hcju/isabelleps huggingface dataset:
B_def: shows "B = (\<lambda>n. A \<inter> (T ^^ (n + 1)) --` A)"
univ_poly_units: fixes K :: "'a list set" assumes "subfield K (poly_ring R)" shows "Units (K [X]\<^bsub>poly_ring R\<^esub>) = {[k] |k. k \<in> K - {\<zero>\<^bsub>poly_ring R\<^esub>}}"
mult_2_2: fixes x :: "real_alg_2"   and y :: "real_alg_2" assumes "invariant_2 x"   and "invariant_2 y" shows "invariant_2 (mult_2 x y)"
lin_red_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "lin_red F p q \<equiv> \<exists>f\<in>F. red_single p q f (0::'a)"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "equivalence L"   and "weak_partial_order_axioms L" shows "weak_partial_order L"
M2Zg: shows "m2z\<^bsub>C\<^esub>g |\<in>| ((YFtor C @@ Y) @@ X)"
j: shows "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
reds_preserves_shconf: fixes es :: "char list exp list"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and es' :: "char list exp list"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "P \<turnstile> \<langle>es,(h, l, sh),b\<rangle> [\<rightarrow>] \<langle>es',(h', l', sh'),b'\<rangle>"   and "P,E,h,sh \<turnstile> es [:] Ts"   and "P,h \<turnstile>\<^sub>s sh \<surd>" shows "P,h' \<turnstile>\<^sub>s sh' \<surd>"
C_meet_is_meet: fixes z :: "C"   and x :: "C"   and y :: "C" shows "(z \<sqsubseteq> C_meet\<cdot>x\<cdot>y) = (z \<sqsubseteq> x \<and> z \<sqsubseteq> y)"
uniformly_Cauchy_imp_Cauchy: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'b" assumes "uniformly_Cauchy_on X f"   and "x \<in> X" shows "Cauchy (\<lambda>n. f n x)"
list_commute: fixes \<Gamma> :: "'a formula multiset" shows "mset (map (\<lambda>d. disj_of_clause a \<^bold>\<or> disj_of_clause d) F) + \<Gamma> \<Rightarrow>\<^sub>n = mset (map (\<lambda>d. disj_of_clause d \<^bold>\<or> disj_of_clause a) F) + \<Gamma> \<Rightarrow>\<^sub>n"
conj_disj_distrib: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "conj x (disj y z) = disj (conj x y) (conj x z)"
map_S_id: fixes x :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) S" shows "map_S id id id id id id x = x"
landau_symbol_at_top_imp_at: fixes f :: "real \<Rightarrow> 'f"   and a :: "real"   and g :: "real \<Rightarrow> 'f" assumes "(\<lambda>x. f (a - inverse x)) \<in> L' at_top (\<lambda>x. g (a - inverse x))"   and "(\<lambda>x. f (a + inverse x)) \<in> L' at_top (\<lambda>x. g (a + inverse x))" shows "f \<in> L (at a) g"
ordLeq_ordIso_trans: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "ordLeq3 r r'"   and "ordIso2 r' r''" shows "ordLeq3 r r''"
chamber_in_subcomplex: fixes Y :: "'a set set"   and C :: "'a set" assumes "folding_g.ChamberSubcomplex Y"   and "C \<in> Y"   and "folding_g.chamber C" shows "SimplicialComplex.maxsimp Y C"
abs_induct: fixes P :: "('b, 'c) residual \<Rightarrow> bool"   and x :: "('b, 'c) residual" assumes "\<And>y. P (abs_residual y)" shows "P x"
\<omega>_limit_pointE: fixes x :: "'a"   and p :: "'a"   and b :: "real" assumes "\<omega>_limit_point x p" obtains s :: "nat \<Rightarrow> real" where "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>" and "(flow0 x \<circ> s) \<longlonglongrightarrow> p" and "\<forall>n. b \<le> s n"
simps_4: fixes v :: "'a"   and va :: "'a"   and vb :: "'b" shows "first_srcNet (AllowPortFromTo v va vb) = srcNet (AllowPortFromTo v va vb)"
dominators_def: fixes g :: "'a"   and n :: "'node" shows "gE2.dominators g n \<equiv> {m \<in> set (\<alpha>n2 g). gE2.dominates g m n}"
