hcju/isabelleps huggingface dataset:
is_RGraph: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "NFlow c s t f" shows "RGraph c s t (residualGraph c f)"
add_strict_left_mono: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subset># b" shows "c + a \<subset># c + b"
lprod_3_5: fixes zr :: "'a"   and z :: "'a"   and R :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "(zr, z) \<in> R" shows "([x, y, zr], [x, y, z]) \<in> lprod R"
sum: fixes g :: "'b \<Rightarrow> 'a"   and S :: "'b set" shows "directional_derivative k a (sum g S) = (\<Sum>aa\<in>S. directional_derivative k a (g aa))"
span_base: fixes a :: "'b"   and S :: "'b set" assumes "a \<in> S" shows "a \<in> span S"
simps: fixes a1 :: "'a"   and a2 :: "('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" shows "((a1, a2) \<in> seq ars) = ((\<exists>a. a1 = a \<and> a2 = []) \<or> (\<exists>a \<alpha> b ss. a1 = a \<and> a2 = (\<alpha>, b) # ss \<and> (a, \<alpha>, b) \<in> ars \<and> (b, ss) \<in> seq ars))"
e: fixes fs1' :: "'a list" assumes "fs1 <~~> fs1'"   and "fs1' [\<sim>] fs2" shows "P"
prod_le_def: fixes P1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P2 :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "prod_le P1 P2 = (\<lambda>(p1, p2) (q1, q2). P1 p1 q1 \<and> P2 p2 q2)"
valid_indexE: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" obtains "isb__ = []" and "ds = []"   | "is" :: "nat list" and dsa :: "nat list" and i :: "nat" and d :: "nat" where "isb__ = i # is" and "ds = d # dsa" and "is \<lhd> dsa" and "i < d"
CCexp_pre_simps_7: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and n :: "Arity" shows "ccExp_syn n (scrut ? e1 : e2) = ccExp_syn 0 scrut \<squnion> (ccExp_syn n e1 \<squnion> ccExp_syn n e2) \<squnion> edom (Aexp_syn' 0 scrut) G\<times> (edom (Aexp_syn' n e1) \<union> edom (Aexp_syn' n e2))"
simps: fixes a1 :: "'tp"   and a2 :: "('fsym, 'psym) lit list" shows "(a1 \<turnstile>2 a2) = ((\<exists>\<sigma> c. a1 = \<sigma> \<and> a2 = c \<and> infTp \<sigma>) \<or> (\<exists>c \<sigma>. a1 = \<sigma> \<and> a2 = c \<and> (\<forall>x xa. x \<in> set c \<longrightarrow> xa \<in> nv2L x \<longrightarrow> tpOfV xa = \<sigma> \<longrightarrow> isGuard xa (grdOf c x xa))))"
approx_hrabs: fixes x :: "real star"   and y :: "real star" assumes "x \<approx> y" shows "\<bar>x\<bar> \<approx> \<bar>y\<bar>"
fa_closed: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "hensel p f a" shows "UP_cring.to_fun (padic_int p) f a \<in> carrier (padic_int p)"
ample_wellfounded: fixes q :: "'state"   and A :: "'action set"   and a :: "'action" assumes "ample_set q A"   and "A \<subset> {a. en a q}"   and "a \<in> A" shows "src (ex a q) q"
ALGxy_det_alternativ: fixes A :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and x :: "'c"   and y :: "'c" shows "ALGxy_det A qs init x y = (\<Sum>i | i < length qs \<and> qs ! i \<in> {y, x}. ALG'_det A qs init i y + ALG'_det A qs init i x)"
conv: shows "((\<lambda>y. LINT x|M. f x * indicat_real {..y} x) \<longlongrightarrow> x) at_top"
vec_to_iarray_morph: fixes A :: "('a, 'b) vec"   and B :: "('a, 'b) vec" shows "(A = B) = (vec_to_iarray A = vec_to_iarray B)"
WTrt_env_mono: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and E' :: "char list \<Rightarrow> ty option" assumes "P,E,h \<turnstile> e : T"   and "E \<subseteq>\<^sub>m E'" shows "P,E',h \<turnstile> e : T"
lubH_le_flubH: fixes H :: "'a set" assumes "H = {x \<in> A. (x, f x) \<in> r}" shows "(lub H cl, f (lub H cl)) \<in> r"
mset_eq_setD: fixes xs :: "'c list"   and ys :: "'c list" assumes "mset xs = mset ys" shows "set xs = set ys"
