hcju/isabelleps huggingface dataset:
fixp_induct: fixes P :: "(('b option, 'd, 'e) gpv \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool \<Rightarrow> ('b option \<times> 'f \<times> 'g) spmf) \<Rightarrow> bool"   and joint_oracle :: "'f \<Rightarrow> 'g \<Rightarrow> 'd \<Rightarrow> (('e option \<times> 'f) option \<times> ('e option \<times> 'g) option) pmf"   and callee1 :: "'f \<Rightarrow> 'd \<Rightarrow> ('e option \<times> 'f) spmf" assumes "spmf.admissible (\<lambda>exec_until_stop. P (curry (curry (curry exec_until_stop))))"   and "P (\<lambda>exec_until_stop gpv s1 s2. return_pmf None)"   and "\<And>b. P b \<Longrightarrow> P (\<lambda>a aa aaa aaaa. if aaaa then the_gpv a \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf (x, aa, aaa) | IO out rpv \<Rightarrow> joint_oracle aa aaa out \<bind> (\<lambda>a. case a of (None, ba) \<Rightarrow> return_pmf None | (Some (r1, s1'), Some (r2, s2')) \<Rightarrow> case (r1, r2) of (None, None) \<Rightarrow> b (Generative_Probabilistic_Value.Done None) s1' s2' True | (None, _) \<Rightarrow> b (Generative_Probabilistic_Value.Done None) s1' s2' True | (Some r1', None) \<Rightarrow> b (rpv r1') s1' s2' False | (Some r1', Some r2') \<Rightarrow> b (rpv r1') s1' s2' True)) else the_gpv a \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf (None, aa, aaa) | IO out rpv \<Rightarrow> callee1 aa out \<bind> (\<lambda>a. case a of (None, s1') \<Rightarrow> b (Generative_Probabilistic_Value.Done None) s1' aaa False | (Some r1', s1') \<Rightarrow> b (rpv r1') s1' aaa False)))" shows "P (exec_until_stop joint_oracle callee1)"
null_sets_def: fixes M :: "'d measure" shows "null_sets M = {N \<in> sets M. emeasure M N = 0}"
list_of_lconcat: fixes xss :: "'a llist llist" assumes "lfinite xss"   and "\<forall>xs\<in>lset xss. lfinite xs" shows "list_of (lconcat xss) = concat (list_of (lmap list_of xss))"
cong_reflexivity: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Cong A B A B"
neq_Leaf_if_l_gt0: fixes t :: "'b tree" assumes "0 < l t" shows "t \<noteq> Leaf"
D: fixes x :: "real" shows "norm (pbernpoly n x) \<le> D"
preserved_code: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a" shows "preserved f h h' = (h \<turnstile> ok f \<and> h' \<turnstile> ok f \<and> |h \<turnstile> f|\<^sub>r = |h' \<turnstile> f|\<^sub>r \<or> \<not> h \<turnstile> ok f \<and> \<not> h' \<turnstile> ok f)"
all_lifted_red_crit: fixes q :: "'q" assumes "q \<in> Q" shows "calculus Bot_FL Inf_FL (\<lambda>N1 N2. entails_q q (\<Union> (\<G>_F_L_q q ` N1)) (\<Union> (\<G>_F_L_q q ` N2))) (Red_I_\<G>_L_q q) (Red_F_\<G>_empty_L_q q)"
analytic_on_holomorphic: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" shows "(f analytic_on S) = (\<exists>T. open T \<and> S \<subseteq> T \<and> f holomorphic_on T)"
mkIde_set: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "mkIde (local.set a) = a"
isCont_arcosh: fixes x :: "real" assumes "1 < x" shows "isCont arcosh x"
eventually_at_right_less: fixes x :: "'d" shows "eventually ((<) x) (at_right x)"
xyS: fixes i :: "'a" assumes "i \<in> K" shows "x i \<in> S \<and> y i \<in> S"