i_contract_def: fixes f :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) Ring_scheme"   and R' :: "('d, 'f) Ring_scheme"   and J :: "'d set" shows "i_contract f R R' J = invim f (carrier R) J"
finite_image_set: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "finite {x. P x}" shows "finite {f x |x. P x}"
assms_6: shows "gram_schmidt_fs.reduced n (map of_int_hom.vec_hom fs) \<alpha> i"
start_remove_decomp: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<in> set xs" shows "\<exists>as bs. xs = as @ x # bs \<and> start_remove xs x ys = rev ys @ as @ remove_cycles bs x [x]"
remove_vertex_path_from_to_add: fixes x :: "'a"   and v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H.remove_vertex x\<^esub> w" shows "v \<leadsto>xs\<leadsto>\<^bsub>remove_vertex v\<^esub> w"
simps: fixes c :: "nat"   and d :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "shift2 c d qs = (if c \<le> d then qs else shift2 c (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}))))"
prems_2: shows "lead_coeff ([:- b, 1:] * r) = 1"
sorted_list_of_multiset_mset: fixes xs :: "'a list" shows "sorted_list_of_multiset (mset xs) = sort xs"
resultant_swap: fixes f :: "'a poly"   and g :: "'a poly" shows "resultant f g = (- (1::'a)) ^ (degree f * degree g) * resultant g f"
representsE_full: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" obtains "p \<noteq> 0" and "ipoly p x = (0::'a)" and "Polynomial.degree p \<noteq> 0"
map_of_sinter_with: fixes xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a" assumes "sorted (map fst xs)"   and "sorted (map fst ys)" shows "map_of (sinter_with f xs ys) k = (case map_of xs k of None \<Rightarrow> None | Some v \<Rightarrow> map_option (f k v) (map_of ys k))"
fmran'E: fixes y :: "'e"   and m :: "'f \<rightharpoonup>\<^sub>f 'e" assumes "y \<in> fmran' m" obtains x :: "'f" where "fmlookup m x = Some y"
permI: fixes f :: "atom \<Rightarrow> atom" assumes "bij f"   and "\<forall>\<^sub>\<infinity>x. f x = x"   and "\<And>a. sort_of (f a) = sort_of a" shows "f \<in> perm"
g: fixes x :: "real"   and y :: "real" assumes "a \<le> x"   and "x \<le> y"   and "y \<le> b" shows "g x \<le> g y"
FL_valid_eqvt: fixes P :: "'state"   and x :: "('b, 'pred, 'act, 'effect) formula"   and p :: "perm" assumes "P \<Turnstile> x" shows "p \<bullet> P \<Turnstile> p \<bullet> x"
I_inj_on_S: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and l :: "nat"   and u :: "nat" assumes "integrate_insert_commute M a s"   and "l < length s \<and> u < length s \<and> I (s ! l) = I (s ! u)" shows "l = u"
unfold_congs_2: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "m r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "m_update f r = m_update f' r'"
word_of_nat_less_eq_iff: fixes m :: "nat"   and n :: "nat" shows "(word_of_nat m \<le> word_of_nat n) = (take_bit LENGTH('k) m \<le> take_bit LENGTH('k) n)"
simps_4: fixes I :: "('e, 'f, 'd) interp"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" shows "dterm_sem I (Plus t1 t2) = (\<lambda>v. dterm_sem I t1 v + dterm_sem I t2 v)"
decode_state_at_dom: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and k :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "dom (\<Phi>\<^sub>S\<inverse> \<Pi> \<A> k) = set (\<Pi>\<^sub>\<V>)"
con: fixes C :: "'a set" assumes "C \<in> \<T>" shows "convex C"
Seq_append_OrdDom: fixes k :: "hf"   and s1 :: "hf"   and s2 :: "hf" assumes "Ord k"   and "OrdDom s1"   and "OrdDom s2" shows "OrdDom (seq_append k s1 s2)"
