hcju/isabelleps huggingface dataset:
ev_sconst: fixes P :: "'a stream \<Rightarrow> bool"   and x :: "'a" shows "ev P (sconst x) = P (sconst x)"
range_eqI: fixes b :: "'e"   and f :: "'f \<Rightarrow> 'e"   and x :: "'f" assumes "b = f x" shows "b \<in> range f"
Retracts_def: fixes s :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and t :: "'b set"   and k :: "'b \<Rightarrow> 'a" shows "Retracts s h t k \<equiv> (continuous_on s h \<and> h ` s = t) \<and> continuous_on t k \<and> k ` t \<subseteq> s \<and> (\<forall>y. y \<in> t \<longrightarrow> h (k y) = y)"
eq_fC: shows "f ` C = f ` B \<union> f ` (C - B)"
whileAnno_def: fixes b :: "'e set"   and I :: "'e set"   and V :: "('e \<times> 'e) set"   and c :: "('e, 'f, 'g) com" shows "whileAnno b I V c = While b c"
c'_def: shows "c' \<equiv> \<lambda>k. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k (real_of_int b) - fs k (real_of_int a))"
no_zero_height_Exec_derivs: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" assumes "(M, l, s, 0, h, v) \<in> Exec" shows "False"
gfp_transfer_bounded: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" assumes "P (f top)"   and "\<And>x. P x \<Longrightarrow> P (f x)"   and "\<And>M. \<lbrakk>decseq M; \<And>i. P (M i)\<rbrakk> \<Longrightarrow> P (Inf (range M))"   and "\<And>M. \<lbrakk>decseq M; \<And>i. P (M i)\<rbrakk> \<Longrightarrow> \<alpha> (Inf (range M)) = (INF i. \<alpha> (M i))"   and "inf_continuous f"   and "inf_continuous g"   and "\<And>x. P x \<Longrightarrow> \<alpha> (f x) = g (\<alpha> x)"   and "\<And>x. g x \<le> \<alpha> (f top)" shows "\<alpha> (gfp f) = gfp g"
fminus_finsert2: fixes A :: "'b fset"   and a :: "'b"   and B :: "'b fset" shows "A |-| finsert a B = A |-| {|a|} |-| B"
real_sqrt_le_1_iff: fixes x :: "real" shows "(sqrt x \<le> 1) = (x \<le> 1)"
finite_Atleast_Atmost_nat: fixes f :: "'e \<Rightarrow> 'd" shows "finite {f x |x. x \<in> UNIV}"
a_kernel_def': fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" shows "a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
rel: shows "TargetTerm TP \<R>\<lbrakk>\<cdot>\<rbrakk>T<TRel> TargetTerm TQ"
finite_fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s: fixes x :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list" shows "finite (fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s x)"
cobounded2: fixes b :: "'c"   and a :: "'c" shows "b \<le> max a b"
holo: fixes a :: "real"   and S :: "complex set" assumes "0 < a" shows "pre_zeta a holomorphic_on S"
inject_2: fixes x21 :: "'g"   and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) env option"   and y21 :: "'g"   and y22 :: "'h \<Rightarrow> ('f, 'g, 'h) env option" shows "(Env x21 x22 = Env y21 y22) = (x21 = y21 \<and> x22 = y22)"
SourcesS9_L2: shows "Sources level2 sS9 = {}"
rtrancl_refl: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" shows "r\<^sup>*\<^sup>* a a"
I: fixes r :: "'a sequent list \<times> 'a sequent"   and R :: "('a sequent list \<times> 'a sequent) set"   and seq :: "'a sequent" assumes "r \<in> R" shows "extendRule seq r \<in> R*"
asym: fixes a :: "'a Word.word"   and b :: "'a Word.word" assumes "a <s b"   and "b <s a" shows "False"
integral_of_real: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set" assumes "f integrable_on S" shows "integral S (\<lambda>x. of_real (f x)) = of_real (integral S f)"
ro_ifexI: fixes a :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and b :: "'b ifex" assumes "(a, b) \<in> bf_ifex_rel" shows "ro_ifex b"
dg_Par_is_arrD_2: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> B" shows "S\<lparr>ArrDom\<rparr> = A"
