder.bdd_above less_eq A"
discrCf_G: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and cf' :: "('test, 'atom, 'choice) cmd \<times> 'state" assumes "discrCf cf"   and "cf' \<in> G cf" shows "discrCf cf'"
disc_1: fixes x1 :: "('a, 'b) hd" shows "is_Hd (Hd x1)"
assms: shows "p \<succeq> q \<and> (q, p) \<notin> relation"
\<psi>_in_terms_of_\<epsilon>: fixes x :: "'a"   and g :: "'b"   and y :: "'b" assumes "Adj.C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<psi> x g = Adj.\<epsilon> x \<cdot>\<^sub>C F g"
step: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "class.ccpo Sup less_eq less"   and "x \<in> ccpo.iterates Sup less_eq f" shows "f x \<in> ccpo.iterates Sup less_eq f"
orthogonal_subspace_decomp_unique: fixes x :: "'a"   and y :: "'a"   and x' :: "'a"   and y' :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "x + y = x' + y'"   and "x \<in> span S"   and "x' \<in> span S"   and "y \<in> span T"   and "y' \<in> span T"   and "\<And>a b. \<lbrakk>a \<in> S; b \<in> T\<rbrakk> \<Longrightarrow> orthogonal a b" shows "x = x' \<and> y = y'"
Inf_upset_id: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and dual :: "'a \<Rightarrow> 'a" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Inf \<circ> ord.upset less_eq = id"
1: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau> (B $ Bv 0)) B"
rm_in_set_of: fixes t :: "tree" assumes "t \<noteq> Tip" shows "rm t \<in> set_of t"
assms: shows "mat_of_cblinfun a = mat_of_cblinfun b"
assms_3: fixes x :: "'n"   and k :: "real" assumes "x \<in> s" shows "norm (f k x) \<le> h x"
less: fixes y :: "'a list"   and k :: "nat" assumes "length y < length xs" shows "T_brauns k y = 4 * length y"
Inr_not_Inl: fixes b :: "'b"   and a :: "'a" shows "Inr b \<noteq> Inl a"
polychain_Cons: fixes y :: "'a \<times> 'a"   and ys :: "('a \<times> 'a) list" shows "polychain (y # ys) = (if ys = [] then True else snd y = fst (ys ! 0) \<and> polychain ys)"
theNFT_def: fixes cl :: "('d, 'e, 'f) cmd list" shows "theNFT cl \<equiv> {n. n < length cl \<and> \<not> finished (cl ! n)}"
domains_simp: fixes f :: "'a" assumes "C.arr f" shows "C.domains f = {C.dom f}"
pw_Sup_inres: fixes r :: "'a" shows "inres (Sup (f ` X)) r = (\<exists>M\<in>f ` X. inres M r)"
elem_set: fixes x :: "'b"   and xo :: "'b option" shows "(x \<in> set_option xo) = (xo = Some x)"
S5Basic_10: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<or> \<^bold>\<box>\<psi>) \<^bold>\<equiv> \<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<box>\<psi> in v]"
frac_lt_1: fixes x :: "'b" shows "frac x < (1::'b)"
l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog" shows "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes set_child_nodes set_child_nodes_locs get_parent get_owner_document get_disconnected_nodes set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove \<equiv> remove_child = l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child get_child_nodes set_child_nodes get_owner_document get_disconnected_nodes set_disconnected_nodes \<and> remove_child_locs = l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child_locs set_child_nodes_locs set_disconnected_nodes_locs \<and> remove = l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove get_child_nodes set_child_nodes get_parent get_owner_document get_disconnected_nodes set_disconnected_nodes"
