hcju/isabelleps huggingface dataset:
height_up\<^sub>i_merge: fixes l :: "'b btree"   and a :: "'b"   and r :: "'b btree"   and t :: "'b btree"   and ls :: "('b btree \<times> 'b) list"   and x :: "'b"   and rs :: "('b btree \<times> 'b) list"   and tt :: "'b btree" assumes "height_up\<^sub>i (Up\<^sub>i l a r) = height t" shows "height (Node (ls @ (t, x) # rs) tt) = height (Node (ls @ (l, a) # (r, x) # rs) tt)"
inf_sup_ord_4: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and y :: "'g"   and x :: "'g" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
null: fixes i :: "'a"   and x :: "'b" assumes "i \<in> I"   and "x \<in> space (M i)" shows "{x} \<in> null_sets (M i)"
select_convs_1: fixes cx :: "int"   and cy :: "int"   and more :: "'a" shows "cx \<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = cx"
simps_3: fixes a :: "ereal" shows "(a < - \<infinity>) = False"
obtain_function_on_ext_funcset: fixes A :: "'c set"   and M :: "'d multiset" assumes "finite A"   and "size M = card A" shows "\<exists>f\<in>A \<rightarrow>\<^sub>E set_mset M. image_mset f (mset_set A) = M"
assms_3: fixes i :: "nat" assumes "i < length xl" shows "intT (tpOfV (xl ! i)) (al ! i)"
rtrancl_trancl_absorb: fixes R :: "('c \<times> 'c) set" shows "(R\<^sup>*)\<^sup>+ = R\<^sup>*"
f_asymp_equiv: shows "f \<sim>[at_top] (\<lambda>x. x)"
drop_Suc: fixes n :: "nat"   and xs :: "'b list" shows "drop (Suc n) xs = drop n (tl xs)"
InvariantEquivalentZLAfterExhaustiveUnitPropagate: fixes state :: "State"   and Phi :: "Literal list list" assumes "exhaustiveUnitPropagate_dom state"   and "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantEquivalentZL (getF state) (getM state) Phi"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqQ (getQ state)" shows "let state' = exhaustiveUnitPropagate state in InvariantEquivalentZL (getF state') (getM state') Phi"
rcs_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and a :: "'e" shows "H \<bullet>\<^bsub>G\<^esub> a = {b. \<exists>h\<in>H. h \<cdot>\<^bsub>G\<^esub> a = b}"
fields: shows "q = \<lparr>lenf = lenf, front = front, status = status, rear = rear, lenr = lenr\<rparr>"
ring_one: fixes R :: "('d, 'e) Ring_scheme" assumes "Ring R" shows "1\<^sub>r\<^bsub>R\<^esub> \<in> carrier R"
IH: shows "rel_stacktraces F2 st1 (Subx.box_stack f st2) (Some g)"
div_mod_decomp: fixes A :: "nat"   and n :: "nat" shows "A = A div n * n + A mod n"
index_of_const_at_0: shows "index_of_const 0 = 0"
T7: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ F \<and> Unchanged v \<longrightarrow> \<circle>F" shows "|~ (F \<and> \<circle>\<box>F) = \<box>F"
cofinite_card: shows "card {f. CK_nf_pos f} = 5"
simps_2: fixes p :: "String.literal"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "Formula.future_bounded (formula.Let p \<phi> \<psi>) = (Formula.future_bounded \<phi> \<and> Formula.future_bounded \<psi>)"
that: fixes a :: "complex" assumes "(\<lambda>z. if z = 1 then a else (z - 1) * deriv zeta z - deriv zeta z / zeta z) analytic_on {s. 1 \<le> Re s}" shows "thesis"
diff_numeral_special_9: 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 "minus one one = zero"
division_split_right_inj: fixes \<D> :: "'a set set"   and S :: "'a set"   and K1 :: "'a set"   and c :: "real"   and k :: "'a"   and K2 :: "'a set" assumes "\<D> division_of S"   and "K1 \<inter> {x. c \<le> x \<bullet> k} = K2 \<inter> {x. c \<le> x \<bullet> k}"   and "K1 \<in> \<D>"   and "K2 \<in> \<D>"   and "K1 \<noteq> K2" shows "interior (K1 \<inter> {x. c \<le> x \<bullet> k}) = {}"
rcis_Ex: fixes z :: "complex" shows "\<exists>r a. z = rcis r a"
keys_monic: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "keys (monic p) = keys p"
i: shows "\<forall>M\<ge>N1. P1 \<in> inferred_clause_sets R S M"