matrix_match_def: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zer :: "'a"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inver :: "'a \<Rightarrow> 'a"   and A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "plus_mult id f zer g inver" shows "plus_mult.matrix_match A1 A2 B1 B2 \<equiv> Matrix_Legacy.mat (mult.row_length A1) (length A1) A1 \<and> Matrix_Legacy.mat (mult.row_length A2) (length A2) A2 \<and> Matrix_Legacy.mat (mult.row_length B1) (length B1) B1 \<and> Matrix_Legacy.mat (mult.row_length B2) (length B2) B2 \<and> length A1 = mult.row_length A2 \<and> length B1 = mult.row_length B2 \<and> A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []"
assms_1: shows "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>A. norm (f n x) \<le> M n"
rank_eq_0_iff: fixes x :: "V" shows "(rank x = 0) = (x = 0)"
assms_1: shows "n < \<^bold>|w\<^bold>|"
assms_1: fixes x :: "'a" assumes "x \<sqinter> nc \<noteq> (0::'a)" shows "nc \<cdot> ((x \<sqinter> nc) \<cdot> nc) = nc"
co2: assumes "dfree t2"   and "Vagree \<nu> \<nu>' (FVT t2)"   and "Iagree I J {Inl x |x. x \<in> SIGT t2}" shows "sterm_sem I t2 (fst \<nu>) = sterm_sem J t2 (fst \<nu>')"
permute_set_def: fixes p :: "perm"   and X :: "'a set" shows "p \<bullet> X = {p \<bullet> x |x. x \<in> X}"
label_incr_simp_rev: fixes m :: "label"   and j :: "nat"   and i :: "nat"   and n :: "label" assumes "m \<oplus> j + i = n \<oplus> i" shows "m \<oplus> j = n"
strict_monoI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less"   and "\<And>x y. less x y \<Longrightarrow> f x < f y" shows "order.strict_mono less f"
null_sets_point_measure_iff: fixes X :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> ennreal" shows "(X \<in> null_sets (point_measure A f)) = (X \<subseteq> A \<and> (\<forall>x\<in>X. f x = 0))"
telescope_sums: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "f \<longlonglongrightarrow> c" shows "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
asEnv_equal_imp_alphaEnv: fixes qrho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and qrho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "qGoodEnv qrho"   and "asEnv qrho = asEnv qrho'" shows "qrho &= qrho'"
gbinomial_altdef_of_nat: fixes a :: "'a"   and k :: "nat" shows "a gchoose k = (\<Prod>i = 0..<k. (a - of_nat i) / of_nat (k - i))"
wf_ciD3'_ci_app: fixes ci :: "('a, 'b) check_instr"   and ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'b"   and stk :: "'a val list"   and loc :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'a frame list"   and pc' :: "nat" assumes "ci_app ci ins P h stk loc C M pc frs"   and "pc \<le> pc'" shows "ci_app ci ins P h stk loc C M pc' frs"
less_succ: fixes r :: "('a \<times> 'a) set"   and i :: "'a"   and j :: "'a" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "((j, succ r i) \<in> r) = ((j, i) \<in> r \<or> j = succ r i)"
assms_6: fixes A :: "'a trm" assumes "P A" shows "P (Fst A)"
complex_eq_cancel_iff2: fixes x :: "real"   and y :: "real"   and xa :: "real" shows "(Complex x y = cor xa) = (x = xa \<and> y = 0)"
INTEG_E: fixes v :: "val" assumes "val_type v = INTEG" obtains i :: "int" where "v = IntVal i"
iprod_assoc: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "\<langle>x *\<^sub>s xs,ys\<rangle> = x * \<langle>xs,ys\<rangle>"
l_subset_y: shows "set_of l \<subset> set_of y"
infsetsum_mono_neutral_right: fixes f :: "'d \<Rightarrow> real"   and A :: "'d set"   and g :: "'d \<Rightarrow> real"   and B :: "'d set" assumes "f abs_summable_on A"   and "g abs_summable_on B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"   and "B \<subseteq> A"   and "\<And>x. x \<in> A - B \<Longrightarrow> f x \<le> 0" shows "infsetsum f A \<le> infsetsum g B"
purge_def: fixes G :: "'U set"   and ucl :: "('U \<times> 'C) list" shows "purge G ucl \<equiv> filter (\<lambda>(u, c). u \<notin> G) ucl"
measurable_density_eq1: fixes g :: "'d \<Rightarrow> 'e"   and Mg :: "'d measure"   and f :: "'d \<Rightarrow> ennreal"   and Mg' :: "'e measure" shows "(g \<in> density Mg f \<rightarrow>\<^sub>M Mg') = (g \<in> Mg \<rightarrow>\<^sub>M Mg')"
