hcju/isabelleps huggingface dataset:
callee_invariant_restrict_relp: fixes S :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and callee1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and callee2 :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf"   and I1 :: "'a \<Rightarrow> bool"   and I2 :: "'b \<Rightarrow> bool" assumes "rel_fun S (rel_fun C (rel_spmf (rel_prod R S))) callee1 callee2"   and "callee_invariant callee1 I1"   and "callee_invariant callee2 I2" shows "rel_fun (S \<upharpoonleft> I1 \<otimes> I2) (rel_fun C (rel_spmf (rel_prod R (S \<upharpoonleft> I1 \<otimes> I2)))) callee1 callee2"
discriminant_pos_ex: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a \<noteq> 0"   and "0 < discrim a b c" shows "\<exists>x y. x \<noteq> y \<and> a * x\<^sup>2 + b * x + c = 0 \<and> a * y\<^sup>2 + b * y + c = 0"
h1: shows "M_mat (get_signs (reduce_system p (qs, M_mat signs subsets, subsets, signs))) (get_subsets (reduce_system p (qs, M_mat signs subsets, subsets, signs))) = M_mat (reduction_signs p qs signs subsets (M_mat signs subsets)) (reduction_subsets p qs signs subsets (M_mat signs subsets))"
i_expand_Suc_def: shows "(\<odot>\<^bsub>iSuc\<^esub>) \<equiv> \<lambda>f k n. if n mod Suc k = 0 then f (n div Suc k) else NoMsg"
Hermite_of_row_preserves_below: fixes i :: "'a"   and a :: "'a"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "(('b, 'c) vec, 'a) vec"   and b :: "'c" assumes "i < a" shows "Hermite_of_row_i ass res A i $ a $ b = A $ a $ b"
cases: fixes x :: "cospan_shape.Arr" obtains "x = cospan_shape.AA"   | "x = cospan_shape.BB"   | "x = cospan_shape.TT"   | "x = cospan_shape.AT"   | "x = cospan_shape.BT"   | "x = cospan_shape.Null"
\<nu>: shows "\<nu> \<in> elts \<omega>1"
scf_list_subset: fixes scf :: "nat \<Rightarrow> nat"   and xs :: "'c list" shows "set (scf_list scf xs) \<subseteq> set xs"
add_frac_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R"   and "w \<in> carrier R"   and "y \<noteq> \<zero>"   and "z \<noteq> \<zero>" shows "x \<oslash> y \<oplus> w \<oslash> z = (x \<otimes> z \<oplus> w \<otimes> y) \<oslash> (y \<otimes> z)"
ws_subint1_induct: fixes G :: "prog"   and I :: "qtname"   and P :: "qtname \<Rightarrow> bool" assumes "is_iface G I"   and "ws_prog G"   and "\<And>I i. iface G I = Some i \<and> (\<forall>J\<in>set (isuperIfs i). (I, J) \<in> subint1 G \<and> P J \<and> is_iface G J) \<Longrightarrow> P I" shows "P I"
wellFormedSimp: fixes p :: "(name \<times> name) list"   and \<sigma> :: "('d list \<times> 'e list) list" shows "wellFormedSubst (p \<bullet> \<sigma>) = wellFormedSubst \<sigma>"
strict_not_refl_weak: fixes x :: "'a pmf"   and y :: "'a pmf" assumes "x \<in> \<P> outcomes \<and> y \<in> \<P> outcomes" shows "((y, x) \<notin> relation) = (x \<succeq> y \<and> (y, x) \<notin> relation)"
simps_6: fixes x11 :: "int"   and x12 :: "int list"   and x31 :: "int"   and x32 :: "int"   and x33 :: "int list" shows "Le x11 x12 \<noteq> NDvd x31 x32 x33"
disc_eq_case_1: fixes tm :: "('a, 'b) tm" shows "is_Hd tm = (case tm of Hd uu_ \<Rightarrow> True | App uu_ uua_ \<Rightarrow> False)"
lzip_simps_3: fixes x :: "'c"   and xs :: "'c llist"   and y :: "'d"   and ys :: "'d llist" shows "lzip (LCons x xs) (LCons y ys) = LCons (x, y) (lzip xs ys)"
assms_1: shows "negligible {x \<in> S - T. f x \<noteq> (0::'a)}"
simps_12: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm" shows "push_forall (AllQ (And \<phi> \<psi>)) = and (push_forall (AllQ \<phi>)) (push_forall (AllQ \<psi>))"