\<Phi>o_preserves_ide: fixes a :: "'t" assumes "S'.ide a" shows "S.ide (\<Psi>o a)"
eval_fm_All2_Eats: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and e :: "(name, hf) finfun"   and A :: "fm" assumes "atom i \<sharp> (t, u)" shows "eval_fm e (All2 i (Eats t u) A) = (eval_fm e (A(i::=u)) \<and> eval_fm e (All2 i t A))"
components_are_iso: fixes a :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.ide a" shows "B.iso \<a>\<^sub>B[fst a, fst (snd a), snd (snd a)]"
maxdeg_def: fixes A :: "(('b \<Rightarrow>\<^sub>0 'a) \<Rightarrow>\<^sub>0 'c) set" shows "maxdeg A = Max (poly_deg ` A)"
mat_mult: fixes m :: "(('c, 'e) vec, 'b) vec"   and m' :: "(('c, 'd) vec, 'e) vec" shows "($) (m ** m') = mat_mult_row m m'"
induct: fixes x :: "'b t_type"   and X :: "'b t_type"   and P :: "'b t_type \<Rightarrow> bool" assumes "x \<in> t_ins_set X"   and "P X"   and "\<And>x y yl yr ts. \<lbrakk>\<lparr>t_type.folding = False, item = x, subtrees = Branch y yl yr # ts\<rparr> \<in> t_ins_set X; P \<lparr>t_type.folding = False, item = x, subtrees = Branch y yl yr # ts\<rparr>; x \<le> y\<rbrakk> \<Longrightarrow> P \<lparr>t_type.folding = False, item = x, subtrees = yl # Branch y yl yr # ts\<rparr>"   and "\<And>x y yl yr ts. \<lbrakk>\<lparr>t_type.folding = False, item = x, subtrees = Branch y yl yr # ts\<rparr> \<in> t_ins_set X; P \<lparr>t_type.folding = False, item = x, subtrees = Branch y yl yr # ts\<rparr>; \<not> x \<le> y\<rbrakk> \<Longrightarrow> P \<lparr>t_type.folding = False, item = x, subtrees = yr # Branch y yl yr # ts\<rparr>"   and "\<And>x ts. \<lbrakk>\<lparr>t_type.folding = False, item = x, subtrees = Leaf # ts\<rparr> \<in> t_ins_set X; P \<lparr>t_type.folding = False, item = x, subtrees = Leaf # ts\<rparr>\<rbrakk> \<Longrightarrow> P \<lparr>t_type.folding = True, item = x, subtrees = Branch x Leaf Leaf # ts\<rparr>"   and "\<And>x xt y yl yr ts. \<lbrakk>\<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr> \<in> t_ins_set X; P \<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr>; x \<le> y\<rbrakk> \<Longrightarrow> P \<lparr>t_type.folding = True, item = x, subtrees = Branch y xt yr # ts\<rparr>"   and "\<And>x xt y yl yr ts. \<lbrakk>\<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr> \<in> t_ins_set X; P \<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr>; \<not> x \<le> y\<rbrakk> \<Longrightarrow> P \<lparr>t_type.folding = True, item = x, subtrees = Branch y yl xt # ts\<rparr>" shows "P x"
domintros_2: fixes m :: "nat"   and l :: "nat list" assumes "ackloop_dom (Suc 0 # m # l)" shows "ackloop_dom (0 # Suc m # l)"
fold_def: shows "fold \<equiv> \<lambda>uu uua. rec_list (\<lambda>f. id) (\<lambda>x xs xsa f. xsa f \<circ> f x) uua uu"
sint_greater_eq: fixes w :: "'c word" shows "- (2 ^ (LENGTH('c) - Suc 0)) \<le> sint w"
simps_8: fixes e :: "expr"   and F :: "char list"   and Cs :: "char list list"   and A :: "char list set option" shows "\<D> (e\<bullet>F{Cs}) A = \<D> e A"
nbt_subset_carrier: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and x :: "'a" assumes "rational_preference carrier relation"   and "x \<in> carrier" shows "no_better_than x carrier relation \<subseteq> carrier"