simps_2: fixes \<phi> :: "'b TESL_atomic"   and \<Phi> :: "'b TESL_atomic list"   and n :: "nat" shows "\<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = \<lbrakk> \<phi> \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> \<inter> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>"
IH: fixes e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "Vars.var list" assumes "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub>)\<cdot>r \<noteq> \<bottom>" shows "\<exists>\<Delta> a. \<Gamma> : e \<Down>\<^bsub>S\<^esub> \<Delta> : a"
dom_glb: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "d x \<le> d y"   and "d x \<le> d z" shows "d x \<le> d y \<cdot> d z"
x_1: shows "(\<lambda>xa. ereal \<bar>birkhoff_sum f xa x\<bar>) \<longlonglongrightarrow> \<infinity>"
var_swap_ref3: fixes b :: "'b"   and a :: "'b" shows "(''y'' ::= (\<lambda>s. s ''z'')) ; rel_R \<lceil>\<lambda>s. s ''x'' = b \<and> s ''y'' = a\<rceil> \<lceil>\<lambda>s. s ''x'' = b \<and> s ''y'' = a\<rceil> \<subseteq> rel_R \<lceil>\<lambda>s. s ''z'' = a \<and> s ''x'' = b\<rceil> \<lceil>\<lambda>s. s ''x'' = b \<and> s ''y'' = a\<rceil>"
scc_of_in_sccs_verts: fixes u :: "'a" assumes "u \<in> verts H" shows "H.scc_of u \<in> H.sccs_verts"
horizontal_homs_axioms: shows "horizontal_homs (\<cdot>\<^sub>B) src\<^sub>B trg\<^sub>B"
apex_def: shows "\<mu>.cod.apex \<equiv> C.dom \<mu>.cod.leg0"
minus_mult_div_eq_mod: fixes a :: "'a"   and b :: "'a" shows "a - b * (a div b) = a mod b"
ih: shows "wfP (\<lambda>xs ys. length ys = n \<and> length xs = n \<and> ext gt ys xs)"
prems: shows "invar \<langle>l, (x, n), r\<rangle>"
compatible_with_structure_def: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" shows "compatible_with_structure x = (\<forall>t1 t2 s1 s2. x t1 s1 \<longrightarrow> x t2 s2 \<longrightarrow> x (t1 \<cdot> t2) (s1 \<cdot> s2))"
insertion_var: fixes var :: "nat"   and L :: "'b list"   and x :: "'b" assumes "var < length L" shows "insertion (nth_default (0::'b) (L[var := x])) (Var var) = x"
elapsed: shows "\<forall>i. eq_elapsed (xs !! i) (xs !! Suc i)"
mult_fract: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "Fraction_Field.Fract a b * Fraction_Field.Fract c d = Fraction_Field.Fract (a * c) (b * d)"
r_cancel: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<otimes> c = b \<otimes> c"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a = b"
intros_44: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and vs :: "val list"   and s :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "length vs = length pns"   and "length Ts = length pns" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(map Val vs),s,True\<rangle> \<rightarrow> \<langle>blocks (pns, Ts, vs, body),s,False\<rangle>"
order_subst1: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and b :: "'b"   and c :: "'b" assumes "a \<le> f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
IH: fixes x :: "nat"   and e :: "cexpr" shows "free_vars (cexpr_comp_aux x e e') \<subseteq> free_vars e' - {x} \<union> free_vars e"
lemma_4_12_i_ii: fixes a :: "'a" assumes "a \<noteq> (1::'a)" shows "filterof {a} = UNIV"
a1: shows "s' = snd (fst (shift_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
join_def: fixes M :: "'c measure measure" shows "join M = measure_of (space (select_sets M)) (sets (select_sets M)) (\<lambda>B. \<integral>\<^sup>+ M'. emeasure M' B \<partial>M)"
Maxr_lo: fixes x :: "nat"   and y :: "nat" assumes "Suc 0 < x"   and "Suc 0 < y" shows "Maxr loR [x, y] x = lo x y"