pdevs_apply_filter_slist: fixes P :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "(nat, 'd) slist"   and i :: "nat" shows "pdevs_apply (Pdevs (filter_slist P xs)) i = (if P i (pdevs_apply (Pdevs xs) i) then pdevs_apply (Pdevs xs) i else (0::'d))"
axioms_1: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_onto A n f" shows "vsv f"
thetaSD_def: shows "thetaSD \<equiv> {(c, d) |c d. proper c \<and> proper d \<and> discr c \<and> discr d}"
eqvtTrans: fixes Rel :: "('a \<times> 'a) set"   and Rel' :: "('a \<times> 'a) set" assumes "eqvt Rel"   and "eqvt Rel'" shows "eqvt (Rel O Rel')"
simps_1: fixes x2 :: "'e"   and y2 :: "'e" shows "(Some x2 = Some y2) = (x2 = y2)"
path_to_left_lang: fixes q :: "hf" assumes "q \<in> accessible" shows "path_to q \<in> left_lang q"
coprime_imp_coprime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and d :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "algebraic_semidom.coprime times one c d"   and "\<And>e. \<lbrakk>\<not> dvd.dvd times e one; dvd.dvd times e a; dvd.dvd times e b\<rbrakk> \<Longrightarrow> dvd.dvd times e c"   and "\<And>e. \<lbrakk>\<not> dvd.dvd times e one; dvd.dvd times e a; dvd.dvd times e b\<rbrakk> \<Longrightarrow> dvd.dvd times e d" shows "algebraic_semidom.coprime times one a b"
P4: shows "Bet B A B' \<and> Cong A B' B A"
assms_2: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
the1I2: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<exists>!x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
B: fixes x :: "real" assumes "x \<in> {- pi..pi} - {- e<..<e}" shows "\<bar>inverse (sin (x / 2))\<bar> \<le> B"
arr_char: fixes f :: "(unit, unit) concrete_category.arr" shows "arr f = (Dom f \<in> {} \<and> f = MkIde (Dom f))"
sep_conjD: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "(P \<and>* Q) h" shows "\<exists>x y. x ## y \<and> h = x + y \<and> P x \<and> Q y"
real_alg_1: fixes x :: "real_alg_2" shows "real_of_1 (real_alg_1 x) = real_of_2 x"
assms_5: shows "t \<le> v * v\<^sup>T \<sqinter> g"
A_trusts_NS2_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and NA :: "nat"   and NB :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_public"   and "Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace> \<in> parts (knows Spy evs)"   and "Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs" shows "Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs"
eq2: shows "punit.higher (rep_list p) (punit.lt (rep_list p)) = 0"
path_polynomial_function: fixes g :: "real \<Rightarrow> 'a" assumes "polynomial_function g" shows "path g"
filterlim_pow_at_top: fixes n :: "nat"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter" assumes "0 < n"   and "filterlim f at_top F" shows "LIM x F. f x ^ n :> at_top"
that: fixes e' :: "nat \<Rightarrow> real" assumes "x = aform_vals e' (aforms_err_to_aforms n X)"   and "e' \<in> UNIV \<rightarrow> {- 1..1}"   and "\<And>i. i < n \<Longrightarrow> e' i = e i" shows "thesis"
P_sup_def: fixes s :: "'s \<times> ('c \<Rightarrow> real) set"   and P :: "('s \<times> ('c \<Rightarrow> real) set) stream \<Rightarrow> bool" shows "R_G.P_sup s P = (\<Squnion>cfg\<in>R_G.cfg_on s. emeasure (R_G.T cfg) {x \<in> space R_G.St. P x})"
almost_all_eventually_provable_suffix: fixes \<phi> :: "'b ltl"   and \<G>' :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" assumes "\<PP>\<^sub>\<infinity> \<phi> \<G>' w" shows "\<PP>\<^sub>\<infinity> \<phi> \<G>' (suffix i w)"
subprob_algebra_cong: fixes M :: "'a measure"   and N :: "'a measure" assumes "sets M = sets N" shows "subprob_algebra M = subprob_algebra N"
apply_cltn2_preserve_set_Col: fixes S :: "proj2 set"   and C :: "cltn2" assumes "proj2_set_Col S" shows "proj2_set_Col {apply_cltn2 p C |p. p \<in> S}"