columns_eq_cols_upt_k: fixes A :: "(('a, 'c) vec, 'b) vec" shows "cols_upt_k A (ncols A - 1) = columns A"
assms_2: shows "F \<subseteq> dgrad_p_set d m"
subs: fixes x :: "'a" assumes "x \<in> S" shows "closed_segment a x \<subseteq> S"
subset_image_inj: fixes S :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and T :: "'d set" shows "(S \<subseteq> f ` T) = (\<exists>U\<subseteq>T. inj_on f U \<and> S = f ` U)"
Knaster_Tarski_idem_extremes_1: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "Mono\<^bsub>L\<^esub> f"   and "Idem\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "\<top>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> f \<top>\<^bsub>L\<^esub>"
elem_le_sum_list: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "nat"   and ns :: "'a list" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "k < length ns" shows "less_eq (ns ! k) (monoid_add.sum_list plus zero ns)"
shiftl_less_t2n: fixes x :: "'b word"   and m :: "nat"   and n :: "nat" assumes "x < 2 ^ (m - n)"   and "m < LENGTH('b)" shows "x << n < 2 ^ m"
subprob_space_distr: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'a spmf"   and M' :: "'b measure" assumes "f \<in> measure_spmf p \<rightarrow>\<^sub>M M'"   and "space M' \<noteq> {}" shows "subprob_space (distr (measure_spmf p) M' f)"
intros: fixes t :: "dag"   and nodes :: "ref set"   and p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref" assumes "set_of t \<subseteq> nodes"   and "Dag p low high t"   and "t \<noteq> Tip" shows "t \<in> Dags nodes low high"
simps: fixes g :: "('a, 'b) generator"   and s :: "'b" shows "list.unfoldr g s = (if list.has_next g s then let (a, s') = list.next g s in a # list.unfoldr g s' else [])"
1: shows "prim_P w v e\<^sup>T * e = bot"
vec_to_iarray_plus: fixes a :: "('b, 'c) vec"   and b :: "('b, 'c) vec" shows "vec_to_iarray (a + b) = vec_to_iarray a + vec_to_iarray b"
distinct_5: fixes x31 :: "'d"   and x32 :: "'d ifex"   and x33 :: "'d ifex" shows "Falseif \<noteq> IF x31 x32 x33"
eq_bottom_iff: fixes x :: "'a" shows "(x = \<bottom>) = (x \<sqsubseteq> \<bottom>)"
distinct_43: fixes x5 :: "char list" shows "Reject \<noteq> Call x5"
low_dim_interior: fixes S :: "'a set" assumes "aff_dim S \<noteq> int DIM('a)" shows "interior S = {}"
unique_V_uneq: fixes V :: "('a, 'b) MWLsCom list"   and V' :: "('a, 'b) MWLsCom list" assumes "V \<noteq> []"   and "V' \<noteq> []"   and "set (PPV V) \<inter> set (PPV V') = {}" shows "V \<noteq> V'"
prems: fixes i :: "nat" assumes "i < length (y # ys)" shows "(x # xs) ! i \<subseteq> (y # ys) ! i"
null_set_Int2: fixes B :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "B \<in> null_sets M"   and "A \<in> sets M" shows "B \<inter> A \<in> null_sets M"
Posix_elims_4: fixes s :: "'b list"   and r1 :: "'b rexp"   and r2 :: "'b rexp"   and v :: "'b val" assumes "s \<in> Plus r1 r2 \<rightarrow> v" obtains va :: "'b val" where "v = val.Left va__" and "s \<in> r1 \<rightarrow> va__"   | va :: "'b val" where "v = val.Right va" and "s \<in> r2 \<rightarrow> va" and "s \<notin> lang r1"
reflp_rel_sum: fixes R1 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "reflp R1"   and "reflp R2" shows "reflp (rel_sum R1 R2)"