distinct_9: fixes x2 :: "(val \<times> val) fset" shows "val.Fun x2 \<noteq> Wrong"
imp_vacuous: fixes u :: "'b upred" shows "(false \<Rightarrow> u) = true"
Card_order_iff_ordIso_card_of: fixes r :: "('c \<times> 'c) set" shows "Card_order r = (r =o |Field r|)"
lset_LCons: fixes x :: "'a"   and xs :: "'a llist" shows "lset (LCons x xs) = insert x (lset xs)"
butlast_tl: fixes xs :: "'b list" shows "butlast (tl xs) = tl (butlast xs)"
div_write_new_val_result: fixes i :: "sparc_operation \<times> inst_operand list"   and result :: "word32"   and temp_V :: "word1"   and s :: "'b sparc_state" shows "snd (div_write_new_val i result temp_V s) = False"
second_summand_not_universe: fixes x :: "i"   and y :: "i" assumes "CEMU"   and "x \<oplus> y \<noteq> u" shows "y \<noteq> u"
Vr_potent_eqTr1: fixes v :: "'a \<Rightarrow> ant"   and n :: "ant"   and m :: "ant" assumes "valuation K v"   and "0 \<le> n"   and "0 \<le> m"   and "vp K v\<^bsup> Vr K v n\<^esup> = vp K v\<^bsup> Vr K v m\<^esup>"   and "m = 0" shows "n = m"
loop_no_end: fixes p :: "'a" assumes "is_point p" shows "end_points (p ; p\<^sup>T) = 0"
that: fixes e :: "nat \<times> nat" assumes "e \<in> set p"   and "cf e = resCap p" shows "thesis"
generalized_sfw_def: fixes l :: "('e simple_match \<times> 'f) list"   and p :: "('e, 'g) simple_packet_scheme" shows "generalized_sfw l p \<equiv> find (\<lambda>(m, a). simple_matches m p) l"
Sup_aform: fixes e :: "nat \<Rightarrow> real"   and X :: "'a \<times> 'a pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" shows "aform_val e X \<le> Sup_aform X"
eq_card_imp_inj_on: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "card (f ` A) = card A" shows "inj_on f A"
outer_prod_def: fixes v :: "'b vec"   and w :: "'b 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)"
lpo_eq_wpo: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "lpo_pr s t = wpo s t"
xor_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a XOR b) XOR c = a XOR b XOR c"
DFA_axioms: shows "DFA alphabet init delta accept wellformed Language wf Lang"
of_nat_power: fixes p :: "nat"   and x :: "nat" assumes "p < 2 ^ x"   and "x < LENGTH('a)" shows "word_of_nat p < 2 ^ x"
pres_type_def: fixes step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat"   and A :: "'b set" shows "pres_type step n A = (\<forall>\<tau>\<in>A. \<forall>p<n. \<forall>(q, \<tau>')\<in>set (step p \<tau>). \<tau>' \<in> A)"
induct: fixes P :: "('b \<times> color) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "('b \<times> color) tree"   and a1 :: "'b"   and a2 :: "('b \<times> color) tree" assumes "\<And>t1 a t2 b t3. P (R t1 a t2) b t3"   and "\<And>a t2 b t3. P \<langle>\<rangle> a (B t2 b t3)"   and "\<And>v vc vb a t2 b t3. P (B v vc vb) a (B t2 b t3)"   and "\<And>a t2 b t3 c t4. P \<langle>\<rangle> a (R (B t2 b t3) c t4)"   and "\<And>v vc vb a t2 b t3 c t4. P (B v vc vb) a (R (B t2 b t3) c t4)"   and "\<And>a. P \<langle>\<rangle> a \<langle>\<rangle>"   and "\<And>a vc vb. P \<langle>\<rangle> a (R \<langle>\<rangle> vc vb)"   and "\<And>a va vf ve vc vb. P \<langle>\<rangle> a (R (R va vf ve) vc vb)"   and "\<And>v vc vb a. P (B v vc vb) a \<langle>\<rangle>"   and "\<And>v vc vb a vf ve. P (B v vc vb) a (R \<langle>\<rangle> vf ve)"   and "\<And>v vc vb a vd vi vh vf ve. P (B v vc vb) a (R (R vd vi vh) vf ve)" shows "P a0 a1 a2"
while_rel2_def: fixes mp :: "nat" shows "while_rel2 mp = measure (\<lambda>x. trivCond (x = [])) <*lex*> measure ((-) (mp + 1)) <*lex*> measure (\<lambda>x. trivCond (\<not> x))"
lprefix_refl: fixes xs :: "'c llist" shows "xs \<sqsubseteq> xs"
Up_To_Technique: fixes d :: "'d"   and PP :: "nat set"   and R' :: "('com list \<times> 'com list) set"   and R :: "('com list \<times> 'com list) set" assumes "disj_dlHPP_Bisimulation_Up_To_R' d PP R' R"   and "Arefl R \<inter> Arefl R' \<subseteq> {[]}" shows "SdlHPPB d PP (R \<union> R')"