id_nec2_1: fixes v :: "i"   and \<alpha> :: "'a"   and \<beta> :: "'a" shows "[\<^bold>\<diamond>\<alpha> \<^bold>= \<beta> \<^bold>\<equiv> \<alpha> \<^bold>= \<beta> in v]"
that: fixes q :: "'a"   and m :: "nat" assumes "poly p (Fraction_Field.Fract a b) = Fraction_Field.Fract q (b ^ m)" shows "thesis"
distinct_list_of_dlist: fixes dxs :: "'b dlist" shows "distinct (list_of_dlist dxs)"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
add_1: fixes Y :: "'a multiset"   and thesis :: "bool" assumes "\<And>Y1 Y2. \<lbrakk>Y = Y1 + Y2; (X1, Y1) \<in> multpw ns; (X2, Y2) \<in> multpw ns\<rbrakk> \<Longrightarrow> thesis"   and "(X1 + X2, Y) \<in> multpw ns" shows "thesis"
poly_mapping_sum_monomials: fixes p :: "'c \<Rightarrow>\<^sub>0 'd" shows "(\<Sum>a\<in>keys p. monomial (lookup p a) a) = p"
lincomb_list_snoc: fixes Vs :: "'a vec list"   and x :: "'a vec"   and f :: "nat \<Rightarrow> 'a" assumes "set Vs \<subseteq> carrier_vec n"   and "x \<in> carrier_vec n" shows "lincomb_list f (Vs @ [x]) = lincomb_list f Vs + f (length Vs) \<cdot>\<^sub>v x"
pop_pow_cancel: fixes u :: "'b list"   and k :: "nat"   and v :: "'b list"   and m :: "nat"   and w :: "'b list" assumes "u \<^sup>@ k \<cdot> v = u \<^sup>@ m \<cdot> w"   and "m \<le> k" shows "u \<^sup>@ (k - m) \<cdot> v = w"
inverse_less_iff_less: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less zero b" shows "less (inverse a) (inverse b) = less b a"
fun_add_comm: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F" shows "x \<oplus>\<^bsub>F\<^esub> y = y \<oplus>\<^bsub>F\<^esub> x"
distinct_list_split: fixes xs :: "'c list"   and xa :: "'c list"   and x :: "'c"   and ya :: "'c list"   and xb :: "'c list"   and yb :: "'c list" assumes "distinct xs"   and "xs = xa @ x # ya"   and "xs = xb @ x # yb" shows "xa = xb \<and> ya = yb"
asym: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a \<subset># b"   and "b \<subset># a" shows "False"
parallelogram_law: fixes x :: "'b"   and y :: "'b" shows "(norm (x + y))\<^sup>2 + (norm (x - y))\<^sup>2 = 2 * ((norm x)\<^sup>2 + (norm y)\<^sup>2)"
one_eq_divide_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "(one = divide a b) = (b \<noteq> zero \<and> a = b)"
derives_append: fixes a :: "symbol list"   and u :: "symbol list"   and b :: "symbol list"   and v :: "symbol list" assumes "derives a u"   and "derives b v"   and "is_sentence a"   and "is_sentence b" shows "derives (a @ b) (u @ v)"
l_get_parent_wf_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_parent_wf_axioms type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent \<equiv> ((\<forall>h ptr children child. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<longrightarrow> child \<in> set children \<longrightarrow> h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr) \<and> (\<forall>h P ptr. heap_is_wellformed h \<longrightarrow> (\<forall>parent. (\<forall>children child. h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r children \<longrightarrow> child \<in> set children \<longrightarrow> P (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)) \<longrightarrow> P parent) \<longrightarrow> P ptr)) \<and> (\<forall>h P ptr. heap_is_wellformed h \<longrightarrow> (\<forall>child. (\<forall>parent child_node. cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child \<longrightarrow> h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent \<longrightarrow> P parent) \<longrightarrow> P child) \<longrightarrow> P ptr) \<and> (\<forall>h child_node parent. heap_is_wellformed h \<longrightarrow> h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent \<longrightarrow> (parent, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node) \<in> parent_child_rel h)"