simulation_jAction_eq: fixes M :: "('a, 'p, 'b) KripkeStructure"   and M' :: "('a, 'p, 'c) KripkeStructure"   and f :: "'b \<Rightarrow> 'c"   and w :: "'b" assumes "kripke M"   and "sim M M' f"   and "w \<in> worlds M" shows "jAction M w = jAction M' (f w)"
order_less_imp_not_eq2: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "(y = x) = False"
zero_in_Polys: fixes X :: "'a set" shows "0 \<in> P[X]"
blinfun_eqI: fixes x :: "'b \<Rightarrow>\<^sub>L 'c"   and y :: "'b \<Rightarrow>\<^sub>L 'c" assumes "\<And>i. blinfun_apply x i = blinfun_apply y i" shows "x = y"
Gamma_legendre_duplication_aux: fixes z :: "'a" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0"   and "z + (1::'a) / (2::'a) \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "Gamma z * Gamma (z + (1::'a) / (2::'a)) = exp (((1::'a) - (2::'a) * z) * of_real (ln 2)) * Gamma ((1::'a) / (2::'a)) * Gamma ((2::'a) * z)"
quotient_of_denom_pos: fixes r :: "rat"   and p :: "int"   and q :: "int" assumes "quotient_of r = (p, q)" shows "0 < q"
simps_1: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "R_scalar_mult R" shows "R_scalar_mult.RSpan R smult [] = 0"
3: fixes x :: "Unary_PCF.type"   and xa :: "Unary_PCF.type"   and xb :: "char list \<times> Unary_PCF.type"   and xc :: "expr"   and xd :: "nat \<Rightarrow> char list \<times> Unary_PCF.type"   and xe :: "(char list \<times> Unary_PCF.type) list"   and xda :: "char list \<times> Unary_PCF.type"   and xdb :: "expr"   and xdc :: "nat \<Rightarrow> char list \<times> Unary_PCF.type"   and xdd :: "(char list \<times> Unary_PCF.type) list"   and xaa :: "expr list" assumes "i < ar T \<and> j < ar (T !- i)"   and "x = T !- i"   and "xa = x !- j"   and "xb = (''f'', T)"   and "xc = \<langle>xb\<rangle>"   and "xd = (\<lambda>k. (variant k ''x'', T !- k))"   and "xe = nth_drop i (map xd [0..<ar T])"   and "xda = (''z'', \<pi> x j 0)"   and "xdb = \<langle>xda\<rangle>"   and "xdc = (\<lambda>k. (variant k ''y'', \<pi> x j (k + 1)))"   and "xdd = map xdc [0..<ar xa]"   and "xaa = xdb # map Var xdd" shows "E x j ::: \<epsilon> x j"
replicate_Suc: fixes n :: "nat"   and x :: "'m" shows "replicate (Suc n) x = x # replicate n x"
acc_modi_le_Protected: fixes a :: "acc_modi" assumes "a \<le> Protected" shows "a = Private \<or> a = Package \<or> a = Protected"
cod_eqI: fixes b :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "ide b"   and "seq b f" shows "cod f = b"
ssubst_def: fixes t :: "tm"   and V :: "name set"   and F :: "name \<Rightarrow> tm" shows "ssubst t V F = Finite_Set.fold (\<lambda>i. subst i (F i)) t V"
RedCall: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Cs :: "char list list"   and M :: "char list"   and Ts' :: "ty list"   and T' :: "ty"   and pns' :: "char list list"   and body' :: "expr"   and Ds :: "char list list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "expr"   and Cs' :: "char list list"   and vs :: "val list"   and bs :: "expr"   and new_body :: "expr"   and E :: "char list \<Rightarrow> ty option" assumes "hp s a = \<lfloor>(C, S)\<rfloor>"   and "P \<turnstile> last Cs