hcju/isabelleps huggingface dataset:
simps_6: fixes x :: "nat"   and P :: "pol"   and y :: "nat"   and Q :: "pol" shows "Pinj x P \<langle>+\<rangle> Pinj y Q = (if x = y then mkPinj x (P \<langle>+\<rangle> Q) else if y < x then mkPinj y (Pinj (x - y) P \<langle>+\<rangle> Q) else mkPinj x (Pinj (y - x) Q \<langle>+\<rangle> P))"
curry_conv: fixes f :: "'d \<times> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "curry f a b = f (a, b)"
varTrue: fixes b :: "BExpr"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and r :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<beta> :: "(Location \<times> Location) set"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "evalB b (fst t)"   and "\<Phi> (r, t, \<beta>)"   and "\<forall>w \<gamma>. r \<equiv>\<^sub>\<gamma> w \<longrightarrow> (b, \<Phi>, \<gamma>, s, w) \<in> var" shows "(b, \<Phi>, \<beta>, s, t) \<in> var"
no_head_generates: fixes b :: "'a list" assumes "b \<in> \<C>" shows "\<langle>no_head_gen \<C> b\<rangle> = no_head \<C> b"
produces_imp_neg_notin_lits: fixes N :: "'a literal multiset set"   and C :: "'a literal multiset"   and A :: "'a" assumes "production N C = {A}" shows "Neg A \<notin># C"
OrdCheck: fixes v :: "i"   and x :: "\<kappa>" shows "[\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<box>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>,x\<rparr> in v] = (proper x \<and> (case rep x of \<omega>\<nu> y \<Rightarrow> True | \<alpha>\<nu> set \<Rightarrow> False))"
fps_is_right_unit_iff_zeroth_is_right_unit: fixes f :: "'b fps" shows "(\<exists>g. 1 = g * f) = (\<exists>k. (1::'b) = k * f $ 0)"
assms_2: fixes x :: "real" shows "((\<lambda>x. f x - g x) has_real_derivative 0) (at x)"
lower_asymptotic_density_one_Inter: fixes A :: "nat \<Rightarrow> nat set" assumes "\<And>n. lower_asymptotic_density (A n) = 1" shows "\<exists>B. lower_asymptotic_density B = 1 \<and> (\<forall>n. \<exists>N. B \<inter> {N..} \<subseteq> A n)"
safe_free_flowing_def: fixes cfg :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> bool \<times> nat set \<times> 'd) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" shows "safe_free_flowing cfg \<equiv> let (ts, m, \<S>) = cfg in \<forall>i<length ts. let (p, is, \<theta>, x, \<D>, \<O>, \<R>) = ts ! i in map owned ts,i\<turnstile> (is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
diff: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "additive f" shows "f (x - y) = f x - f y"
smallest_norm_exists: fixes M :: "'b set" assumes "convex M"   and "closed M"   and "M \<noteq> {}" shows "\<exists>k. is_arg_min norm (\<lambda>t. t \<in> M) k"
op_uprio_empty_def: shows "op_uprio_empty \<equiv> Map.empty"
ConseqMGT: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and P' :: "'d \<Rightarrow> 'd set"   and c :: "('d, 'e, 'f) com"   and Q' :: "'d \<Rightarrow> 'd set"   and A' :: "'d \<Rightarrow> 'd set"   and P :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) c (Q' Z),(A' Z)"   and "\<And>s. s \<in> P \<Longrightarrow> s \<in> P' s \<and> (\<forall>t. t \<in> Q' s \<longrightarrow> t \<in> Q) \<and> (\<forall>t. t \<in> A' s \<longrightarrow> t \<in> A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
left_neutral: fixes a :: "'c \<times> 'd" shows "sup_prod bot_prod a = a"
space_measure_of_eq: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" assumes "sigma_algebra \<Omega> M" shows "space (measure_of \<Omega> M \<mu>) = \<Omega>"
order_consistent_sym: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "order_consistent r s" shows "order_consistent s r"
