hcju/isabelleps huggingface dataset:
gen_shleg_poly_n: shows "local.gen_shleg_poly n = Polynomial.smult (fact n) local.shleg_poly"
Extend1: shows "l2 = l1 @ l12 \<and> l12 \<noteq> []"
going_to_within_def: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'d filter"   and A :: "'c set" shows "f going_to F within A = inf (filtercomap f F) (principal A)"
left_idem: fixes a :: "'b word"   and b :: "'b word" shows "signed.min a (signed.min a b) = signed.min a b"
beta: shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* u"
DERIV_ident: fixes F :: "'a filter" shows "((\<lambda>x. x) has_field_derivative (1::'a)) F"
\<omega>_nf_exists: fixes \<alpha> :: "V" assumes "Ord \<alpha>" obtains \<alpha>s :: "V list" where "list.set \<alpha>s \<subseteq> ON" and "sorted_wrt (\<lambda>x y. y \<le> x) \<alpha>s" and "\<alpha> = \<omega>_sum \<alpha>s"
select_convs_17: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "sip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = sip"
invertible_left_inverse2: fixes u :: "'a"   and v :: "'a" assumes "invertible u"   and "u \<in> G"   and "v \<in> G" shows "local.inverse u \<cdot> (u \<cdot> v) = v"
card_partition_on_size1_eq_1: fixes A :: "'a set"   and k :: "nat" assumes "finite A"   and "card A \<le> k" shows "card {P. partition_on A P \<and> card P \<le> k \<and> (\<forall>X\<in>P. card X = 1)} = 1"
Un_empty: fixes A :: "'a set"   and B :: "'a set" shows "(A \<union> B = {}) = (A = {} \<and> B = {})"
simpdvd_numbound0: fixes t :: "MIR.num"   and d :: "int" assumes "numbound0 t" shows "numbound0 (snd (simpdvd d t))"
\<psi>_natural: fixes f :: "'c"   and x :: "'c"   and x' :: "'c"   and g :: "'d"   and y' :: "'d"   and y :: "'d"   and h :: "'d" assumes "\<guillemotleft>f : x \<rightarrow>\<^sub>C x'\<guillemotright>"   and "\<guillemotleft>g : y' \<rightarrow>\<^sub>D y\<guillemotright>"   and "\<guillemotleft>h : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "f \<cdot>\<^sub>C \<psi> x h \<cdot>\<^sub>C F g = \<psi> x' ((G f \<cdot>\<^sub>D h) \<cdot>\<^sub>D g)"
wt_subst_TAtom_subterms_subst: fixes \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "\<forall>x\<in>fv t. \<exists>a. \<Gamma> (Var x) = Var a"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<theta> ` fv t)" shows "subterms (t \<cdot> \<theta>) = subterms t \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
preserves_arr: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.arr f" shows "S.arr (S.trg f)"
F: shows "simAbs ` set (simTrans a (runJP mkAutoSim t a)) = {sim_equiv_class a (t' \<leadsto> s) |t' s. t' \<leadsto> s \<in> jkbpC \<and> jview a t' = jview a t}"
frameImpNilStatEq: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "\<langle>\<epsilon>, \<Psi>\<rangle> \<hookrightarrow>\<^sub>F \<langle>\<epsilon>, \<Psi>'\<rangle> = \<Psi> \<hookrightarrow> \<Psi>'"
distinct_113: shows "LessOrEqual \<noteq> ShiftRightZeros"
scalar_product_SNOC: fixes xs :: "nat list"   and bs :: "bool list"   and a :: "nat"   and "is" :: "int list" assumes "length xs = length bs" shows "scalar_product (map_index (\<lambda>i m. m + 2 ^ a * map of_bool bs ! i) xs) is = scalar_product xs is + 2 ^ a * scalar_product (map of_bool bs) is"
simps_2: fixes uv :: "'b"   and uw :: "real"   and l :: "'b kdt"   and r :: "'b kdt" shows "KD_Tree.complete (Node uv uw l r) = (KD_Tree.complete l \<and> KD_Tree.complete r \<and> height l = height r)"