has least M = (Ts', T', pns', body') via Ds"   and "P \<turnstile> (C,Cs @\<^sub>p Ds) selects M = (Ts, T, pns, body) via Cs'"   and "length vs = length pns"   and "length Ts = length pns"   and "bs = blocks (this # pns, Class (last Cs') # Ts, Ref (a, Cs') # vs, body)"   and "new_body = (case T' of Class D \<Rightarrow> \<lparr>D\<rparr>bs | _ \<Rightarrow> bs)" shows "P,E \<turnstile> \<langle>Call (ref (a, Cs)) None M (map Val vs),s\<rangle> \<rightarrow> \<langle>new_body,s\<rangle>"
assms_5: shows "(\<lambda>x. restrict x J) -` A \<inter> Pi\<^sub>E I S = (\<lambda>x. restrict x J) -` B \<inter> Pi\<^sub>E I S"
OTNIDaux4: fixes x :: "'b"   and y :: "'b" shows "{x, y, x} = {y, x}"
nMRtoNone: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "p \<noteq> []"   and "applied_rule_rev C x p = \<bottom>" shows "C (list2FWpolicy p) x = \<bottom>"
virt_to_phys_unchanged_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word36"   and val :: "word8" assumes "low_equal s1 s2" shows "\<forall>va. virt_to_phys va (mmu s2) ((mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))) = virt_to_phys va (mmu s2) ((mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None)))"
subgroup_subgroup_generated: fixes G :: "('a, 'b) monoid_scheme"   and S :: "'a set" assumes "Group.group G" shows "subgroup (carrier (subgroup_generated G S)) G"
counit_in_hom_2: shows "\<guillemotleft>S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) : S.UP f \<star>\<^sub>S S.UP g \<Rightarrow>\<^sub>S S.trg (S.UP f)\<guillemotright>"
snd_Gauss_Jordan_det_P_is_Gauss_Jordan: fixes A :: "(('b, 'c) vec, 'c) vec" shows "snd (Gauss_Jordan_det_P A) = Gauss_Jordan A"
assms_5: shows "\<forall>U\<in>set Us. IrrFinGroupRepresentation G (\<cdot>) U"
comp_simp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "C.seq (fst g) (fst f)"   and "C.seq (snd g) (snd f)" shows "CC.comp g f = (fst g \<cdot> fst f, snd g \<cdot> snd f)"
l_def: shows "l = (if p = 0 then 0 else cindex_poly (- R) R q p)"
\<Lambda>_PBD_axioms: shows "\<Lambda>_PBD \<V> \<B> 1 \<K>"
refl_on_Un: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and B :: "'b set"   and s :: "('b \<times> 'b) set" assumes "refl_on A r"   and "refl_on B s" shows "refl_on (A \<union> B) (r \<union> s)"
less_multiset_ext\<^sub>D\<^sub>M_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a multiset"   and N :: "'a multiset" shows "less_multiset_ext\<^sub>D\<^sub>M R M N = (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> R k a)))"
D: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and a :: "'b" assumes "class.left_complemented_monoid_algebra times inf imp_l less_eq less one" shows "less_eq a one"
bisim1_call_pcD: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and aMvs :: "'addr \<times> String.literal \<times> 'addr val list" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)"   and "call1 e' = \<lfloor>aMvs\<rfloor>" shows "pc < length (compE2 e)"
intros_38: fixes es :: "e list"   and n :: "nat"   and i :: "nat"   and vs :: "v list" assumes "const_list es"   and "length es = n" shows "\<lparr>[Local n i vs es]\<rparr> \<leadsto> \<lparr>es\<rparr>"