relcompp_apply: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'b"   and c :: "'f" shows "(R OO S) a c = (\<exists>b. R a b \<and> S b c)"
dghm_tdghm_comp_NTMap_app: fixes x :: "V"   and \<HH> :: "V" assumes "x \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN>)\<lparr>NTMap\<rparr>\<lparr>x\<rparr> = \<HH>\<lparr>ArrMap\<rparr>\<lparr>\<NN>\<lparr>NTMap\<rparr>\<lparr>x\<rparr>\<rparr>"
vint_in_Vset_\<omega>2: shows "\<int>\<^sub>\<circ> \<in>\<^sub>\<circ> Vset (\<omega> + \<omega>)"
unat_minus: fixes x :: "'b word" shows "unat (- x) = (if x = 0 then 0 else 2 ^ size x - unat x)"
monotonic_fun_mult: fixes r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and E1 :: "'b multiset"   and E2 :: "'b multiset" assumes "\<And>t s. (t, s) \<in> r \<Longrightarrow> (f t, f s) \<in> r"   and "(E1, E2) \<in> mult r" shows "(image_mset f E1, image_mset f E2) \<in> mult r"
less_oFix_0D: fixes x :: "ordinal"   and F :: "ordinal \<Rightarrow> ordinal" assumes "x < oFix F 0"   and "mono F" shows "x < F x"
init_fin_bisim_simps_3: fixes final1 :: "'a \<Rightarrow> bool"   and final2 :: "'b \<Rightarrow> bool"   and bisim :: "'c \<Rightarrow> 'a \<times> 'd \<Rightarrow> 'b \<times> 'e \<Rightarrow> bool"   and t :: "'c"   and x1 :: "'a"   and m1 :: "'d"   and s2 :: "status"   and x2 :: "'b"   and m2 :: "'e" shows "FWbisimulation_base.init_fin_bisim final1 final2 bisim t ((Finished, x1), m1) ((s2, x2), m2) = (s2 = Finished \<and> bisim t (x1, m1) (x2, m2) \<and> final1 x1 \<and> final2 x2)"
assms: shows "y \<in>\<^sub>\<circ> set {x}"
Max_wrt_among_mono: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" assumes "le x y"   and "x \<in> Max_wrt_among le A"   and "y \<in> A" shows "y \<in> Max_wrt_among le A"
roots_inclI': fixes p :: "'a list"   and m :: "'a multiset" assumes "p \<in> carrier (poly_ring R)"   and "\<And>a. \<lbrakk>a \<in> carrier R; p \<noteq> []\<rbrakk> \<Longrightarrow> alg_mult p a \<le> count m a" shows "roots p \<subseteq># m"
nvars_CONS: fixes x :: "'x"   and \<AA> :: "'i" shows "#\<^sub>V (CONS x \<AA>) = #\<^sub>V \<AA>"
zero_less_iff_neq_zero: fixes n :: "'k multiset" shows "({#} \<subset># n) = (n \<noteq> {#})"
Neg_Disj_I: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> Neg A"   and "H \<turnstile> Neg B" shows "H \<turnstile> Neg (A OR B)"
finite_Inter: fixes M :: "'b set set" assumes "\<exists>A\<in>M. finite A" shows "finite (\<Inter> M)"
minus_coset_filter: fixes A :: "'b set"   and xs :: "'b list" shows "A - List.coset xs = set (filter (\<lambda>x. x \<in> A) xs)"
parPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi" assumes "P \<leadsto>[Rel] Q"   and "(P, Q) \<in> Rel"   and "\<And>P Q R. (P, Q) \<in> Rel \<Longrightarrow> (P \<parallel> R, Q \<parallel> R) \<in> Rel'"   and "\<And>P Q a. (P, Q) \<in> Rel' \<Longrightarrow> (<\<nu>a>P, <\<nu>a>Q) \<in> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "P \<parallel> R \<leadsto>[Rel'] Q \<parallel> R"
inverse_closed: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "group_ow U f z inverse"   and "a \<in> U" shows "inverse a \<in> U"
dg_length: shows "vcard \<CC> = 4\<^sub>\<nat>"
compl_zero: shows "NOT (0::'c) = - (1::'c)"
SimpleLevi: fixes p :: "'b list"   and s :: "'b list"   and a :: "'b"   and list :: "'b list" shows "(p @ s = a # list) = (p = [] \<and> s = a # list \<or> (\<exists>t. p = a # t \<and> t @ s = list))"
parent_child_rel_def: shows "parent_child_rel \<equiv> l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_parent_child_rel get_child_nodes"