Borsuk_maps_homotopic_in_connected_component_eq: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "compact S"   and "a \<notin> S"   and "b \<notin> S"   and "2 \<le> DIM('a)" shows "homotopic_with_canon (\<lambda>x. True) S (sphere (0::'a) 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) (\<lambda>x. (x - b) /\<^sub>R norm (x - b)) = connected_component (- S) a b"
P_sup_lfp: fixes Q :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'s stream \<Rightarrow> 'a"   and M :: "'a measure"   and s :: "'s" assumes "sup_continuous Q"   and "f \<in> St \<rightarrow>\<^sub>M M"   and "\<And>P. Measurable.pred M P \<Longrightarrow> Measurable.pred M (Q P)" shows "P_sup s (\<lambda>x. lfp Q (f x)) = (\<Squnion>i. P_sup s (\<lambda>x. (Q ^^ i) \<bottom> (f x)))"
trans: fixes r :: "'n"   and s :: "'n"   and t :: "'n" assumes "r = s"   and "s = t" shows "r = t"
simps_2: fixes v :: "real" shows "R v + MInfty = R v"
square_diff_one_factored: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.ring_1 one times plus zero minus uminus" shows "minus (times x x) one = times (plus x one) (minus x one)"
sumZero: fixes P :: "pi" shows "P \<oplus> \<zero> \<sim>\<^sub>l P"
headnz: fixes p :: "'b poly"   and n :: "nat"   and m :: "nat" assumes "isnpolyh p n"   and "p \<noteq> 0\<^sub>p" shows "headn p m \<noteq> 0\<^sub>p"
del_list_notin_Cons: fixes x :: "'c"   and xs :: "'c list" assumes "Sorted_Less.sorted (x # xs)" shows "List_Ins_Del.del_list x xs = xs"
ak_bk: shows "to_nat a < k \<or> to_nat b < k"
steps: fixes i :: "nat" shows "(f i, f (Suc i)) \<in> {(x, y). x < y \<and> y \<le> b}"
delete_shadow_root_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'aj" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
vlrestriction_vconst_on: fixes A :: "V"   and c :: "V"   and B :: "V" shows "vconst_on A c \<restriction>\<^sup>l\<^sub>\<circ> B = vconst_on B c \<restriction>\<^sup>l\<^sub>\<circ> A"
scalar_prod_cons: fixes ze :: "'b"   and pl :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ti :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and as :: "'b list"   and b :: "'b"   and bs :: "'b list" shows "scalar_prodI ze pl ti (a # as) (b # bs) = pl (ti a b) (scalar_prodI ze pl ti as bs)"
pol_coeff_le: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and n :: "nat" assumes "pol_coeff S c"   and "n \<le> fst c" shows "pol_coeff S (n, snd c)"
FCI_def: fixes \<Gamma> :: "'a Gamma"   and \<V> :: "'a V_rec"   and Tr :: "'a list set" shows "FCI \<Gamma> \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>\<^esub>. \<forall>v\<in>V\<^bsub>\<V>\<^esub> \<inter> \<nabla>\<^bsub>\<Gamma>\<^esub>. \<beta> @ [v] @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> (\<exists>\<alpha>' \<delta>'. set \<delta>' \<subseteq> N\<^bsub>\<V>\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>\<^esub> \<and> \<beta> @ [c] @ \<delta>' @ [v] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
assms_4: fixes a :: "nat"   and obj :: "obj" assumes "h a = \<lfloor>obj\<rfloor>" shows "fst obj \<noteq> C"
length_Suc_conv: fixes xs :: "'d list"   and n :: "nat" shows "(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
list_all_eq_def: shows "list_all_eq = list_all id"
map_nth_append1: fixes xs :: "'a list"   and d :: "nat"   and ys :: "'a list" assumes "length xs = d" shows "map ((!) (xs @ ys)) [0..<d] = xs"