relcompI: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c"   and s :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d" assumes "r a b"   and "s b c" shows "(r OO s) a c"
already_in_span: fixes A :: "'a vec set"   and v :: "'a vec" assumes "A \<subseteq> carrier_vec n"   and "v \<in> local.span A" shows "local.span A = local.span (A \<union> {v})"
asympartp_sympartp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "asympartp (sympartp r) = bot"
distinct_17: fixes x31 :: "fm"   and x32 :: "fm"   and x51 :: "name"   and x52 :: "fm" shows "x31 OR x32 \<noteq> SyntaxN.Ex x51 x52"
word_log2_zero_eq: shows "word_log2 0 = 0"
msed_rel_invL: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'b"   and M :: "'b multiset"   and N :: "'e multiset" assumes "rel_mset R (add_mset a M) N" shows "\<exists>N1 b. N = add_mset b N1 \<and> R a b \<and> rel_mset R M N1"
mono_bal_i: fixes h :: "nat"   and n :: "nat"   and n' :: "nat"   and h' :: "nat" assumes "int h \<le> \<lceil>c * log 2 (real n + 1)\<rceil>"   and "n \<le> n'"   and "h' \<le> h" shows "int h' \<le> \<lceil>c * log 2 (real n' + 1)\<rceil>"
valuation_for_plan_def: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list" shows "valuation_for_plan \<Pi> \<pi> \<equiv> let vs = \<Pi>\<^sub>\<V>; ops = \<Pi>\<^sub>\<O>; \<tau> = trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>; t = length \<pi>; t' = length \<tau>; \<A>\<^sub>V = valuation_for_state_variables \<Pi> \<pi> \<tau>; \<A>\<^sub>O = valuation_for_operator_variables \<Pi> \<pi> \<tau>; V = {State k (index vs v) |k v. k \<in> {0..<t + 1} \<and> v \<in> set vs} in Let {Operator k (index ops op) |k op. k \<in> {0..<t} \<and> op \<in> set ops} (override_on (override_on \<A>\<^sub>0 \<A>\<^sub>V V) \<A>\<^sub>O)"
assms_2: shows "xi \<noteq> repelling_fixed_point f"
ctr_simps_in_iff: fixes A :: "'b set"   and U :: "'b set" shows "(\<forall>x\<in>A. x \<in> U) = (A \<subseteq> U)"
pow_nat: fixes i :: "int"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "0 \<le> i" shows "x [^]\<^bsub>G\<^esub> nat i = x [^]\<^bsub>G\<^esub> i"
not_interp_to_interp_imp_less: fixes A :: "'a"   and C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "A \<notin> interp C"   and "A \<in> interp D" shows "C < D"
card_eq_dim: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "card B = dim V"   and "finite B" shows "independent B = (V \<subseteq> span B)"
drop_exp_base: fixes n :: "nat"   and x :: "'a"   and m :: "nat" shows "drop n (exp_base x m) = exp_base x (m - n)"
elims: fixes x :: "('b, 'a) ta_rule"   and y :: "'a" assumes "rhsl x = y" obtains q :: "'b" and l :: "'a" and qs :: "'b list" where "x = q \<rightarrow> l qs" and "y = l"
qfN_mult1: fixes a :: "int"   and N :: "int"   and b :: "int"   and c :: "int"   and d :: "int" shows "(a\<^sup>2 + N * b\<^sup>2) * (c\<^sup>2 + N * d\<^sup>2) = (a * c + N * b * d)\<^sup>2 + N * (a * d - b * c)\<^sup>2"
lpp_focus: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "g \<in> P[insert x X]" shows "i.lpp (focus X g) = except (i.lpp g) {x}"
sums_def: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" shows "f sums s = (\<lambda>n. sum f {..<n}) \<longlonglongrightarrow> s"
min_eq_find_min_Some: fixes xs :: "('c, 'd) bintree option list"   and a :: "'c" shows "(Binomial_Queue.min xs = Some a) = (\<exists>t. find_min xs = Some t \<and> priority t = a)"
preserves_reflects_arr: fixes f :: "'a \<times> 'a" shows "arr (cmp\<^sub>U\<^sub>P f) = B.VV.arr f"