continuous_on_components_gen: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>C. C \<in> components S \<Longrightarrow> openin (top_of_set S) C \<and> continuous_on C f" shows "continuous_on S f"
unit_inv_inv: fixes x :: "'a" assumes "unit x" shows "local.inv (local.inv x) = x"
hyps_1: shows "a \<rightarrow>\<^sup>+ y"
unlabel_Cons_2: fixes b :: "'e \<times> 'd"   and A :: "('e \<times> 'd) list" shows "unlabel (b # A) = snd b # unlabel A"
ennreal_mult_right_cong: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" assumes "a \<noteq> 0 \<Longrightarrow> b = c" shows "b * a = c * a"
stabiliser_subgroup: fixes x :: "Vertex" shows "subgroup (tetrahedral.stabiliser x) tetrahedral_group"
rel_compp: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_Fac (A OO B) = rel_Fab A OO rel_Fbc B"
bound_invI: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list"   and outside :: "bool" assumes "LLL_invariant upw i fs"   and "f_bound outside i fs"   and "g_bound fs" shows "LLL_bound_invariant outside upw i fs"
map_option_case: fixes f :: "'e \<Rightarrow> 'd"   and y :: "'e option" shows "map_option f y = (case y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>x\<rfloor> \<Rightarrow> \<lfloor>f x\<rfloor>)"
GDD_axioms: shows "GDD \<V> \<B> \<G> \<Lambda>"
lookupDef_to_\<alpha>n: fixes g :: "'g"   and n :: "'node"   and v :: "'var"   and v' :: "'var"   and n' :: "'node"   and def :: "Def" assumes "lookupDef g n v = (v', n', def)"   and "n \<in> set (\<alpha>n g)" shows "n' \<in> set (\<alpha>n g)"
rep_eq: fixes x :: "'b trm" shows "fvs x = ptrm_fvs (rep_trm x)"
symmetric_reachable': fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>with_proj (list_digraph G)\<^esub> w" shows "w \<rightarrow>\<^sup>*\<^bsub>with_proj (list_digraph G)\<^esub> v"
well_order_implies_wqo: fixes r :: "('a \<times> 'a) set" assumes "well_order r" shows "wqo_on (\<lambda>a b. (a, b) \<in> r) UNIV"
left_right_impl_times: fixes a :: "'b"   and b :: "'b" shows "a * (a r\<rightarrow> b) = (a l\<rightarrow> b) * a"
distrib_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> (b + c) = a \<cdot> b + a \<cdot> c"
w: shows "w = monom 1 1 ^ CARD('a) ^ d mod v"
power_aform_err: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat"   and p :: "nat" assumes "x \<in> aform_err e X"   and "fst (fst X) \<in> float"   and "snd X \<in> float"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "x ^ n \<in> aform_err e (power_aform_err p X n)"
axiom_funcong_def: fixes i :: "String.literal"   and l :: "tm list"   and l' :: "tm list" shows "axiom_funcong i l l' \<equiv> if equal_length l l' then Thm (foldr Imp (zip_eq l l') (Atom (Rl STR ''='' [Fn i l, Fn i l']))) else Thm Truth"
Units_r_inv: fixes x :: "'c" assumes "x \<in> Units H" shows "x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> x = \<one>\<^bsub>H\<^esub>"
wbt_balanceR: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree"   and r' :: "('b \<times> nat) tree"   and a' :: "'b" assumes "wbt \<langle>l, (a, n), r\<rangle>"   and "wbt r'"   and "size r = size r' + 1" shows "wbt (balanceR l a' r')"
topological_basis_open: fixes "open" :: "'b set \<Rightarrow> bool"   and B :: "'b set set"   and X :: "'b set" assumes "class.topological_space open"   and "topological_space.topological_basis open B"   and "X \<in> B" shows "open X"
le_onorm: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "bounded_linear f" shows "norm (f x) / norm x \<le> onorm f"
nth_append_length_plus: fixes xs :: "'b list"   and ys :: "'b list"   and n :: "nat" shows "(xs \<cdot> ys) ! (\<^bold>|xs\<^bold>| + n) = ys ! n"