fold_union_elem: fixes x :: "'a"   and xss :: "'a set list"   and xs :: "'a set" assumes "x \<in> fold (\<union>) xss xs" obtains ys :: "'a set" where "x \<in> ys" and "ys \<in> set xss \<union> {xs}"
triv_phis_base: fixes g :: "'g" shows "uninst_code.ssa.trivial_phis (\<lambda>_. p_g g s1_phis g) g = uninst_code.ssa.trivial_phis (p_g g s1_phis) g"
hom_frag_minus: fixes h :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and S :: "'b set"   and T :: "'c set"   and a :: "'b \<Rightarrow>\<^sub>0 int" assumes "h \<in> hom (free_Abelian_group S) (free_Abelian_group T)"   and "Poly_Mapping.keys a \<subseteq> S" shows "h (- a) = - h a"
t_list_of_A_log_bound: fixes t :: "'b tree" assumes "braun t" shows "t_list_of_A t \<le> 2 * nat \<lceil>log 2 (real (size t + 1))\<rceil> * size t"
tr: shows "(s, a, s') \<in> automaton.trans (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o)"
find_lasso_def: shows "find_lasso \<equiv> find_ce_spec \<bind> (\<lambda>ce. case ce of None \<Rightarrow> RETURN None | Some (Vr, Vl) \<Rightarrow> reconstruct_lasso Vr Vl \<bind> (\<lambda>l. RETURN (Some l)))"
kruskal_ref_spec: shows "(kruskal getEdges_impl endpoints_impl, uncurry0 (SPEC (\<lambda>r. distinct r \<and> set r \<subseteq> superE \<and> MSF (\<alpha> ` set r)))) \<in> unit_assn\<^sup>k \<rightarrow>\<^sub>a id_assn"
left_right_inverse_matrix_unique: fixes A :: "'b matrix"   and Y :: "'b matrix"   and X :: "'b matrix" assumes "left_inverse_matrix A Y"   and "right_inverse_matrix A X" shows "X = Y"
card_Diff_subset_Int: fixes A :: "'e set"   and B :: "'e set" assumes "finite (A \<inter> B)" shows "card (A - B) = card A - card (A \<inter> B)"
induct: fixes P :: "'d set list \<Rightarrow> 'd list \<Rightarrow> bool"   and a0 :: "'d set list"   and a1 :: "'d list" assumes "P [] []"   and "\<And>A AS a as. P AS as \<Longrightarrow> P (A # AS) (a # as)"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
assms_2: shows "is_arg_min norm (\<lambda>t. t \<in> M) r"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "f (multA x y) = multB (f x) (f y)"
nv_OfWax_2: fixes x :: "Preliminaries.var"   and \<sigma>' :: "'d" shows "x \<notin> nvT (rOfWax \<sigma>')"
IH_1: fixes t' :: "'a tree" assumes "Root_Balanced_Tree.split_min l = (x, t')"   and "l \<noteq> \<langle>\<rangle>" shows "\<Phi> t' - \<Phi> l \<le> 6 * e * real (height l)"
vector_choose_size: fixes c :: "real" assumes "0 \<le> c" obtains x :: "'b" where "norm x = c"
set_trace_conv: fixes d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and i :: "nat"   and xs :: "'b list" shows "set (trace d i xs) = (if xs = [] then {} else insert (deltas d xs i) (set (butlast (trace d i xs))))"
Prefix1_is_CSP: fixes c :: "'d \<Rightarrow> 'c"   and S :: "'d \<Rightarrow> 'c set" shows "is_CSP_process (R (true \<turnstile> \<lambda>(A, A'). do_I c (S (alpha_rp.more A)) (A, A') \<and> alpha_rp.more A' = alpha_rp.more A))"
multrow_mat_carrier: fixes n :: "nat"   and k :: "nat"   and a :: "'c" shows "multrow_mat n k a \<in> carrier_mat n n"
PNT5_imp_PNT4: assumes "\<psi> \<sim>[at_top] (\<lambda>x. x)" shows "\<theta> \<sim>[at_top] (\<lambda>x. x)"
row_space_subset_span_basis_row_space: fixes A :: "(('a, 'b) vec, 'c) vec" shows "row_space A \<subseteq> vec.span (basis_row_space A)"
NullT_leaf_array: fixes C :: "Arraytype" shows "NullT \<le> ArrT C"
bal_i_def: fixes n :: "nat"   and h :: "nat" shows "bal_i n h = (int h \<le> \<lceil>c * log 2 (real n + 1)\<rceil>)"