ev_suntil: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and \<psi> :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" assumes "(\<phi> suntil \<psi>) \<omega>" shows "ev \<psi> \<omega>"
cone_def: fixes s :: "'c set" shows "cone s = (\<forall>x\<in>s. \<forall>c\<ge>0. c *\<^sub>R x \<in> s)"
not_le_imp_less: fixes x :: "'d"   and y :: "'d" assumes "\<not> x \<le> y" shows "y < x"
4: shows "[l] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<preceq> [l] \<langle>g\<rangle> [m] \<and> [l] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<preceq> [l] \<langle>g\<rangle> ms"
pop_min_def: fixes \<sigma> :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)" shows "pop_min \<sigma> \<equiv> let (wl, res) = \<sigma> in ASSERT (wl \<noteq> {}) \<bind> (\<lambda>_. RES (least_map (path_weight' \<circ> res) wl) \<bind> (\<lambda>v. RETURN (v, wl - {v}, res)))"
OUTfromChCorrect_data7: shows "OUTfromChCorrect data7"
gmnl_gmd: fixes d :: "('d, 'f) Digit" assumes "is_measured_digit d" shows "gmnl (digitToNlist d) = gmd d"
override_A_def: fixes m2 :: "'c \<Rightarrow> 'd decision option"   and m1 :: "'c \<Rightarrow> 'd decision option" shows "m2 \<Oplus>\<^sub>A m1 = (\<lambda>x. case m1 x of \<bottom> \<Rightarrow> m2 x | \<lfloor>allow a\<rfloor> \<Rightarrow> \<lfloor>allow a\<rfloor> | \<lfloor>deny a\<rfloor> \<Rightarrow> case m2 x of \<bottom> \<Rightarrow> \<lfloor>deny a\<rfloor> | \<lfloor>allow b\<rfloor> \<Rightarrow> \<lfloor>allow b\<rfloor> | \<lfloor>deny \<alpha>\<rfloor> \<Rightarrow> \<lfloor>deny a\<rfloor>)"
comp2_def: fixes h :: "'i \<Rightarrow> 'j \<Rightarrow> 'h"   and f :: "'f \<Rightarrow> 'i"   and g :: "'g \<Rightarrow> 'j" shows "h o2 (f,g) \<equiv> \<lambda>x y. h (f x) (g y)"
assms_2: shows "coincides_on Mkt Mkt2 (stocks Mkt)"
antilinear_o_antilinear: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "antilinear f"   and "antilinear g" shows "clinear (g \<circ> f)"
eint_min_ineq: fixes c :: "eint"   and d :: "eint"   and b :: "eint"   and e :: "eint" assumes "min c d \<le> b"   and "e < c"   and "e < d" shows "e < b"
left_idem: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "signed.max a (signed.max a b) = signed.max a b"
set_id_def: fixes U :: "'b set" shows "set_id U = (\<lambda>s\<in>Pow U. \<lparr>set_dom = s, set_func = \<lambda>x\<in>s. x, set_cod = s\<rparr>)"
sumr_diff_mult_const2: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and r :: "'a" shows "sum f {..<n} - of_nat n * r = (\<Sum>i<n. f i - r)"
test_subid: fixes p :: "'a" assumes "test p" shows "p \<le> (1::'a)"
prems: fixes a1 :: "'a"   and a2 :: "'a" assumes "a1 \<in> insert a A"   and "a2 \<in> insert a A"   and "a1 \<noteq> a2" shows "keys (f a1) \<inter> keys (f a2) = {}"
cring_simprules_26: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
mstfe_prefix_input_in_V: fixes V :: "'e list set"   and M1 :: "('e, 'f, 'g) FSM"   and M2 :: "('e, 'f, 'g) FSM"   and vs :: "('e \<times> 'f) list"   and xs :: "('e \<times> 'f) list" assumes "minimal_sequence_to_failure_extending V M1 M2 vs xs" shows "map fst vs \<in> V"