nomatch: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c" assumes "\<not> matches \<gamma> m a p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided"
strict_implies_order: fixes a :: "'t"   and b :: "'t" assumes "a \<prec>\<^sub>t b" shows "a \<preceq>\<^sub>t b"
Rep: fixes x :: "'a" shows "Rep x \<in> {a. sort_of a = s}"
Cons1_1: fixes a :: "(real \<times> real \<times> real) list"   and b :: "(real \<times> real \<times> real) list"   and c :: "(real \<times> real \<times> real) list"   and d :: "(real \<times> real \<times> real) list" assumes "separateAtoms L = (a, b, c, d)" shows "(\<forall>l\<in>set L. evalUni (substNegInfinityUni l) x) = ((\<forall>a\<in>set a. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c = 0) \<and> (\<forall>a\<in>set b. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c < 0) \<and> (\<forall>a\<in>set c. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<le> 0) \<and> (\<forall>a\<in>set d. case a of (a, ba, c) \<Rightarrow> \<exists>x. \<forall>y<x. a * y\<^sup>2 + ba * y + c \<noteq> 0))"
\<pi>_nat_cancel: fixes x :: "int" shows "\<pi> (real (nat x)) = \<pi> (real_of_int x)"
ch2ch_Rep_cfunR: fixes Y :: "nat \<Rightarrow> 'a"   and f :: "'a \<rightarrow> 'b" assumes "chain Y" shows "chain (\<lambda>i. f\<cdot>(Y i))"
in_keys_plusI2: fixes t :: "'c"   and q :: "'c \<Rightarrow>\<^sub>0 'd"   and p :: "'c \<Rightarrow>\<^sub>0 'd" assumes "t \<in> keys q"   and "t \<notin> keys p" shows "t \<in> keys (p + q)"
PRelQ: fixes \<phi> :: "'c"   and Q :: "('a, 'b, 'c) psi" assumes "(\<phi>, Q) mem CsQ" shows "\<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> Eq \<Psi> P Q"
Q2: shows "Quotient R2 Abs2 Rep2 T2"
aform_val_msum_aform: fixes f :: "'a \<times> 'a pdevs"   and n :: "nat"   and e :: "nat \<Rightarrow> real"   and g :: "'a \<times> 'a pdevs" assumes "degree_aform f \<le> n" shows "aform_val e (msum_aform n f g) = aform_val e f + aform_val (\<lambda>i. e (i + n)) g"
NSLIMSEQ_le: fixes f :: "nat \<Rightarrow> real"   and l :: "real"   and g :: "nat \<Rightarrow> real"   and m :: "real" assumes "f \<longlonglongrightarrow>\<^sub>N\<^sub>S l"   and "g \<longlonglongrightarrow>\<^sub>N\<^sub>S m"   and "\<exists>N. \<forall>n\<ge>N. f n \<le> g n" shows "l \<le> m"
distinct_83: fixes x6 :: "nat"   and x8 :: "int" shows "Passwd x6 \<noteq> Pub_AgrK x8"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel))"
E: fixes X' :: "'v set" assumes "X' \<subseteq> X" shows "E' `` X' = \<^bold>E `` X'"
ideal_generated_pair_exists_pq1: fixes 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 a :: "'b"   and b :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "ring.ideal_generated plus zero uminus times {a, b} = UNIV" shows "\<exists>p q. plus (times p a) (times q b) = one"
ridmap_surjec: fixes A :: "('c, 'd) Ring_scheme" assumes "Ring A" shows "surjec\<^bsub>A,A\<^esub> ridmap A"
finite_measure_restricted: fixes S :: "'a set" assumes "S \<in> sets M" shows "finite_measure (density M (indicator S))"
no_call_in_path_set_conv: fixes x :: "'b"   and p :: "'b list" shows "no_call_in_path x p = (x \<notin> set p)"
neval_det: fixes x :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and n :: "nat"   and y :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and y' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "abs.neval x n y"   and "abs.neval x n y'" shows "y = y'"