nccD_5: fixes X :: "'c set" assumes "ncc X" shows "convex X"
that: fixes z :: "'a" assumes "P z"   and "r\<^sup>*\<^sup>* z x"   and "\<And>y. r y z \<Longrightarrow> \<not> P y" shows "thesis"
\<pi>_pos: fixes x :: "real" assumes "2 \<le> x" shows "0 < \<pi> x"
order_linear: fixes a :: "'a" shows "order a [:- a, 1::'a:] = Suc 0"
hs_it_next_def: fixes it :: "nat \<times> ('b \<times> unit) list \<times> ('b, unit) hashtable" shows "hs_it_next it \<equiv> hm_it_next it \<bind> (\<lambda>((x, uu_), it). return (x, it))"
e_transcendental_aux_bound: fixes n :: "nat" obtains C :: "real" where "0 \<le> C" and "\<And>x. x \<in> closed_segment 0 (of_nat n) \<Longrightarrow> cmod (\<Prod>k = 1..n. x - of_nat k) \<le> C"
simps_1: fixes seq :: "('d, 'e, 'f) com \<Rightarrow> ('d, 'e, 'f) com \<Rightarrow> ('d, 'e, 'f) com" shows "sequence seq [] = Skip"
convex_ereal_max: fixes s :: "'b set"   and f :: "'b \<Rightarrow> ereal"   and g :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on s f"   and "Lower_Semicontinuous.convex_on s g" shows "Lower_Semicontinuous.convex_on s (\<lambda>x. max (f x) (g x))"
b_def: shows "b = max \<bar>l\<bar> \<bar>u\<bar>"
valid_C: fixes s :: "'s" assumes "s \<in> S" shows "simple ct s \<in> valid_cfg"
run_state_get: fixes s :: "'b" shows "run_state State_Monad.get s = (s, s)"
ListRed2: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v :: "val" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>] \<langle>es',s',b'\<rangle>" shows "P \<turnstile> \<langle>Val v # es,s,b\<rangle> [\<rightarrow>] \<langle>Val v # es',s',b'\<rangle>"
simps_1: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and a :: "'e" shows "foldl f a [] = a"
sym_preproc_aux_term2_def: fixes d :: "'a \<Rightarrow> nat" shows "sym_preproc_aux_term2 d = {((gs1, ks1, ts1, fs1), gs2, ks2, ts2, fs2). gs1 = gs2 \<and> dgrad_set_le d (pp_of_term ` set ts1) (pp_of_term ` (Keys (set gs2) \<union> set ts2))}"
map_graph_preserves_subgraph: fixes A :: "('e, 'f) labeled_graph"   and B :: "('e, 'f) labeled_graph"   and f :: "('f \<times> 'g) set" assumes "subgraph A B" shows "subgraph (map_graph f A) (map_graph f B)"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state return bind get put" shows "monad return bind"
hyps_1: shows "(v, w) \<in> (E \<union> U)\<^sup>+"
polyexp_fps_Cons: fixes x :: "'b \<times> nat \<times> 'b"   and xs :: "('b \<times> nat \<times> 'b) list" shows "polyexp_fps (x # xs) = (case x of (a, k, b) \<Rightarrow> fps_of_poly (smult a (fps_monom_poly b k)) / (1 - fps_const b * fps_X) ^ (k + 1)) + polyexp_fps xs"
integrable_measure: fixes X :: "'b \<Rightarrow> 'c set"   and I :: "'b set"   and M :: "'c pmf" assumes "disjoint_family_on X I"   and "countable I" shows "integrable (count_space I) (\<lambda>i. measure_pmf.prob M (X i))"
partition_tailrec_def: fixes p :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "partition_tailrec p xs = foldr (select_p_tuple p) xs ([], [])"
proots_card_fcompose_bij_eq: fixes q1 :: "'a poly"   and q2 :: "'a poly"   and A :: "'a set"   and B :: "'a set"   and p :: "'a poly" assumes "bij_betw (\<lambda>x. poly q1 x / poly q2 x) A B"   and "p \<noteq> 0"   and "\<forall>x\<in>A. poly q2 x \<noteq> (0::'a)"   and "max (degree q1) (degree q2) \<le> 1"   and "\<forall>c. q1 \<noteq> smult c q2"   and "infinite UNIV" shows "card (proots_within p B) = card (proots_within (fcompose p q1 q2) A)"
vbrelation: fixes a :: "V"   and b :: "V" shows "vpairs (set {\<langle>a, b\<rangle>}) = set {\<langle>a, b\<rangle>}"