fls_one_subdegree: shows "fls_subdegree 1 = 0"
f_def: shows "f = (\<lambda>x. enn2real (h x))"
simps_1: fixes S :: "'c set" shows "sstart S [] = streams S"
arrI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<phi>.B.arr f"
simps_4: fixes \<phi> :: "'b pltl" shows "next_free (X\<^sub>p \<phi>) = False"
PiM_iter: shows "distr (M \<Otimes>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M)) (Pi\<^sub>M UNIV (\<lambda>i. M)) (\<lambda>(s, \<omega>). case_nat s \<omega>) = Pi\<^sub>M UNIV (\<lambda>i. M)"
some: fixes a :: "'a"   and b :: "'a" assumes "x = Some a"   and "y = Some b"   and "a \<noteq> b" shows "thesis"
dom_eqI': fixes a :: "'b"   and f :: "'b" assumes "a \<in> C'.domains f" shows "a = C'.dom f"
assms_1: shows "(r \<restriction>\<^sub>\<circ> A)\<lparr>a\<rparr> = b"
rel_is_equivalence: shows "equivalence (Sigma {U. is_open\<^sub>Y U \<and> f x \<in> U} \<O>\<^sub>Y) {(x, y). stfx.rel x y}"
sg4: shows "i2 (Suc t + d) = i1 (Suc t)"
arcsin_eq_iff: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> 1"   and "\<bar>y\<bar> \<le> 1" shows "(arcsin x = arcsin y) = (x = y)"
iorder_equiv: fixes x :: "'a itrm"   and y :: "'a itrm" assumes "x \<simeq> y" shows "iorder x = iorder y"
dual\<^sub>s\<^sub>t_bvars: fixes A :: "('b, 'a) strand_step list" shows "bvars\<^sub>s\<^sub>t (dual\<^sub>s\<^sub>t A) = bvars\<^sub>s\<^sub>t A"
listsetI_nth: fixes xs :: "'a list"   and Xs :: "'a set list" assumes "length xs = length Xs"   and "\<forall>n<length xs. xs ! n \<in> Xs ! n" shows "xs \<in> listset Xs"
uncurry_def: fixes f :: "'i \<Rightarrow> 'j \<Rightarrow> 'k" shows "uncurry f \<equiv> \<lambda>(a, b). f a b"
suitable_prime_bz_def: fixes f :: "int poly" shows "suitable_prime_bz f \<equiv> let lc = lead_coeff f in int (find_prime (\<lambda>n. let p = int n in comm_monoid_mult_class.coprime lc p \<and> separable_impl p f))"
matrix_vector_mult_basis: fixes A :: "((real, 'b) vec, 'a) vec"   and k :: "'b" shows "A *v \<e> k = column k A"
conj_idem: fixes conj :: "'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice"   and chaos :: "'a::refinement_lattice"   and a :: "'a::refinement_lattice" assumes "conjunction (conj::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (chaos::'a::refinement_lattice)" shows "(conj::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) (a::'a::refinement_lattice) a = a"
nn_integral_const: fixes M :: "'a measure"   and c :: "ennreal" shows "\<integral>\<^sup>+ x. c \<partial>M = c * emeasure M (space M)"
mult_adjugate_det: fixes A :: "('c, 'd) sq_matrix" shows "A * Square_Matrix.adjugate A = diag (Square_Matrix.det A)"
tarski_absolute_space_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "tarski_absolute_space C B \<equiv> tarski_first5 C B \<and> tarski_absolute_space_axioms B"
sublist_split_part: fixes lo :: "nat"   and hi :: "nat"   and p :: "nat"   and xs :: "'c list" assumes "lo \<le> hi"   and "lo < p"   and "p < hi"   and "hi < length xs" shows "WB_Sort.sublist xs lo (p - 1) @ xs ! p # WB_Sort.sublist xs (p + 1) hi = WB_Sort.sublist xs lo hi"
remdups_fwd_split_exact: fixes Acc :: "'b set"   and xs :: "'b list"   and ys :: "'b list"   and x :: "'b"   and zs :: "'b list" assumes "remdups_fwd_acc Acc xs = ys @ x # zs" shows "\<exists>us vs. xs = us @ x # vs \<and> x \<notin> Acc \<and> x \<notin> set ys \<and> remdups_fwd_acc Acc us = ys \<and> remdups_fwd_acc (Acc \<union> set ys \<union> {x}) vs = zs"
prime_power_inj'': fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and q :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q" shows "(power.power one times p m = power.power one times q n) = (m = 0 \<and> n = 0 \<or> p = q \<and> m = n)"