sub: shows "\<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF \<phi>)) \<subseteq> - U"
H: shows "Hermite_of_upt_row_i A k ass res $ i $ y \<noteq> (0::'b)"
hyps: shows "\<not> is_final (steps0 (1, [], Oc \<up> x) tcopy_begin n)"
fds_converges_cmult_left: fixes f :: "'b fds"   and s :: "'b"   and c :: "'b" assumes "fds_converges f s" shows "fds_converges (fds_const c * f) s"
simps_2: fixes U :: "'a set"   and u :: "'a"   and r :: "'a list" shows "atU_s U (u # r) = (u \<in> U)"
hyps: shows "P ((\<Prod>i = 0..<b. monom 1 1) mod f) n"
id2: shows "comm_monoid_mult_class.coprime f'' g'' = coprime_m f g"
Un_upper2: fixes B :: "'b set"   and A :: "'b set" shows "order.greater_eq (A \<union> B) B"
drop_0: fixes xs :: "'k list" shows "drop 0 xs = xs"
product_run_embed_limit_finiteness_snd: fixes \<delta>\<^sub>1 :: "'b \<Rightarrow> 'd \<Rightarrow> 'b"   and \<delta>\<^sub>2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>1 :: "'b"   and q\<^sub>2 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and S' :: "('c \<times> 'd \<times> 'c) set" assumes "finite (range (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w))" shows "(limit (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w) \<inter> \<Union> (embed_transition_snd ` S') = {}) = (limit (run\<^sub>t \<delta>\<^sub>2 q\<^sub>2 w) \<inter> S' = {})"
rel_simps_65: shows "- (1::'a) < (1::'a)"
finite_converse: fixes r :: "('d \<times> 'c) set" shows "finite (r\<inverse>) = finite r"
HEndPhase0_dblock_mbal: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p" shows "\<forall>br\<in>allBlocksRead s p. mbal br < mbal (dblock s' p)"
INF_eqI: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and x :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq x (f i)"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq y (f i)) \<Longrightarrow> less_eq y x" shows "Inf (f ` A) = x"
IH: fixes x3a :: "'a sorter"   and Rs :: "('a \<times> 'a) set set" assumes "x3a \<in> range f"   and "finite Rs" shows "card ((\<lambda>R. eval_sorter R x3a) ` Rs) \<le> 2 ^ count_wc_queries Rs x3a"
bound_def: fixes z :: "'a"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "bound z n f \<equiv> \<forall>m>n. f m = z"
convex_on_ereal_subset: fixes t :: "'b set"   and f :: "'b \<Rightarrow> ereal"   and s :: "'b set" assumes "Lower_Semicontinuous.convex_on t f"   and "s \<subseteq> t" shows "Lower_Semicontinuous.convex_on s f"
height_balL: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "hbt l"   and "hbt r"   and "height l = height r + m + 1" shows "height (balL l a r) \<in> {height r + m + 1, height r + m + 2}"
square_free_heuristic: fixes f :: "int poly"   and p :: "int" assumes "square_free_heuristic f = Some p" shows "comm_monoid_mult_class.coprime (lead_coeff f) p \<and> separable_impl p f \<and> prime p"
merge_list_correct: fixes ls :: "'a list list"   and as :: "'a list list" assumes "\<And>l. l \<in> set ls \<Longrightarrow> distinct l \<and> sorted l"   and "\<And>l. l \<in> set as \<Longrightarrow> distinct l \<and> sorted l" shows "distinct (merge_list as ls) \<and> sorted (merge_list as ls) \<and> set (merge_list as ls) = set (concat (as @ ls))"
Chain_sub: fixes D :: "('h, 'i) Order_scheme"   and X :: "'h set" assumes "Order D"   and "Chain D X" shows "X \<subseteq> carrier D"
assms_2: shows "v \<cdot> u \<le>p z"
divisor_count_eq_0_iff: fixes n :: "nat" shows "(divisor_count n = 0) = (n = 0)"
conv_galois_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ; y \<le> z) = (y \<le> - (x\<^sup>T ; - z))"
bv_to_nat_zero_imp_empty: fixes w :: "bit list" assumes "bv_to_nat w = 0" shows "norm_unsigned w = []"