Elem_Elem_Fun: fixes F :: "ZF"   and U :: "ZF"   and V :: "ZF"   and p :: "ZF" assumes "Elem F (Fun U V)"   and "Elem p F" shows "isOpair p \<and> Elem (Fst p) U \<and> Elem (Snd p) V"
sorted_list_of_set_imp_less_sets: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs < ys"   and "sorted xs"   and "sorted ys" shows "list.set xs \<lless> list.set ys"
sums_ereal_positive: fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" shows "f sums (SUP n. sum f {..<n})"
nat_power_power_left: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and k :: "nat"   and n :: "'a" assumes "class.nat_power one times plus zero nat_power"   and "0 < m" shows "nat_power (m ^ k) n = power.power one times (nat_power m n) k"
le_cINF_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and u :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_below less_eq (f ` A)" shows "less_eq u (Inf (f ` A)) = (\<forall>x\<in>A. less_eq u (f x))"
collapse_1: fixes sum :: "'g + 'h" assumes "isl sum" shows "Inl (projl sum) = sum"
less_def: fixes x :: "nat"   and y :: "nat" shows "nat.less x y = (x \<le> y \<and> x \<noteq> y)"
set_of_graph_rules_def: fixes Rs :: "(('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) set" shows "set_of_graph_rules Rs \<equiv> \<forall>R\<in>Rs. graph_rule R"
fst_snd_flip: fixes xy :: "'d \<times> 'e" shows "fst xy = (snd \<circ>\<circ> case_prod) (\<lambda>x y. (y, x)) xy"
eval_expression_no_jump: fixes Env :: "env"   and s0 :: "state"   and e :: "expr"   and v :: "val"   and s1 :: "state"   and j :: "jump"   and T :: "ty" assumes "prg Env\<turnstile>s0 \<midarrow>e-\<succ>v\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)"   and "Env\<turnstile>e\<Colon>-T"   and "wf_prog (prg Env)" shows "fst s1 \<noteq> Some (Jump j)"
lm126: fixes x :: "'b"   and n :: "nat"   and l :: "'b list" shows "set (rotateLeft (length (x # rotateLeft n l) - n mod length (x # rotateLeft n l)) (x # rotateLeft n l)) = {x} \<union> set l"
mono_invE: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "f x < f y" obtains "less_eq x y"
is_none_map_option: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d option" shows "Option.is_none (map_option f x) = Option.is_none x"
rec_ci_z_def: shows "rec_ci_z \<equiv> [Goto 1]"
A: assumes "\<lbrakk>M; map Send T\<rbrakk>\<^sub>c \<theta>"   and "t \<in> set T" shows "\<lbrakk>M; [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rbrakk>\<^sub>c \<theta>"
qSwapAll_igSubstAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "igSubstClsSTR MOD"   and "igFreshClsSTR MOD"   and "igAbsRenSTR MOD" shows "(qGood qX \<longrightarrow> (\<forall>ys y1 y. qAFresh ys y1 qX \<longrightarrow> qInit MOD (qX #[[y1 \<and> y]]_ys) = igSubst MOD ys (igVar MOD ys y1) y (qInit MOD qX))) \<and> (qGoodAbs qA \<longrightarrow> (\<forall>ys y1 y. qAFreshAbs ys y1 qA \<longrightarrow> qInitAbs MOD (qA $[[y1 \<and> y]]_ys) = igSubstAbs MOD ys (igVar MOD ys y1) y (qInitAbs MOD qA)))"
rat_to_normalized_int_poly_4: fixes p :: "rat poly"   and d :: "rat"   and q :: "int poly" assumes "rat_to_normalized_int_poly p = (d, q)" shows "degree q = degree p"
pt_fun_app_eq: fixes pi :: "('b \<times> 'b) list"   and f :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> f x = (pi \<bullet> f) (pi \<bullet> x)"