path_eventually_conforms_to_\<sigma>_map_n: fixes P :: "'a llist" assumes "lset P \<subseteq> S - W"   and "valid_path P"   and "path_conforms_with_strategy p P well_ordered_strategy" shows "\<exists>n. path_conforms_with_strategy p (ldropn n P) (path_strategies P $ n)"
hom_smult_mat: fixes a :: "real"   and A :: "real mat" shows "of_real_hom.mat_hom (a \<cdot>\<^sub>m A) = of_real a \<cdot>\<^sub>m of_real_hom.mat_hom A"
x: shows "(berlekamp_mat u)\<^sup>T *\<^sub>v x = x"
bind_return_pmf: fixes x :: "'e"   and f :: "'e \<Rightarrow> 'd pmf" shows "return_pmf x \<bind> f = f x"
constr_rep_2: fixes x3 :: "'d"   and x2 :: "name list"   and x1 :: "'d" shows "Rep_action (x3\<lparr>\<nu>*x2\<rparr>\<langle>x1\<rangle>) = Out_Rep x3 x2 x1"
if_distribR: fixes b :: "bool"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(if b then f else g) x = (if b then f x else g x)"
section_def: fixes f :: "'a" shows "C.section f \<equiv> \<exists>g. C.ide (g \<cdot> f)"
2: shows "substT (substT (rawpsubstT r (zip (map Var vs) xs)) (Var u) x) t u = substT (substT (rawpsubstT r (zip (map Var vs) xs)) (Var v) x) t v"
ins_rbt_sorted: fixes t :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a"   and x :: "'c" assumes "rbt_sorted t" shows "rbt_sorted (rbt_ins f k x t)"
FactGroup_nonempty: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and X :: "'c set" assumes "group_hom G H h"   and "X \<in> carrier (G Mod kernel G H h)" shows "X \<noteq> {}"
tendsto_upperbound: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b"   and F :: "'c filter"   and a :: "'b" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq (f i) a"   and "F \<noteq> \<bottom>" shows "less_eq x a"
inv_matrix_vector_mul_left: fixes A :: "(('a, 'b) vec, 'c) vec"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" assumes "invertible A" shows "(A *v x = A *v y) = (x = y)"
borel_measurable_scaleR: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
order_gt_0_iff_finite: shows "(0 < order (G\<lparr>carrier := P\<rparr>)) = finite (carrier (G\<lparr>carrier := P\<rparr>))"
simps_3: fixes n :: "num" shows "numeral (num.Bit1 n) = numeral n + numeral n + (1::'a)"
rel_prod_conv: fixes R1 :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
rPar1: fixes xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* C"   and "distinct xvec" shows "Prop C \<Psi> (P \<parallel> !P) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' (P' \<parallel> !P))"
prm_disagreement_def: fixes \<pi> :: "'b prm"   and \<sigma> :: "'b prm" shows "ds \<pi> \<sigma> \<equiv> {x. \<pi> $ x \<noteq> \<sigma> $ x}"
img_snd: fixes a :: "'a"   and b :: "'b"   and S :: "('a \<times> 'b) set" assumes "(a, b) \<in> S" shows "b \<in> snd ` S"
per_lemma_comm: fixes w :: "'b list"   and r :: "'b list"   and s :: "'b list" assumes "w \<le>p r \<cdot> w"   and "w \<le>p s \<cdot> w"   and "\<^bold>|s\<^bold>| + \<^bold>|r\<^bold>| - gcd \<^bold>|s\<^bold>| \<^bold>|r\<^bold>| \<le> \<^bold>|w\<^bold>|" shows "s \<cdot> r = r \<cdot> s"
ex_3_different_points: fixes z :: "complex_homo" shows "\<exists>z1 z2. z \<noteq> z1 \<and> z1 \<noteq> z2 \<and> z \<noteq> z2"
input2state_empty: fixes x1 :: "nat" shows "input2state [] $ x1 = None"
post_delete: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "invar t" shows "post_del t (delete x t)"
ahm_empty_correct: shows "ahm_\<alpha> (ahm_empty ()) = Map.empty"