\<epsilon>': shows "\<epsilon>' = D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)"
algebraic_int_imp_algebraic: fixes x :: "'a" assumes "algebraic_int x" shows "algebraic x"
simps: fixes l :: "color list"   and a :: "nat" shows "valid_point l a = ((\<exists>i. a = i \<and> i + 2 < length l \<and> l ! i = R \<and> l ! (i + 1) = R \<and> l ! (i + 2) = R) \<or> (\<exists>i. a = i \<and> 1 \<le> i \<and> i + 1 < length l \<and> l ! (i - 1) = R \<and> l ! i = R \<and> l ! (i + 1) = R) \<or> (\<exists>i. a = i \<and> 2 \<le> i \<and> i < length l \<and> l ! (i - 2) = R \<and> l ! (i - 1) = R \<and> l ! i = R) \<or> (\<exists>i. a = i \<and> i < length l \<and> l ! i = B))"
poly_roots_normalize: fixes p :: "'a poly" shows "poly_roots (normalize p) = poly_roots p"
real_root_eq_1_iff: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "(root n x = 1) = (x = 1)"
preserves_arr: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.hseq (fst f) (snd f)"
scut: fixes p :: "'state"   and a :: "'action" assumes "p \<in> nodes"   and "en a p"   and "a \<notin> cuts" shows "scut p (ex a p)"
function_one_def: fixes S :: "'a set"   and R :: "('b, 'c) ring_scheme" shows "function_one S R = (\<lambda>x\<in>S. \<one>\<^bsub>R\<^esub>)"
I: fixes c :: "real"   and x :: "'a" assumes "0 < c"   and "x \<in> G" shows "upper_asymptotic_density (V c x) \<le> real_cond_exp M Invariants f x / c"
elims_2: fixes x :: "('state, 'effect) L_state"   and xa :: "'pred" assumes "x \<turnstile>\<^sub>L xa" obtains uu_ :: "'effect" and uv_ :: "'effect fs_set" and P :: "'state" and \<phi> :: "'pred" where "x = AC (uu_, uv_, P)" and "xa = \<phi>" and "P \<turnstile> \<phi>"
eval_Method_i_i_i_o_o_o_o_conv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and M :: "char list" shows "pred.eval (Method_i_i_i_o_o_o_o P C M) = (\<lambda>(Ts, T, m, D). P \<turnstile> C sees M: Ts\<rightarrow>T = m in D)"
closed_orbit_flow0: fixes x :: "'a"   and t :: "real" assumes "closed_orbit x" shows "closed_orbit (flow0 x t)"
simps_4: fixes test :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b regex"   and s :: "'b regex" shows "match test (Times r s) = match test r OO match test s"
f1: shows "Deliver (i, Store e mo i) \<in> set (history j)"
verit_la_disequality: fixes a :: "'g"   and b :: "'g" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
vsv_appE: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "r\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" obtains "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
E_tl: fixes v :: "trm"   and fs :: "(char list \<times> trm) list"   and fs' :: "(char list \<times> trm) list"   and l :: "char list" assumes "v \<in> value"   and "fs [\<longmapsto>] fs'" shows "(l, v) \<Colon> fs [\<longmapsto>] (l, v) \<Colon> fs'"
measure_eq_sum_singleton: fixes S :: "'b set"   and M :: "'b measure" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. x \<in> S \<Longrightarrow> emeasure M {x} \<noteq> \<infinity>" shows "Sigma_Algebra.measure M S = (\<Sum>x\<in>S. Sigma_Algebra.measure M {x})"
hyps_1: fixes b :: "name"   and ba :: "Lambda_Z.term" shows "subst b ba\<^sup>\<bullet> t\<^sub>1\<^sup>\<bullet> \<rightarrow>\<^sub>\<beta>\<^sup>* (subst b ba t\<^sub>1)\<^sup>\<bullet>"
rel_fset_image: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d fset"   and B :: "'e fset"   and Q :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "rel_fset P A B"   and "\<And>a b. \<lbrakk>a |\<in>| A; b |\<in>| B; P a b\<rbrakk> \<Longrightarrow> Q (f a) (g b)" shows "rel_fset Q (f |`| A) (g |`| B)"