swap_compose: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good X" shows "X #[x \<and> y]_zs #[x' \<and> y']_zs' = X #[x' \<and> y']_zs' #[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']_zs"
has_derivatives_less_lemma: fixes v :: "real \<Rightarrow> real"   and v' :: "real \<Rightarrow> real"   and T :: "real set"   and y :: "real \<Rightarrow> real"   and y' :: "real \<Rightarrow> real"   and t0 :: "real"   and f :: "real \<Rightarrow> real \<Rightarrow> real"   and t :: "real" assumes "(v has_vderiv_on v') T"   and "(y has_vderiv_on y') T"   and "\<And>t. \<lbrakk>t \<in> T; t0 < t\<rbrakk> \<Longrightarrow> v' t - f t (v t) < y' t - f t (y t)"   and "v t0 \<le> y t0"   and "v t0 = y t0 \<Longrightarrow> v' t0 < y' t0"   and "t0 < t"   and "t0 \<in> T"   and "t \<in> T"   and "is_interval T" shows "v t < y t"
inv_inv_next_tame0: fixes p :: "nat" shows "invariant Invariants.inv next_tame0\<^bsub>p\<^esub>"
Cod_f: shows "\<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = d"
Suc_1: fixes A2 :: "'a mat"   and A3 :: "'a mat"   and A4 :: "'a mat" assumes "A2 = 0\<^sub>m n m"   and "A3 \<in> carrier_mat m n"   and "A4 \<in> carrier_mat m m" shows "det (four_block_mat A1 A2 A3 A4) = det A1 * det A4"
invar_r_ge_k: fixes k :: "'a"   and v :: "real"   and l :: "'a kdt"   and r :: "'a kdt" assumes "invar (Node k v l r)" shows "\<forall>p\<in>set_kdt r. v < p $ k"
merge_subset: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "local.merge i \<subseteq> local.merge j"
min_for_def: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "linorder_list0.min_for le a b = (if le a b then a else b)"
1: shows "\<forall>l\<in>reachable_s PROB s. \<exists>as. l = exec_plan s as \<and> as \<in> valid_plans PROB"
pivotal_exists: fixes b :: "alt" shows "\<exists>i. pivotal i b"
qSwap_preserves_alpha: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm"   and zs :: "'e"   and z1 :: "'d"   and z2 :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X \<or> FixVars.qGood X'"   and "X #= X'" shows "X #[[z1 \<and> z2]]_zs #= X' #[[z1 \<and> z2]]_zs"
assms: shows "range_vars \<theta> \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M"
Par2F: fixes Q :: "pi"   and \<alpha> :: "Early_Semantics.freeRes"   and Q' :: "pi"   and P :: "pi" assumes "Q \<longmapsto>\<^sub>e \<alpha> \<prec> Q'" shows "P \<parallel> Q \<longmapsto>\<^sub>e \<alpha> \<prec> P \<parallel> Q'"
bv_to_nat_helper: fixes b :: "bit"   and bs :: "bit list" shows "bv_to_nat (b # bs) = bitval b * 2 ^ length bs + bv_to_nat bs"
falsifies\<^sub>c\<^sub>s_empty: fixes C :: "fterm literal set" assumes "unification"   and "falsifies\<^sub>c [] C" shows "C = {}"
assms_1: fixes \<pi> :: "nat \<Rightarrow> nat" assumes "\<pi> permutes A" shows "g \<pi> permutes X"
mono: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and gt' :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "ext ext"   and "gt \<le> gt'" shows "ext gt \<le> ext gt'"
rv_X: fixes i :: "'b" assumes "i \<in> I" shows "random_variable borel (X i)"
normal_invE_1: fixes N :: "'a set" assumes "N \<lhd> add_monoid R" shows "subgroup N (add_monoid R)"
aff_dim_openin: fixes T :: "'a set"   and S :: "'a set" assumes "openin (top_of_set T) S"   and "affine T"   and "S \<noteq> {}" shows "aff_dim S = aff_dim T"
v1: assumes "((\<lambda>x. sqrt (Re (cinner x x))) has_derivative (\<lambda>xa. Re (cinner x xa) * inverse (sqrt (Re (cinner x x))))) (at x)" shows "((\<lambda>x. sqrt (Re (cinner x x))) has_derivative (\<lambda>y. Re (cinner x y) / sqrt (Re (cinner x x)))) (at x)"