horiz_vector_helper_neg: fixes p :: "real poly"   and qs :: "real poly list"   and x :: "real"   and sign :: "rat list"   and I :: "nat list" assumes "p \<noteq> 0"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec_copr qs x"   and "list_constr I (length qs)" shows "(poly (prod_list (retrieve_polys qs I)) x < 0) = (z I sign = - 1)"
subcls1_induct_aux: fixes P :: "'a prog"   and C :: "String.literal"   and wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and Q :: "String.literal \<Rightarrow> bool" assumes "is_class P C"   and "wf_prog wf_md P"   and "Q Object"   and "\<And>C D fs ms. \<lbrakk>C \<noteq> Object; is_class P C; class P C = \<lfloor>(D, fs, ms)\<rfloor> \<and> wf_cdecl wf_md P (C, D, fs, ms) \<and> P \<turnstile> C \<prec>\<^sup>1 D \<and> is_class P D \<and> Q D\<rbrakk> \<Longrightarrow> Q C" shows "Q C"
iMin_singleton: fixes a :: "'b" shows "iMin {a} = a"
N: fixes k :: "nat" assumes "N \<le> k" shows "\<bar>(LINT x|lebesgue_on {a..b}. (if x \<le> a + 1 / real (Suc k) then real (Suc k) * d * (a + 1 / real (Suc k) - x) else 0)\<^sup>2) - (LINT x|lebesgue_on {a..b}. (if x = a then d\<^sup>2 else 0))\<bar> < e\<^sup>2"
least_lem: fixes f :: "nat \<Rightarrow> com list \<times> state"   and c :: "com"   and cs :: "com list"   and s :: "state" assumes "f 0 = (c # cs, s)"   and "\<forall>i. f i \<rightarrow> f (Suc i)"   and "\<exists>i. fst (f i) = cs" shows "\<exists>k. fst (f k) = cs \<and> ([c], s) \<rightarrow>\<^sup>* ([], snd (f k))"
net_node_reachable_is_node: fixes st :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and ii :: "nat"   and R\<^sub>i :: "nat set"   and I :: "'d node_action \<Rightarrow> bool" assumes "st \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) I" shows "\<exists>ns R. st = NodeS ii ns R"
map_is_TVarb: fixes \<Gamma>' :: "binding list"   and \<Gamma> :: "binding list"   and i :: "nat"   and T :: "POPLmarkRecord.type" assumes "map is_TVarB \<Gamma>' = map is_TVarB \<Gamma>"   and "\<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB T\<rfloor>" shows "\<exists>T. \<Gamma>'\<langle>i\<rangle> = \<lfloor>TVarB T\<rfloor>"
equal_div_mod2: fixes j :: "nat"   and a :: "nat"   and i :: "nat" shows "(j div a = i div a \<and> j mod a = i mod a) = (j = i)"
orelse_thms_2: fixes y :: "tr" shows "(FF orelse y) = y"
unity\<^sub>F\<^sub>M\<^sub>C_def: shows "\<I> = mkarr \<^bold>\<I>"
is_digit_char_def: fixes c :: "char" shows "is_digit_char c = (c = CHR ''0'' \<or> c = CHR ''5'' \<or> c = CHR ''1'' \<or> c = CHR ''6'' \<or> c = CHR ''2'' \<or> c = CHR ''7'' \<or> c = CHR ''3'' \<or> c = CHR ''8'' \<or> c = CHR ''4'' \<or> c = CHR ''9'')"
FWdelay_bisimulation_final_base_flip: shows "FWdelay_bisimulation_final_base final2 r2 final1 r1 (\<lambda>t. flip (bisim t)) \<tau>move2 \<tau>move1"
snd_convol: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b" shows "snd \<circ> BNF_Def.convol f g = g"
tab_simps_4: shows "local.dom ((\<phi> \<star> f) \<cdot> \<rho>) = g"
ha: fixes x :: "'a"   and s :: "'s" assumes "x \<in> S s" shows "healthy (wp (a x))"
infsetsum_Re: fixes f :: "'c \<Rightarrow> complex"   and M :: "'c set" assumes "f abs_summable_on M" shows "(\<Sum>\<^sub>ax\<in>M. Re (f x)) = Re (infsetsum f M)"