f_def: fixes y :: "nat"   and x :: "nat" shows "f y x = set_to_nat (nat_to_set (c_fst x) \<union> {z \<in> nat_to_set (c_snd x). z < y})"
contract_success: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<phi> \<notin> Cn {}" shows "\<phi> \<notin> K \<div> \<phi>"
induces_extended_monoidal_category: shows "extended_monoidal_category (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D"
heap_is_wellformed_children_distinct: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "distinct children"
assms_2: shows "b < emeasure M {x \<in> space M. upper_asymptotic_density {n. P x n} < a}"
elims_2: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "inv_loop6_exit x xa" obtains xb :: "nat" and l :: "cell list" and r :: "cell list" where "x = xb" and "xa = (l, r)" and "\<exists>i j. i + j = xb \<and> 0 < j \<and> (l, r) = (Oc \<up> i, Oc # Bk \<up> j @ Oc \<up> j)"
inject_1: fixes x1 :: "'f option"   and y1 :: "'f option" shows "(Resumption.resumption.Done x1 = Resumption.resumption.Done y1) = (x1 = y1)"
intro: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" assumes "anonymous_sds agents alts sds"   and "neutral_sds agents alts sds" shows "an_sds agents alts sds"
continuous_on_inverse_open: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "open S"   and "continuous_on S f"   and "DIM('b) \<le> DIM('a)"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x" shows "continuous_on (f ` S) g"
IH: fixes n :: "nat" shows "totient (n ^ Suc m) = n ^ m * totient n"
ctxt_supt: fixes C :: "('d, 'e) ctxt"   and s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "C \<noteq> \<box>"   and "s = C\<langle>t\<rangle>" shows "s \<rhd> t"
prj_simps_12: shows "C.seq trg_f.prj\<^sub>0 \<p>\<^sub>0[trg.leg0, trg.leg1 \<cdot> trg_f.prj\<^sub>1]"
gram_schmidt_result_4: fixes ws :: "'a vec list"   and n :: "nat"   and us :: "'a vec list" assumes "set ws \<subseteq> carrier_vec n"   and "distinct ws"   and "\<not> module.lin_dep class_ring (module_vec TYPE('a) n) (set ws)"   and "us = gram_schmidt n ws" shows "length us = length ws"
parCasesF: fixes P :: "pi"   and Q :: "pi"   and \<alpha> :: "freeRes"   and PQ' :: "pi"   and F :: "freeRes \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<parallel> Q \<longmapsto> \<alpha> \<prec> PQ'"   and "\<And>P'. P \<longmapsto> \<alpha> \<prec> P' \<Longrightarrow> F \<alpha> (P' \<parallel> Q)"   and "\<And>Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> F \<alpha> (P \<parallel> Q')"   and "\<And>P' Q' a b. \<lbrakk>P \<longmapsto> a<b> \<prec> P'; Q \<longmapsto> a[b] \<prec> Q'\<rbrakk> \<Longrightarrow> F (\<tau>) (P' \<parallel> Q')"   and "\<And>P' Q' a b. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; Q \<longmapsto> a<b> \<prec> Q'\<rbrakk> \<Longrightarrow> F (\<tau>) (P' \<parallel> Q')"   and "\<And>P' Q' a x. \<lbrakk>P \<longmapsto> a<x> \<prec> P'; Q \<longmapsto> a<\<nu>x> \<prec> Q'; x \<sharp> P; x \<sharp> C\<rbrakk> \<Longrightarrow> F (\<tau>) (<\<nu>x>(P' \<parallel> Q'))"   and "\<And>P' Q' a x. \<lbrakk>P \<longmapsto> a<\<nu>x> \<prec> P'; Q \<longmapsto> a<x> \<prec> Q'; x \<sharp> Q; x \<sharp> C\<rbrakk> \<Longrightarrow> F (\<tau>) (<\<nu>x>(P' \<parallel> Q'))" shows "F \<alpha> PQ'"
div_closed: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "field R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "y \<noteq> \<zero>\<^bsub>R\<^esub>" shows "x \<oslash>\<^bsub>R\<^esub> y \<in> carrier R"