ordering_oSuc: fixes A :: "ordinal set"   and x :: "ordinal" shows "OrdinalVeblen.ordering A (oSuc x) = (LEAST z. z \<in> A \<and> OrdinalVeblen.ordering A x < z)"
iterate_is_empty_by_size_def: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat" shows "iterate_is_empty_by_size it = (iterate_size_abort it 1 = 0)"
monom_mult: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and n :: "nat"   and m :: "nat" assumes "UP_ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "monom (UP R) (a \<otimes>\<^bsub>R\<^esub> b) (n + m) = monom (UP R) a n \<otimes>\<^bsub>UP R\<^esub> monom (UP R) b m"
pp_is_le_rel_canc_left: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a"   and u :: "'a"   and v :: "'a" assumes "ordered_powerprod_lin.is_le_rel rel" shows "rel (t + u) (t + v) = rel u v"
to_tree_order_same_root: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root_ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_root_node ptr \<rightarrow>\<^sub>r root_ptr"   and "h \<turnstile> Shadow_DOM.to_tree_order root_ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to" shows "h \<turnstile> Shadow_DOM.get_root_node ptr' \<rightarrow>\<^sub>r root_ptr"
mk_minsky_inc: fixes v :: "nat" shows "mk_minsky (\<lambda>vs vs'. vs' = (\<lambda>x. if x = v then Suc (vs v) else vs x))"
cet_op: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and oper :: "pdf_operator"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c e : t"   and "op_type oper t = Some t'" shows "\<Gamma> \<turnstile>\<^sub>c oper $$\<^sub>c e : t'"
harm_expansion_bigo_simple2: shows "(\<lambda>n. harm n - (ln (real n) + euler_mascheroni)) \<in> O(\<lambda>n. 1 / real n)"
monom_def: shows "MPoly_Type.monom \<equiv> map_fun id (map_fun id MPoly) Poly_Mapping.single"
perm_6: fixes pi :: "(name \<times> name) list"   and x1 :: "name"   and x2 :: "('d, 'e, 'f) psi" shows "pi \<bullet> \<lparr>\<nu>x1\<rparr>x2 = \<lparr>\<nu>(pi \<bullet> x1)\<rparr>(pi \<bullet> x2)"
WN_onE: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and a :: "'b" assumes "WN_on r A"   and "a \<in> A" obtains b :: "'b" where "(a, b) \<in> r\<^sup>!"
normalize_positive_src_ports_def: shows "normalize_positive_src_ports = normalize_positive_ports_step (is_Src_Ports, src_ports_sel) Src_Ports"
assms: assumes "\<phi> \<pi>l" shows "\<psi> \<pi>l"
sup_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
HComplex_eq_1: fixes x :: "real star"   and y :: "real star" shows "(HComplex x y = 1) = (x = 1 \<and> y = 0)"
embed_measure_def: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'a" shows "embed_measure M f = measure_of (f ` space M) {f ` A |A. A \<in> sets M} (\<lambda>A. emeasure M (f -` A \<inter> space M))"
assms: shows "sp F xs (replicate n None) = Ok \<Sigma>'"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and negation_type :: "'c negation_type" shows "P (case negation_type of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = ((\<forall>x1. negation_type = Pos x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. negation_type = Neg x2 \<longrightarrow> P (f2 x2)))"
perm_2: fixes pi :: "(NominalSequents.var \<times> NominalSequents.var) list"   and x2 :: "char list"   and x1 :: "form_list" shows "pi \<bullet> Cpd0 x2 x1 = Cpd0 (pi \<bullet> x2) (pi \<bullet> x1)"
set_sequence_of: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "set (sequence_of A enum f) \<subseteq> B"