numeral_Bit0: fixes n :: "num" shows "numeral (num.Bit0 n) = numeral n + numeral n"
pivot_set_var: shows "\<exists>f. pivot_fun A f (dim_col A) \<and> set (pivot_positions A) = {(i, f i) |i. i < dim_row A \<and> f i < dim_col A}"
obsv_ref_rel_def: shows "obsv_ref_rel \<equiv> {(sa, sc). sa = v_state.truncate sc}"
measurable_cong_simp: fixes M :: "'a measure"   and N :: "'a measure"   and M' :: "'b measure"   and N' :: "'b measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = N"   and "M' = N'"   and "\<And>w. w \<in> space M \<Longrightarrow> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> N \<rightarrow>\<^sub>M N')"
eint_0_iff_2: fixes x :: "int" shows "(0 = eint x) = (x = 0)"
zero_eq_lcm_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(zero = lcm a b) = (a = zero \<or> b = zero)"
loc_sl_def: 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 mxl :: "nat" shows "loc_sl P mxl \<equiv> Listn.sl mxl (Err.sl (SemiType.esl P))"
\<phi>: shows "\<guillemotleft>\<phi> : f \<rightarrow>\<^sub>A f'\<guillemotright> \<and> F \<phi> = \<psi>"
inj_singleton: fixes A :: "'b set" shows "inj_on \<eta> A"
simps: fixes a :: "'state" shows "reachNT a = (a = istate \<or> (\<exists>trn. a = tgtOf trn \<and> reachNT (srcOf trn) \<and> step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn) \<and> \<not> T trn))"
uint64_sdiv_def: fixes x :: "uint64"   and y :: "uint64" shows "uint64_sdiv x y = (if y = 0 then undefined (div) x 0 else Abs_uint64 (Rep_uint64 x sdiv Rep_uint64 y))"
sum0: shows "(\<Sum>(x, y)\<in>local.Z_not_inj. local.weight x y) = (0::'a)"
comp_fun_commute_insort: shows "comp_fun_commute insort"
f: fixes C :: "'a set set" assumes "C \<noteq> {}"   and "subset.chain \<B> C" shows "\<Union> C \<in> \<B>"
map_snd_zip_take: fixes xs :: "'b list"   and ys :: "'a list" shows "map snd (zip xs ys) = take (min (length xs) (length ys)) ys"
varnum_plus: fixes X :: "'a set"   and s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "varnum X (s + t) = max (varnum X s) (varnum X t)"
r: shows "(S', Q') = Smith_1xn_aux i A (S, Q)"
DA_is_deny: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set" assumes "x \<in> dom (Cp (DenyAllFromTo a b \<oplus> DenyAllFromTo b a \<oplus> DenyAllFromTo a b))" shows "Cp (DenyAllFromTo a b \<oplus> DenyAllFromTo b a \<oplus> DenyAllFromTo a b) x = \<lfloor>deny ()\<rfloor>"
refl: fixes a :: "'t \<Rightarrow>\<^sub>0 'c" shows "a \<preceq>\<^sub>p a"
nzrows_def: fixes A :: "'b mat" shows "nzrows A = filter (\<lambda>r. r \<noteq> 0\<^sub>v (dim_col A)) (rows A)"
left_residual: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.left_residuated times imp_l less_eq" shows "less_eq (times x a) b = less_eq x (imp_l a b)"
mset_ran_swap: fixes i :: "int"   and r :: "int set"   and j :: "int"   and a :: "int \<Rightarrow> 'a" assumes "i \<in> r"   and "j \<in> r" shows "mset_ran (swap a i j) r = mset_ran a r"
vsUW: shows "adjuster n w us \<in> local.span (set (us @ ws))"
ANR_from_Un_Int: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T"   and "ANR (S \<union> T)"   and "ANR (S \<inter> T)" shows "ANR S"
convex_contains_open_segment: fixes s :: "'a set" shows "convex s = (\<forall>a\<in>s. \<forall>b\<in>s. open_segment a b \<subseteq> s)"
assms: shows "Rewriting_Sterm.compile rs \<turnstile>\<^sub>s u \<longrightarrow> u'"
elims_2: fixes x :: "nat" assumes "Prime x" obtains xa :: "nat" where "x = xa" and "1 < xa \<and> (\<forall>u<xa. \<forall>v<xa. u * v \<noteq> xa)"