lim_explicit: fixes f :: "nat \<Rightarrow> 'b"   and f0 :: "'b" shows "f \<longlonglongrightarrow> f0 = (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))"
no_overlap: fixes x :: "variable \<times> typ" assumes "x \<in> \<Union> (fv ` snd ` set insts)" shows "x \<notin> fst ` set insts"
remove_set: fixes h :: "'b \<Rightarrow> int"   and t :: "'b Tree"   and e :: "'b" shows "sortedTree h t \<longrightarrow> setOf (remove h e t) = setOf t - eqs h e"
tensor_mat_hermitian: fixes m1 :: "'b mat"   and m2 :: "'b mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2"   and "hermitian m1"   and "hermitian m2" shows "hermitian (tensor_mat m1 m2)"
identify_blocksD_4: fixes k :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and A :: "'b mat" assumes "k < n"   and "(i, j) \<in> set (??.Jordan_Normal_Form_Existence.identify_blocks A k)"   and "Suc j \<noteq> k" shows "A $$ (j, Suc j) \<noteq> (1::'b)"
can_runit: fixes a :: "'c local.term set" assumes "ide a" shows "can \<r>[a]"
simps_1: fixes i :: "nat" shows "tuple_decode 0 i = [i]"
Cdom: fixes f :: "'b" assumes "f \<in> mor" shows "dom f \<in> obj"
simps_4: fixes x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree"   and y41 :: "'d"   and y42 :: "'c"   and y43 :: "('a, 'b, 'c, 'd) Tree" shows "(tAct x41 x42 x43 = tAct y41 y42 y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
power_numeral_even: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and w :: "num" assumes "class.monoid_mult one times" shows "power.power one times z (numeral (num.Bit0 w)) = (let w = power.power one times z (numeral w) in times w w)"
simps_7: fixes as :: "Arity list"   and v :: "Vars.var"   and S :: "stack_elem list" shows "Fstack as (Upd v # S) = Fstack as S"
subset_spec_ivls_def: fixes X :: "'c set"   and Y :: "'c set" shows "subset_spec_ivls X Y = sets_of_coll Y \<bind> (\<lambda>Ys. FORWEAK Ys (RETURN False) (op_subset X) (\<lambda>a b. RETURN (a \<or> b)))"
neighbourhood_base_of: fixes P :: "'c set \<Rightarrow> bool"   and X :: "'c topology" shows "neighbourhood_base_of P X = (\<forall>W x. openin X W \<and> x \<in> W \<longrightarrow> (\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W))"
tuple_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "tuple f g = mkArr (Dom f) (local.set (local.prod (cod f) (cod g))) (\<lambda>x. UP \<langle>DOWN (Fun f x), DOWN (Fun g x)\<rangle>)"
map\<^sub>0_simps_1: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"   and "horizontal_homs.obj C src\<^sub>C a" shows "horizontal_homs.obj D src\<^sub>D (weak_arrow_of_homs.map\<^sub>0 src\<^sub>D F a)"
ring_simprules_11: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z)"
sign_simps_11: fixes ab :: "'a"   and bb :: "'a"   and cb :: "'a" shows "ab - (bb - cb) = ab + cb - bb"
goodXx''x_rho'': shows "qGood X #[[x'' \<and> x]]_xs #[[rho'']]"
that_2: assumes "\<bar>y\<bar> = - y" shows "thesis"