jozsa_algo_result: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "jozsa f n" shows "jozsa.jozsa_algo f n = Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then \<Sum>k<2 ^ n. (- 1) ^ (f k + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1)) else \<Sum>k<2 ^ n. (- 1) ^ (f k + 1 + i div 2 \<cdot>\<^bsub>n\<^esub>  k) / (sqrt 2 ^ n * sqrt 2 ^ (n + 1))))"
f_s: shows "(\<lambda>i. \<integral>\<^sup>+ x. ennreal (norm (f x - s i x)) \<partial>M) \<longlonglongrightarrow> 0"
toFull_fromFull: fixes \<sigma> :: "'tp"   and a :: "univ" shows "toFull \<sigma> (fromFull \<sigma> a) = a"
l_set_mode_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_child_nodes type_wf set_mode set_mode_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_mode type_wf set_mode set_mode_locs \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_set_mode_get_child_nodes_axioms set_mode_locs get_child_nodes_locs"
set_pmf_iff: fixes x :: "'a"   and M :: "'a pmf" shows "(x \<in> set_pmf M) = (pmf M x \<noteq> 0)"
refl_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. x \<^bsub>r\<^esub>\<preceq> x))"
less_eq_bot_Bot: fixes x :: "'a bot" shows "Bot \<le> x"
subst_apply_img_var': fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "x \<in> fv (t \<cdot> \<delta>)"   and "x \<notin> fv t" shows "\<exists>y\<in>fv t. x \<in> fv (\<delta> y)"
prems_1: shows "{$$} \<turnstile>\<^sub>W Rec x (Lam y e') : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
SubstFormP_trivial: fixes i :: "name"   and A :: "fm" assumes "atom i \<sharp> A" shows "{} \<turnstile> SubstFormP \<guillemotleft>Var i\<guillemotright> Zero \<guillemotleft>A\<guillemotright> \<guillemotleft>A\<guillemotright>"
ground_ssubst_eq: fixes V :: "name set"   and t :: "tm" assumes "finite V"   and "supp t = {}" shows "ssubst t V F = t"
carrier_vs_is_self: fixes W :: "'a vec set" shows "carrier (vs W) = W"
total_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "Relation.total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
inject_3: fixes x1 :: "('d, 'e, 'f) psi"   and y1 :: "('d, 'e, 'f) psi" shows "(RTau x1 = RTau y1) = (x1 = y1)"
emp_in: fixes G :: "'b list \<Rightarrow> bool" shows "hullp G \<epsilon>"
hyps: assumes "p \<noteq> 0"   and "c \<noteq> 0" shows "periodic_set (roots (\<lambda>x. poly p (tan (x / c)))) (c * pi)"
jvm_kildall_correct: fixes P :: "(nat \<times> nat \<times> 'b instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog" shows "wf_jvm_prog\<^sub>k P = wf_jvm_prog P"
assms_3: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"
fun_upd_image: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a"   and A :: "'b set" shows "f(x := y) ` A = (if x \<in> A then insert y (f ` (A - {x})) else f ` A)"
C3r: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> y" shows "y \<cdot> x\<^sup>\<star> \<le> y"
star_slide: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<odot> y)\<^sup>\<odot> \<odot> x = x \<odot> (y \<odot> x)\<^sup>\<odot>"
path_split_2: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node"   and as' :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n (as @ a # as') n'" shows "valid_edge prog a"
outer_prod_def: fixes v :: "'a vec"   and w :: "'a vec" shows "outer_prod v w = mat (dim_vec v) 1 (\<lambda>(i, j). v $ i) * mat 1 (dim_vec w) (\<lambda>(i, j). conjugate w $ j)"
simps_3: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.tab \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = tab"
Inf_nat_def: fixes X :: "nat set" shows "\<Sqinter> X = (LEAST n. n \<in> X)"
lowner_le_traceI: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat n n; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> trace (A * \<rho>) \<le> trace (B * \<rho>)" shows "A \<le>\<^sub>L B"
inner_dirsum_double_leftfull_imp_right0: fixes A :: "'a set"   and B :: "'a set" assumes "Group A"   and "B \<noteq> {}"   and "A = A \<oplus> B" shows "B = 0"