l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.append_child Shadow_DOM.insert_before"
abs_conv_abscissa_truncate: fixes m :: "nat"   and f :: "'a fds" shows "abs_conv_abscissa (fds_truncate m f) = - \<infinity>"
ran_restrictD: fixes y :: "'a"   and m :: "'b \<Rightarrow> 'a option"   and A :: "'b set" assumes "y \<in> ran (m |` A)" shows "\<exists>x\<in>A. m x = Some y"
rel: shows "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
def_C': shows "C' = cl_ecl P1 - {L2} \<union> {L'}"
has_vderiv_on_uminus: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and T :: "real set" assumes " D f = f' on T" shows " D (\<lambda>x. - f x) = (\<lambda>x. - f' x) on T"
Fvars_def: fixes t :: "'a" shows "Fvars t = {a. \<not> atom a \<sharp> t}"
one_and_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and one a = modulo a (numeral.numeral one plus (num.Bit0 num.One))"
uminus_numeral_One: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.neg_numeral minus plus zero uminus" shows "uminus (numeral.numeral one plus num.One) = uminus one"
inner_zero_right: fixes x :: "'b" shows "inner x (0::'b) = 0"
simps: fixes a :: "'a set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = insert aa A \<and> finite A))"
sturm_tarski_interval: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < b"   and "poly p a \<noteq> 0"   and "poly p b \<noteq> 0" shows "taq {x. poly p x = 0 \<and> a < x \<and> x < b} q = changes_itv_smods a b p (pderiv p * q)"
C: shows "cfs ! k = (size (acomp a\<^sub>1) + size (acomp a\<^sub>2), s, aval a\<^sub>2 s # aval a\<^sub>1 s # stk)"
Join_constrains: fixes F :: "'d program"   and G :: "'d program"   and A :: "'d set"   and B :: "'d set" shows "(F \<squnion> G \<in> A co B) = (F \<in> A co B \<and> G \<in> A co B)"
F2bd'_Card_order: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "Card_order (r1 +c r2)"
hyp: assumes "\<forall>n'<n. \<not> P n'" shows "P n"
hd_def: fixes list :: "'a list" shows "hd list = (case list of x21 # x22 \<Rightarrow> x21)"
cod_dom: fixes f :: "'a" assumes "Right.arr f" shows "Right.cod (Right.dom f) = Right.dom f"
OUTfromChCorrect_data17: shows "OUTfromChCorrect data17"
delay_le_length: fixes i :: "nat"   and x :: "nat" shows "the (delay i x) \<le> e_length x"
int: fixes S :: "'n set" assumes "S \<in> \<T>" shows "(f has_integral i S) S"
lucas_lehmer_ring_mod_def: fixes m :: "nat" shows "lucas_lehmer_ring_mod m = \<lparr>carrier = {..<m} \<times> {..<m}, monoid.mult = lucas_lehmer_mult m, one = (1, 0), ring.zero = (0, 0), add = lucas_lehmer_add m\<rparr>"
ds_def: fixes r :: "('b \<times> 'b) set"   and S :: "'b set" shows "r \<down>s S = {y. \<exists>x\<in>S. (y, x) \<in> r}"
gen_mono_on_singleton: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "reflp R" shows "gen_mono_on f R {x}"
rewrite_syn_imp_sound: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n rewrite_syn_imp \<phi> = w \<Turnstile>\<^sub>n \<phi>"
simps_5: fixes f1 :: "'k"   and f2 :: "'l \<Rightarrow> 'l list \<Rightarrow> 'k"   and x21 :: "'l"   and x22 :: "'l list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
esize_iff_empty: fixes A :: "'a set" shows "(esize A = 0) = (A = {})"
coboundedI1: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a \<le> c" shows "min a b \<le> c"
hyps: fixes n :: "nat"   and xs :: "'a list" shows "take n (drop m_ xs) = drop m_ (take (n + m_) xs)"