elims_2: fixes x :: "Core.term" assumes "eta_reducible x" obtains uu_ :: "typ" and t :: "Core.term" where "x = Abs uu_ (t__ $ Bv 0)" and "\<not> is_dependent t__ \<or> eta_reducible t__"   | uv_ :: "typ" and v :: "String.literal" and va :: "typ" where "x = Abs uv___ (Ct v__ va__)" and "eta_reducible (Ct v__ va__)"   | uv_ :: "typ" and v :: "variable" and va :: "typ" where "x = Abs uva___ (Fv vb__ vaa__)" and "eta_reducible (Fv vb__ vaa__)"   | uv_ :: "typ" and v :: "nat" where "x = Abs uvb___ (Bv vc__)" and "eta_reducible (Bv vc__)"   | uv_ :: "typ" and v :: "typ" and va :: "Core.term" where "x = Abs uvc___ (Abs vd__ va)" and "eta_reducible (Abs vd__ va)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "String.literal" and vc :: "typ" where "x = Abs uvd___ (ve__ $ Ct vba__ vca__)" and "eta_reducible (ve__ $ Ct vba__ vca__)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "variable" and vc :: "typ" where "x = Abs uve___ (vf__ $ Fv vbb__ vcb__)" and "eta_reducible (vf__ $ Fv vbb__ vcb__)"   | uv_ :: "typ" and v :: "Core.term" and vc :: "nat" where "x = Abs uvf___ (vg__ $ Bv (Suc vcc__))" and "eta_reducible (vg__ $ Bv (Suc vcc__))"   | uv_ :: "typ" and v :: "Core.term" and vb :: "typ" and vc :: "Core.term" where "x = Abs uvg___ (vh__ $ Abs vbc__ vcd__)" and "eta_reducible (vh__ $ Abs vbc__ vcd__)"   | uv_ :: "typ" and v :: "Core.term" and vb :: "Core.term" and vc :: "Core.term" where "x = Abs uv_ (v $ (vb $ vc))" and "eta_reducible (v $ (vb $ vc))"   | t :: "Core.term" and u :: "Core.term" where "x = t $ u" and "eta_reducible t \<or> eta_reducible u"
wf_filterQuery_3: fixes I :: "nat set"   and V :: "nat set"   and n :: "nat"   and Qp :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and QQp :: "(nat set \<times> 'a option list set) set"   and QQn :: "(nat set \<times> 'a option list set) set" assumes "I \<subseteq> V"   and "1 \<le> card I"   and "rwf_query n V Qp Qn"   and "QQp = filterQuery I Qp"   and "QQn = filterQueryNeg I Qn" shows "covering I QQp"
last_plus: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "prod g {m..n} = g n * prod g {m..<n}"
strict_mono_limit_ordinal: fixes f :: "nat \<Rightarrow> ordinal" assumes "OrdinalInduct.strict_mono f" shows "limit_ordinal (oLimit f)"
set_intros_1: fixes x21 :: "'k"   and x22 :: "'k list" shows "x21 \<in> set (x21 # x22)"
simps_2: shows "RELABEL' \<noteq> NONSAT_PUSH'"
assms_2: fixes i :: "'a" assumes "i \<notin> S" shows "Rep_ell2 x i = 0"
Int_ac_3: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> B = B \<inter> A"
simps_6: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" shows "rec_nat f1 f2 0 = f1"
bdd_aboveI: fixes A :: "'k multiset set"   and M :: "'k multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<subseteq># M" shows "subset_mset.bdd_above A"
findc_fresh: fixes a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list" assumes "a \<sharp> \<theta>_c" shows "\<theta>_c cmaps a to None"
takeWhile_sing_root: fixes a :: "'b"   and w :: "'b list" shows "takeWhile (\<lambda>x. x = a) w \<in> [a]*"
list_to_fun_def: fixes ts :: "'d list"   and cs :: "'c list"   and t :: "'d" shows "list_to_fun ts cs t = (case map_of (zip ts cs) t of None \<Rightarrow> 0::'c | Some c \<Rightarrow> c)"
invertible_bezout_matrix_transpose: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and i :: "'d" assumes "is_bezout_ext bezout"   and "a < b"   and "A $h i $h a \<noteq> (0::'b)" shows "invertible (Finite_Cartesian_Product.transpose (bezout_matrix (Finite_Cartesian_Product.transpose A) a b i bezout))"
mon_cD: fixes m :: "'f"   and fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and c :: "'g list multiset" assumes "m \<in> mon_c fg c" shows "\<exists>s. s \<in># c \<and> m \<in> mon_s fg s"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D GF.map GF.cmp"
wt_Methd_is_methd: fixes E :: "env"   and C :: "qtname"   and sig :: "sig"   and T :: "tys" assumes "E\<turnstile>In1l (Methd C sig)\<Colon>T" shows "is_methd (prg E) C sig"