elist_expr_inj_term: fixes t :: "expr list"   and w :: "expr" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
path_is_trancl: fixes E :: "('c \<times> 'c) set"   and u :: "'c"   and l :: "'c list"   and v :: "'c" assumes "path E u l v"   and "l \<noteq> []" shows "(u, v) \<in> E\<^sup>+"
assms_2: shows "(\<lambda>x. sum_upto (\<lambda>n. sum_upto f (x / real n) * g n) (sqrt x) - S x) \<in> O(b)"
heron_step_div_eq_floored: fixes t :: "nat"   and n :: "nat" assumes "0 < t" shows "(t + n div t) div 2 = nat \<lfloor>(real t + real n / real t) / 2\<rfloor>"
pochhammer_1: fixes a :: "'a" shows "pochhammer a 1 = a"
induct: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x1 :: "'a language"   and x2 :: "'a language"   and P :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool" assumes "Plus_Times_pre_cong R x1 x2"   and "\<And>x y. x \<le> y \<Longrightarrow> P x y"   and "\<And>x y z. \<lbrakk>Plus_Times_pre_cong R x y; P x y; Plus_Times_pre_cong R y z; P y z\<rbrakk> \<Longrightarrow> P x z"   and "\<And>x y. R x y \<Longrightarrow> P x y"   and "\<And>x y x' y'. \<lbrakk>Plus_Times_pre_cong R x y; P x y; Plus_Times_pre_cong R x' y'; P x' y'\<rbrakk> \<Longrightarrow> P (Plus x x') (Plus y y')"   and "\<And>x y x' y'. \<lbrakk>Plus_Times_pre_cong R x y; P x y; Plus_Times_pre_cong R x' y'; P x' y'\<rbrakk> \<Longrightarrow> P (Times x x') (Times y y')" shows "P x1 x2"
FOREACHc_rule': fixes S :: "'b set"   and I :: "'b set \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma>0 :: "'c"   and c :: "'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c nres"   and P :: "'c \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. \<lbrakk>I {} \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>it \<sigma>. \<lbrakk>it \<subseteq> S; I it \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "FOREACH\<^sub>C S c f \<sigma>0 \<le> SPEC P"
ev_inv: fixes f :: "'a stream \<Rightarrow> 'b stream"   and P :: "'b stream \<Rightarrow> bool"   and s :: "'a stream" assumes "\<And>s. f (stl s) = stl (f s)" shows "ev P (f s) = ev (\<lambda>x. P (f x)) s"
bij_betw_imageI: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "inj_on f A"   and "f ` A = B" shows "bij_betw f A B"
fref_to_Down_curry_right: fixes f :: "'a \<Rightarrow> 'b nres"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e nres"   and P :: "'c \<times> 'd \<Rightarrow> bool"   and A :: "('a \<times> 'c \<times> 'd) set"   and B :: "('b \<times> 'e) set"   and a :: "'c"   and b :: "'d"   and x' :: "'a" assumes "(f, uncurry g) \<in> [P]\<^sub>f A \<rightarrow> \<langle>B\<rangle>nres_rel"   and "P (a, b)"   and "(x', a, b) \<in> A" shows "f x' \<le> \<Down> B (g a b)"
confluentp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "confluentp r = (r\<inverse>\<inverse>\<^sup>*\<^sup>* OO r\<^sup>*\<^sup>* \<le> r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>*\<^sup>*)"
lossless_iter: fixes guard :: "'a \<Rightarrow> bool"   and body :: "'a \<Rightarrow> 'a spmf"   and n :: "nat"   and s :: "'a" assumes "\<And>s. guard s \<Longrightarrow> lossless_spmf (body s)" shows "lossless_spmf (loop_spmf.iter guard body n s)"
AE_PAR_least: fixes x :: "'a" shows "AE p in PAR x. \<exists>y\<in>I x. select_first x p y"
set_category_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set" shows "set_category S \<AA> \<equiv> category S \<and> set_category_axioms S \<AA>"