map_ident: fixes t :: "'a stream" shows "smap (\<lambda>x. x) t = t"
\<R>_def': shows "\<R> = {region X I r |I r. valid_region X k I r}"
kruskal1_def: shows "kruskal1 \<equiv> obtain_sorted_carrier \<bind> (\<lambda>l. let initial_union_find = per_init V in nfoldli l (\<lambda>_. True) (\<lambda>e (uf, T). ASSERT (e \<in> E) \<bind> (\<lambda>_. a_endpoints e \<bind> (\<lambda>(a, b). ASSERT (a \<in> V \<and> b \<in> V \<and> a \<in> Domain uf \<and> b \<in> Domain uf \<and> T \<subseteq> E) \<bind> (\<lambda>_. if \<not> per_compare uf a b then let uf = per_union uf a b in ASSERT (e \<notin> T) \<bind> (\<lambda>_. RETURN (uf, insert e T)) else RETURN (uf, T))))) (initial_union_find, {}) \<bind> (\<lambda>(per, spanning_forest). RETURN spanning_forest))"
valid_ty_dom_fresh: fixes \<Gamma> :: "binding list"   and X :: "tyvrs" assumes "\<turnstile> \<Gamma> ok" shows "X \<sharp> ty_dom \<Gamma> = X \<sharp> \<Gamma>"
Cons_1: fixes a :: "'a list"   and b :: "'a list"   and us :: "'a list" assumes "ram \<in> set vs"   and "(a, b) = splitAtRec ram us vs" shows "us @ vs = a @ [ram] @ b"
that: fixes a :: "'a" assumes "\<And>n. a \<in> S n" shows "thesis"
finite_dom_map_of: fixes l :: "('a \<times> 'b) list" shows "finite (dom (table_of l))"
simps_2: fixes x :: "'r"   and xs :: "[:'r:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "next\<cdot>(x :# xs)\<cdot>Null = Null"
stutter_reduced_def: fixes \<sigma> :: "nat \<Rightarrow> 'a" shows "\<natural>\<sigma> = \<sigma> \<circ> max_stutter_sampler \<sigma>"
eventually_INF_base: fixes B :: "'b set"   and F :: "'b \<Rightarrow> 'c filter"   and P :: "'c \<Rightarrow> bool" assumes "B \<noteq> {}"   and "\<And>a b. \<lbrakk>a \<in> B; b \<in> B\<rbrakk> \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)" shows "eventually P (Inf (F ` B)) = (\<exists>b\<in>B. eventually P (F b))"
SS: shows "dom (hm_\<alpha> m) \<subseteq> \<Union> {uu_. \<exists>lm hc. uu_ = dom (lm.\<alpha> lm) \<and> rm.\<alpha> m' hc = Some lm}"
sumOutside: fixes g :: "('c \<times> 'd) set"   and f :: "'c \<times> 'd \<Rightarrow> 'e"   and X :: "'c set" assumes "finite g" shows "sum f g = sum f (Outside' X g) + sum f (g || X)"
sisoL_update: fixes cl :: "('test, 'atom, 'choice) cmd list"   and c' :: "('test, 'atom, 'choice) cmd"   and n :: "nat" assumes "sisoL cl"   and "siso c'" shows "sisoL (cl[n := c'])"
rp: shows "(\<xi>, p) \<in> reachable A I"
to_ns_sat: fixes cs :: "('i \<times> constraint) list"   and v' :: "(nat, 'a) mapping" assumes "\<langle>v'\<rangle> \<Turnstile>\<^sub>n\<^sub>s\<^sub>s  (snd ` set (to_ns cs))" shows "\<langle>from_ns v' (map snd (to_ns cs))\<rangle> \<Turnstile>\<^sub>c\<^sub>s (snd ` set cs)"
rel_funI: fixes A :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and B :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and f :: "'k \<Rightarrow> 'm"   and g :: "'l \<Rightarrow> 'n" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
f1: shows "\<not> ((fst instr = sreg_type RDPSR \<or> fst instr = sreg_type RDWIM \<or> fst instr = sreg_type RDTBR \<or> fst instr = sreg_type RDASR \<and> privileged_ASR (get_operand_w5 (snd instr ! 0))) \<and> get_S (cpu_reg_val PSR (snd (fst (get_curr_win () s)))) = 0)"
src_some_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_src (some_trg \<mu>) = some_trg \<mu>"
connected_iff_connected_component: fixes S :: "'a set" shows "connected S = (\<forall>x\<in>S. \<forall>y\<in>S. connected_component S x y)"