bind_update: fixes f :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'k \<Rightarrow> 'a option"   and m :: "'m"   and g :: "'a \<Rightarrow> 'm" shows "bind (update f m) g = update f (bind m g)"
ed_ou_witness_inter: fixes S :: "ed_ou_witness set"   and T :: "ed_ou_witness set" assumes "open S"   and "open T" shows "open (S \<inter> T)"
llist_all2_llength_ltakeWhileD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and Q1 :: "'c \<Rightarrow> bool"   and Q2 :: "'d \<Rightarrow> bool" assumes "llist_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> Q1 x = Q2 y" shows "llength (ltakeWhile Q1 xs) = llength (ltakeWhile Q2 ys)"
bottomD_1: fixes P :: "'a set" assumes "has_bottom P" shows "bottom P \<in> P"
fix_eq: shows "P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>F n,Up\<^sub>F\<rbrakk> = (\<Lambda> X. (\<lambda>fs. Mprefix (Tr\<^sub>F n fs) (\<lambda>e. X (Up\<^sub>F fs e))))\<cdot>(P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>F n,Up\<^sub>F\<rbrakk>)"
compat_act: fixes g :: "'a"   and h :: "'a"   and x :: "'b" shows "g \<in> carrier G \<and> h \<in> carrier G \<longrightarrow> g \<odot> (h \<odot> x) = g \<otimes> h \<odot> x"
canonical_retraction_face_distance_map: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and F :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "A \<in> \<A>"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<in> A"   and "order.greater_eq C F" shows "ChamberComplex.face_distance X F (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A C ` D) = ChamberComplex.face_distance X F D"
ileI1: fixes m :: "eint"   and n :: "eint" assumes "m < n" shows "m + 1 \<le> n"
get_M_pure: fixes ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'u"   and h :: "(_) heap" shows "pure (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h"
\<C>_Low: fixes mem :: "'Var \<Rightarrow> 'Val" shows "\<forall>x\<in>\<C>. dma mem x = Low"
l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr DocumentClass.type_wf DocumentClass.known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
sorted_wrt_take_drop: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and n :: "nat" assumes "sorted_wrt f xs" shows "\<forall>x\<in>set (take n xs). \<forall>y\<in>set (drop n xs). f x y"
int_pow_def2: fixes G :: "('e, 'f) monoid_scheme"   and a :: "'e"   and z :: "int" shows "a [^]\<^bsub>G\<^esub> z = (if z < 0 then inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> nat (- z)) else a [^]\<^bsub>G\<^esub> nat z)"
has_integral_bound_spike_finite: fixes B :: "real"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a" assumes "0 \<le> B"   and "finite S"   and "(f has_integral i) (cbox a b)"   and "\<And>x. x \<in> cbox a b - S \<Longrightarrow> norm (f x) \<le> B" shows "norm i \<le> B * content (cbox a b)"
create_opt_sol_in_LP_1: fixes X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "[A*\<^sub>vx]\<le>b"
insert3_d_def: fixes a :: "'a"   and t :: "'a tree \<times> nat \<times> nat" shows "insert3_d a t = Time_Monad.val (insert3_d_tm a t)"
rename_constrains: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "bij h" shows "(rename h F \<in> h ` A co h ` B) = (F \<in> A co B)"
restrict_correct_aux1_1: fixes m :: "'m1"   and P :: "'u \<times> 'v \<Rightarrow> bool" assumes "invar1 m" shows "\<alpha>2 (restrict P m) = \<alpha>1 m |` {k. \<exists>v. \<alpha>1 m k = Some v \<and> P (k, v)}"
r'_1: shows "r = smult c (map_poly to_fract r')"