disconnected_nodes_eq2_h2: fixes doc_ptr :: "(_) document_ptr" assumes "doc_ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr" shows "|h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h' \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
keys_single: fixes k :: "'b"   and v :: "'c" shows "keys (single k v) = (if v = (0::'c) then {} else {k})"
simps: fixes p :: "'b poly"   and q :: "'b poly" shows "gcd_poly_code_aux p q = (if q = 0 then normalize p else gcd_poly_code_aux q (primitive_part (pseudo_mod p q)))"
loop_correct: fixes n :: "nat" assumes "0 < n" shows "{inv_loop1 n} tcopy_loop {inv_loop0 n}"
A: shows "A = prod_emb I M J (Pi\<^sub>E J E)"
assms_4: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "\<bar>det (matrix (f' x))\<bar> \<le> B"
d_OUT_ge_point: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" shows "f (x, y) \<le> d_OUT f x"
rep_eq: fixes x :: "uint32" shows "Rep_uint (uint_of_uint32 x) = ucast (Rep_uint32 x)"
bra_bra_vec: fixes v :: "complex Matrix.vec" shows "\<langle>|v\<rangle>| = bra_vec v"
IH: fixes ys :: "'a list" assumes "traverse cast_Dyn xs = Some ys" shows "map typeof xs = replicate (length xs) None"
select_convs_6: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "cache \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = cache"
r_pdec1_prim: shows "prim_recfn 1 r_pdec1"
extstrand_subst_hom_2: fixes x :: "('a, 'b) extstrand_step"   and S :: "('a, 'b) extstrand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "x # S \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta> = (x \<cdot>\<^sub>e\<^sub>s\<^sub>t\<^sub>p \<theta>) # (S \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta>)"
length_i: fixes s :: "String.literal" assumes "\<exists>d. (l, i) = (s, [d])" shows "length i = 1"
plus_discr_def: fixes x :: "'a discr"   and y :: "'a discr" shows "x + y \<equiv> Discr (undiscr x + undiscr y)"
Conseq: fixes P :: "'e set"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'f, 'g) com option"   and \<Theta> :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and F :: "'g set"   and c :: "('e, 'f, 'g) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<forall>s\<in>P. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' c Q',A' \<and> s \<in> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
insert_mono: fixes C :: "'a set"   and D :: "'a set"   and a :: "'a" assumes "C \<subseteq> D" shows "insert a C \<subseteq> insert a D"
com_sifum_secure_def: fixes c :: "('Var, 'AExp, 'BExp) Stmt" shows "com_sifum_secure c = c \<sim>\<^bsub>mds\<^sub>s\<^esub> c"
maximal_independent_subset_extend: fixes S :: "('b, 'c) vec set"   and V :: "('b, 'c) vec set" assumes "S \<subseteq> V"   and "vec.independent S" obtains B :: "('b, 'c) vec set" where "S \<subseteq> B" and "B \<subseteq> V" and "vec.independent B" and "V \<subseteq> vec.span B"
g_inv_least: fixes n :: "nat"   and m :: "nat" assumes "n \<le> g m" shows "g_inv n \<le> m"
gm_expr: shows "LINT x|M. f x * enn2real (nn_cond_exp M F gm x) = LINT x|M. f x * enn2real (gm x)"
IDirProds_eq_finprod_PiE: fixes I :: "'d set"   and S :: "'d \<Rightarrow> ('c \<Rightarrow> 'a) set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (S i) (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)" shows "IDirProds (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) S I = (\<lambda>x. finprod (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) x I) ` Pi\<^sub>E I S"
lookup_combine: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m1 :: "('b, 'a) mapping"   and m2 :: "('b, 'a) mapping"   and x :: "'b" shows "Mapping.lookup (Mapping.combine f m1 m2) x = combine_options f (Mapping.lookup m1 x) (Mapping.lookup m2 x)"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier H'" shows "\<exists>y\<in>carrier H'. y \<otimes>\<^bsub>H'\<^esub> x = \<one>\<^bsub>H'\<^esub>"