assms: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P (removeAll x xs)" shows "P xs"
induct: fixes P1 :: "Term.var \<Rightarrow> bool"   and P2 :: "expr \<Rightarrow> bool"   and P3 :: "stmt \<Rightarrow> bool"   and P4 :: "expr list \<Rightarrow> bool"   and var :: "Term.var" assumes "\<And>x. P1 (LVar x)"   and "\<And>x1 x2 x3 x4 x5. P2 x4 \<Longrightarrow> P1 ({x1,x2,x3}x4..x5)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P1 (x1.[x2])"   and "\<And>x1 x2. \<lbrakk>P3 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (InsInitV x1 x2)"   and "\<And>x. P2 (NewC x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (New x1[x2])"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Cast x1 x2)"   and "\<And>x1 x2. P2 x1 \<Longrightarrow> P2 (x1 InstOf x2)"   and "\<And>x. P2 (Lit x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (UnOp x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (BinOp x1 x2 x3)"   and "P2 Super"   and "\<And>x. P1 x \<Longrightarrow> P2 (Acc x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1:=x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (x1 ? x2 : x3)"   and "\<And>x1 x2 x3 x4 x5 x6 x7. \<lbrakk>P2 x4; P4 x7\<rbrakk> \<Longrightarrow> P2 ({x1,x2,x3}x4\<cdot>x5( {x6}x7))"   and "\<And>x1 x2. P2 (Methd x1 x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P2 (Body x1 x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (InsInitE x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Callee x1 x2)"   and "P3 Skip"   and "\<And>x. P2 x \<Longrightarrow> P3 (Expr x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (x1\<bullet> x2)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P3 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (If(x1) x2 Else x3)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x2; P3 x3\<rbrakk> \<Longrightarrow> P3 (x1\<bullet> While(x2) x3)"   and "\<And>x. P3 (Jmp x)"   and "\<And>x. P2 x \<Longrightarrow> P3 (Throw x)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P3 x1; P3 x4\<rbrakk> \<Longrightarrow> P3 (Try x1 Catch(x2 x3) x4)"   and "\<And>x1 x2. \<lbrakk>P3 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 Finally x2)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P3 (FinA x1 x2)"   and "\<And>x. P3 (Init x)"   and "P4 []"   and "\<And>x1 x2. \<lbrakk>P2 x1; P4 x2\<rbrakk> \<Longrightarrow> P4 (x1 # x2)" shows "P1 var"
comp_apply_eq: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and x :: "'f"   and h :: "'g \<Rightarrow> 'd"   and k :: "'f \<Rightarrow> 'g" assumes "f (g x) = h (k x)" shows "(f \<circ> g) x = (h \<circ> k) x"
evnotp2: assumes "0 < h" shows "\<forall>\<^sub>F t in at_right 0. flow0 x0 t \<notin> p2"
1: fixes n :: "nat" shows "complex_set_integrable lebesgue {0<..} (\<lambda>x. complex_of_real x powr (z - 1) / complex_of_real (exp ((real n + a) * x)))"
exhaust: fixes y :: "state" obtains x1 :: "glb \<Rightarrow> nat" and x2 :: "loc \<Rightarrow> nat" where "y = st x1 x2"
ffold_finsert: fixes x :: "'a"   and A :: "'a fset"   and z :: "interface" assumes "x |\<notin>| A" shows "ffold ((\<otimes>) \<circ> p) z (finsert x A) = ((\<otimes>) \<circ> p) x (ffold ((\<otimes>) \<circ> p) z A)"
polyfun_mult: fixes N :: "nat set"   and f :: "(nat \<Rightarrow> 'c) \<Rightarrow> 'c"   and g :: "(nat \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "polyfun N f"   and "polyfun N g" shows "polyfun N (\<lambda>x. f x * g x)"
proj_zero: fixes k :: "'k" shows "proj_poly k 0 = 0"