isPath_tail: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and e :: "nat \<times> nat"   and v :: "nat" shows "isPath u (p @ [e]) v = (isPath u p (fst e) \<and> e \<in> E \<and> snd e = v)"
axiom_distribimp: fixes ux :: "fol fm"   and uy :: "fol fm"   and uz :: "fol fm" shows "\<turnstile> concl (axiom_distribimp ux uy uz)"
list_slice2_nth: fixes m :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "m < length xs div k" shows "list_slice2 xs k ! m = xs \<up> (m * k) \<down> k"
9: shows "(forest_components f \<squnion> e \<squnion> e\<^sup>T) * forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) \<le> forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
Int_Un_eq_2: fixes S :: "'e set"   and T :: "'e set" shows "S \<inter> T \<union> T = T"
LeastI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
LLL_d_pos: fixes fs :: "int vec list"   and k :: "nat" assumes "LLL_invariant_weak fs"   and "k \<le> m" shows "0 < d fs k"
cinner_diff_left: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "cinner (x - y) z = cinner x z - cinner y z"
t_A'_t: fixes n :: "nat" assumes "n < length qs" shows "t_A' n = int (t (s_A' n) (qs ! n) (acts ! n))"
suc_underS: fixes r :: "('c \<times> 'c) set"   and a :: "'c" assumes "wo_rel r"   and "a \<in> Field r" shows "a = suc r (underS r a)"
QF_Suc_Suc: fixes k :: "nat"   and j :: "nat"   and i :: "nat" shows "QF k (Suc j, Suc i) = enum (nxt N (Suc (Max (QF k (Suc j, i))))) ` {..<del k (if Suc j = k then m - Suc (Suc i) else Suc i) (Suc j)}"
smalloD_tendsto: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> (0::'b)) F"
subseqs_powset: fixes xs :: "'b list" shows "set ` set (subseqs xs) = Pow (set xs)"
simple_arg_pos_def: fixes rel :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) set"   and f :: "'c \<times> nat"   and i :: "nat" shows "simple_arg_pos rel f i \<equiv> \<forall>ts. i < snd f \<longrightarrow> length ts = snd f \<longrightarrow> (Fun (fst f) ts, ts ! i) \<in> rel"
axioms_1: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_hd_or_tl ext" shows "ext ext"
writes_bind: fixes S :: "('a, 'b, 'c) prog set"   and f :: "('a, 'b, 'd) prog"   and h :: "'a"   and g :: "'d \<Rightarrow> ('a, 'b, 'e) prog"   and h' :: "'a" assumes "\<And>h2. writes S f h h2"   and "\<And>x h2. \<lbrakk>h \<turnstile> f \<rightarrow>\<^sub>r x; h \<turnstile> f \<rightarrow>\<^sub>h h2\<rbrakk> \<Longrightarrow> writes S (g x) h2 h'" shows "writes S (f \<bind> g) h h'"
emsapss_encode_help8_def: fixes DBzero :: "bit list"   and H :: "bit list" shows "emsapss_encode_help8 DBzero H = DBzero @ H @ BC"
gauss_jordan_inverse_other_direction: fixes A :: "'a mat"   and n :: "nat"   and b :: "'b"   and B :: "'a mat"   and nc :: "nat" assumes "A \<in> Units (ring_mat TYPE('a) n b)"   and "B \<in> carrier_mat n nc" shows "fst (gauss_jordan A B) = 1\<^sub>m n"
code_el_dec: fixes \<C> :: "'b list set"   and c :: "'b list" assumes "code \<C>"   and "c \<in> \<C>" shows "Dec \<C> c = [c]"
that: assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1" shows "thesis"
possible_steps_remove_unreachable: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and l :: "String.literal"   and i :: "value list" assumes "obtainable s r e"   and "\<not> reachable s' e" shows "possible_steps (remove_state s' e) s r l i = possible_steps e s r l i"
Weierstrass_l2_trigonometric_set: fixes f :: "real \<Rightarrow> real"   and e :: "real" assumes "f square_integrable {- pi..pi}"   and "0 < e" obtains n :: "nat" and a :: "nat \<Rightarrow> real" where "l2norm {- pi..pi} (\<lambda>x. f x - (\<Sum>k\<le>n. a k * trigonometric_set k x)) < e"