raw_analz_image_freshK: fixes evs :: "event list" assumes "evs \<in> recur" shows "\<forall>K KK. KK \<subseteq> - range shrK \<longrightarrow> (Key K \<in> analz (Key ` KK \<union> knows Spy evs)) = (K \<in> KK \<or> Key K \<in> analz (knows Spy evs))"
cc_true: shows "I \<Turnstile>m mset CAs \<cdot>cm \<sigma> \<cdot>cm \<eta>"
perm_append1_eq: fixes zs :: "'d list"   and xs :: "'d list"   and ys :: "'d list" shows "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"
real_root_divide: fixes n :: "nat"   and x :: "real"   and y :: "real" shows "root n (x / y) = root n x / root n y"
upper_minus: fixes A :: "'a interval"   and B :: "'a interval" shows "upper (A - B) = upper A - lower B"
UnI2: fixes c :: "'e"   and B :: "'e set"   and A :: "'e set" assumes "c \<in> B" shows "c \<in> A \<union> B"
abs_summable_on_prod_PiE: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)"   and "\<And>x. x \<in> A \<Longrightarrow> f x abs_summable_on B x" shows "(\<lambda>g. \<Prod>x\<in>A. f x (g x)) abs_summable_on Pi\<^sub>E A B"
degree_const: fixes z :: "real"   and x :: "nat" shows "MPoly_Type.degree (Const z) x = 0"
gt_proper_sub: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary t"   and "proper_sub s t" shows "local.kbo_std.gt t s"
closest_valid_2: fixes X :: "'a set"   and I :: "'a \<Rightarrow> Regions.intv"   and k :: "'a \<Rightarrow> nat"   and r :: "('a \<times> 'a) set" assumes "\<forall>x\<in>X. \<not> Regions.isConst (I x)"   and "finite X"   and "Regions.valid_region X k I r"   and "{x \<in> {x \<in> X. Regions.isIntv (I x)}. \<forall>y\<in>{x \<in> X. Regions.isIntv (I x)}. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r} \<noteq> {}" shows "Regions.valid_region X k (\<lambda>x. if x \<notin> {x \<in> {x \<in> X. Regions.isIntv (I x)}. \<forall>y\<in>{x \<in> X. Regions.isIntv (I x)}. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r} then I x else Regions.intv.Const (intv_const (I x) + 1)) {(x, y). (x, y) \<in> r \<and> x \<notin> {x \<in> {x \<in> X. Regions.isIntv (I x)}. \<forall>y\<in>{x \<in> X. Regions.isIntv (I x)}. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r} \<and> y \<notin> {x \<in> {x \<in> X. Regions.isIntv (I x)}. \<forall>y\<in>{x \<in> X. Regions.isIntv (I x)}. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r}}"
untilE: fixes \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "(\<gamma>' \<UU>\<^sub>c \<gamma>) t n" shows "\<exists>n''\<ge>n. \<gamma> t n'' \<and> (\<forall>n'\<ge>n. n' < n'' \<longrightarrow> \<gamma>' t n')"
lunit_eqI: fixes f :: "'d"   and a :: "'d" assumes "\<guillemotleft>f : \<I>\<^sub>D \<otimes>\<^sub>D a \<rightarrow>\<^sub>D a\<guillemotright>"   and "\<I>\<^sub>D \<otimes>\<^sub>D f = (\<iota>\<^sub>D \<otimes>\<^sub>D a) \<cdot>\<^sub>D D.assoc' \<I>\<^sub>D \<I>\<^sub>D a" shows "f = \<l>\<^sub>D[a]"
assms_1: shows "w \<in> set_disconnected_nodes_locs object_ptr"
simps_9: fixes v :: "formula"   and va :: "formula" shows "get_and_list (And v va) = [And v va]"
distinct_remdups_id: fixes xs :: "'b list" assumes "distinct xs" shows "remdups xs = xs"
fsubset_funion_eq: fixes A :: "'c fset"   and B :: "'c fset" shows "(A |\<subseteq>| B) = (A |\<union>| B = B)"
INVS_soundK_all: fixes K :: "nat"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and D :: "Class"   and m' :: "Method"   and T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and MI :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and Anno2 :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and M' :: "Var list \<times> (int \<times> Instr) list \<times> int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "INVS_soundK K G C m l D m' T MI Anno Anno2 M' A B I"