n\<^sub>\<MM>_def: fixes m :: "nat"   and x :: "nat" assumes "n\<^sub>\<MM> \<le> m" shows "\<MM>.token_succeeds x = (m < x \<or> (\<exists>j\<ge>i\<^sub>\<MM>. \<MM>.rank x m = Some j) \<or> \<MM>.token_run x m \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
comp_PRE_def: fixes R :: "('d \<times> 'b) set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'b \<Rightarrow> bool" shows "comp_PRE R P Q S \<equiv> \<lambda>x. S x \<longrightarrow> P x \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> Q x y)"
assms_4: fixes x :: "'b" assumes "x \<in> S" shows "g x = h x"
akra_bazzi_params_nonzero_def: fixes k :: "nat"   and as :: "real list"   and bs :: "real list" shows "akra_bazzi_params_nonzero k as bs \<equiv> (length as = k \<and> length bs = k) \<and> (\<forall>a. a \<in> set as \<longrightarrow> 0 \<le> a) \<and> (\<exists>a\<in>set as. 0 < a) \<and> (\<forall>b. b \<in> set bs \<longrightarrow> b \<in> {0<..<1})"
relation_of_inverse: fixes x :: "('c, 'd) action" shows "action_of (action.relation_of x) = x"
not_S_least: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'a"   and t :: "('a, 'b) Term.term" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "least f" shows "\<not> fst (kbo.kbo w w0 scf least pr_strict pr_weak (Fun f []) t)"
SN_order_pair_def: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" shows "SN_order_pair S NS \<equiv> order_pair S NS \<and> SN_ars S"
top_closed: shows "\<bottom>\<^sub>o\<^sub>w \<in> U"
that_1: fixes x :: "'a" assumes "x \<in> src.carrier" shows "diff k (src.charts_submanifold (U x)) charts2 f"
abs_bound: shows "\<bar>x\<bar> \<in>\<^sub>r abs_bound"
emeasure_subprob_space_less_top: fixes p :: "'c spmf"   and A :: "'c set" shows "emeasure (measure_spmf p) A \<noteq> \<top>"
assms_4: shows "i *\<^sub>R p + j *\<^sub>R q = 0"
T_2: shows "T \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta> = map Var Z"
no_self_dep: shows "\<not> depends_on (insert_messages M \<union> {m}) m m"
c': fixes c :: "'b" shows "f' (K$ c) = cnst c"
Call: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s :: "'d" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> (Call p # cs, css, Normal s) \<rightarrow> ([bdy], (cs, Throw # cs) # css, Normal s)"
all: fixes v :: "'a iarray" assumes "v \<in> set mm" shows "IArray.length v = nc"
one_lam_cong: fixes t1 :: "lam"   and t2 :: "lam"   and a :: "name" assumes " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2" shows " Lam [a].t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* Lam [a].t2"
WTrt_elim_cases_7: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'b J_prog"   and h :: "'a"   and E :: "String.literal \<Rightarrow> ty option"   and D :: "ty"   and e :: "'b expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (Cast D e) T" obtains T :: "ty" where "Ta__ = D" and "J_heap_base.WTrt typeof_addr P h E e T" and "is_type P D"
lower_inv_eq: fixes x :: "'a" assumes "x \<in> carrier \<X>" shows "\<pi>\<^sub>* (\<pi>\<^sup>* x) = x"
zero: shows "\<forall>k<n. coeff p k = (0::'a)"
elims_2: fixes x :: "const"   and xa :: "const" assumes "x \<sqsubseteq> xa" obtains uu_ :: "const" where "x = uu_" and "xa = Any"   | n :: "int" and m :: "int" where "x = Const n" and "xa = Const m" and "n = m"
PiM_cong: fixes I :: "'b set"   and J :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and N :: "'b \<Rightarrow> 'c measure" assumes "I = J"   and "\<And>x. x \<in> I \<Longrightarrow> M x = N x" shows "Pi\<^sub>M I M = Pi\<^sub>M J N"