Henstock_lemma: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and e :: "real" assumes "f integrable_on cbox a b"   and "0 < e" obtains \<gamma> :: "'c \<Rightarrow> 'c set" where "gauge \<gamma>" and "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; \<gamma> fine p\<rbrakk> \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e"
gp_expr: shows "LINT x|M. f x * enn2real (nn_cond_exp M F gp x) = LINT x|M. f x * enn2real (gp x)"
start_to_end: fixes x :: "'a" assumes "path x" shows "start_points x ; end_points x\<^sup>T \<le> x\<^sup>\<star>"
methd_declaredD: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "wf_prog G"   and "is_class G C"   and "methd G C sig = Some m" shows "G\<turnstile>Methd sig  m declared_in declclass m"
polyfun_submatrix_deep_model: fixes i :: "nat"   and j :: "nat" assumes "i < r ^ N_half"   and "j < r ^ N_half" shows "polyfun {..<weight_space_dim} (\<lambda>f. witness_submatrix f $$ (i, j))"
freshEnv_swapEnv_id: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "goodEnv rho"   and "freshEnv xs x1 rho"   and "freshEnv xs x2 rho" shows "rho &[x1 \<and> x2]_xs = rho"
tagged_division_ofD_5: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set" assumes "s tagged_division_of i"   and "(x1, K1) \<in> s"   and "(x2, K2) \<in> s"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
b: shows "berlekamp_resulting_mat u \<in> carrier_mat (degree u) (degree u)"
partGen_def: fixes P :: "'b set set" shows "partGen P \<equiv> {gen P I |I. I \<in> P}"
square_part_le: fixes n :: "nat" shows "square_part n \<le> n"
monofun_def: fixes f :: "'b \<Rightarrow> 'c" shows "monofun f = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> f x \<sqsubseteq> f y)"
find_first_correct_3: fixes x :: "'b"   and xs :: "'b list"   and i :: "nat" assumes "find_first x xs = Some i" shows "x \<notin> set (take i xs)"
poincare_collinear3_iff: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc" shows "poincare_collinear {u, v, w} = (poincare_between u v w \<or> poincare_between v u w \<or> poincare_between v w u)"
weak_bounded_lattice_axioms: shows "weak_bounded_lattice (fpl \<P> \<H>)"
eq: shows "(\<Union>n. {S n..b}) = {a<..b}"
mono: fixes callee :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd, 'e, 'f) gpv"   and x :: "('a, 'b, 'c) gpv \<times> 'd" shows "spmf.mono_body (\<lambda>inline1. case x of (gpv, s) \<Rightarrow> the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). curry inline1 (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv)))))"
hm_correct_5: fixes ma :: "('a, 'b) HashMap.hashmap"   and k :: "'a" assumes "hm.invar ma" shows "hm.lookup k ma = hm.\<alpha> ma k"
hom_zero: fixes x :: "'a"   and y :: "'a" shows "poly2 0 x y = (0::'a)"
csol_def: fixes t0 :: "real"   and x0 :: "'a" shows "csol t0 x0 = (SOME csol. \<forall>t\<in>existence_ivl t0 x0. (csol t, t) \<in> csols t0 x0)"
sets_PiM_I_finite: fixes I :: "'c set"   and E :: "'c \<Rightarrow> 'd set"   and M :: "'c \<Rightarrow> 'd measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
interval_bisection: fixes P :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "P {}"   and "\<And>S T. \<lbrakk>P S; P T; interior S \<inter> interior T = {}\<rbrakk> \<Longrightarrow> P (S \<union> T)"   and "\<not> P (cbox a b)" obtains x :: "'a" where "x \<in> cbox a b" and "\<forall>e>0. \<exists>c d. x \<in> cbox c d \<and> cbox c d \<subseteq> ball x e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
axioms_1: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V" shows "FGModule G smult V"
moebius_mu_def: fixes n :: "nat" shows "moebius_mu n = (if squarefree n then (- (1::'b)) ^ card (prime_factors n) else (0::'b))"