induct: fixes P :: "('c, 'd) BinomialTree list \<Rightarrow> ('c, 'd) BinomialTree list \<Rightarrow> bool"   and a0 :: "('c, 'd) BinomialTree list"   and a1 :: "('c, 'd) BinomialTree list" assumes "\<And>bq. P [] bq"   and "\<And>v va. P (v # va) []"   and "\<And>t1 bq1 t2 bq2. \<lbrakk>rank t1 < rank t2 \<Longrightarrow> P bq1 (t2 # bq2); \<lbrakk>\<not> rank t1 < rank t2; rank t2 < rank t1\<rbrakk> \<Longrightarrow> P (t1 # bq1) bq2; \<lbrakk>\<not> rank t1 < rank t2; \<not> rank t2 < rank t1\<rbrakk> \<Longrightarrow> P bq1 bq2\<rbrakk> \<Longrightarrow> P (t1 # bq1) (t2 # bq2)" shows "P a0 a1"
normalize_quot_zero: fixes a :: "'b" shows "normalize_quot (a, 0::'b) = (0::'b, 1::'b)"
set_rev_mp: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
unfree_trm_def: fixes t :: "'b trm" shows "unfree t = (case t of Var a \<Rightarrow> Some (to_name a) | trm.Const x \<Rightarrow> Map.empty x | trm1 \<cdot> x \<Rightarrow> Map.empty x)"
iso_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "C.ide a"   and "C.ide b"   and "C.ide c" shows "C.iso (C.assoc a b c)"
elims: fixes x :: "('b, 'c) Combinators list"   and y :: "('b, 'c) Combinators list" assumes "removeAllDuplicates x = y" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "y = removeDuplicates xa # removeAllDuplicates xs"   | "x = []" and "y = []"
a_subgroup: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme" assumes "additive_subgroup H G" shows "subgroup H (add_monoid G)"
WHILEI_le_rule: fixes s :: "'c"   and s' :: "'d"   and R :: "('c \<times> 'd) set"   and M :: "'d \<Rightarrow> 'c nres"   and I :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres" assumes "(s, s') \<in> R"   and "\<And>W s s'. \<lbrakk>\<And>s s'. (s, s') \<in> R \<Longrightarrow> W s \<le> M s'; (s, s') \<in> R\<rbrakk> \<Longrightarrow> ASSERT (I s) \<bind> (\<lambda>_. if b s then f s \<bind> W else RETURN s) \<le> M s'" shows "WHILE\<^bsup>I\<^esup> b f s \<le> M s'"
prv_imp_neg_imp_neg_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv (neg \<phi>)" shows "prv (imp \<chi> (neg (imp \<chi> \<phi>)))"
swap: shows "LLL_Impl.basis_reduction_swap m i state' = state''"
select_convs_2: fixes init :: "'a set"   and trans :: "('a \<times> 'a) set"   and more :: "'b" shows "TS.trans \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = trans"
ts_b_def_3: shows "\<S>\<bullet>\<C>' \<turnstile> xs : [] _> ts_b"
Sup_insert: fixes a :: "'a"   and A :: "'a set" shows "\<Squnion> insert a A = a \<squnion> \<Squnion> A"
cSup_eq_maximum: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and X :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x z" shows "Sup X = z"
mon_env_map_env: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and w :: "('c, 'f) label list list" shows "mon_env fg (map ENV w) = mon_ww fg w"
true_Union_grounding_of_cls_iff: fixes I :: "'a set"   and N :: "'a literal multiset set" shows "(I \<TTurnstile>s (\<Union>C\<in>N. {C \<cdot> \<sigma> |\<sigma>. is_ground_subst \<sigma>})) = (\<forall>\<sigma>. is_ground_subst \<sigma> \<longrightarrow> I \<TTurnstile>s N \<cdot>cs \<sigma>)"
add: fixes f :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'b" assumes "additive f" shows "f (x + y) = f x + f y"
l_set_disconnected_nodes_get_tag_name_axioms_def: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_tag_name_axioms set_disconnected_nodes_locs get_tag_name_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
open_Var_open_Var_close_Var: fixes i :: "nat"   and j :: "nat"   and xT :: "char list \<times> Unary_PCF.type"   and yU :: "char list \<times> Unary_PCF.type"   and t :: "expr"   and zV :: "char list \<times> Unary_PCF.type" assumes "i \<noteq> j"   and "xT \<noteq> yU"   and "fresh yU t" shows "open_Var i yU (open_Var j zV (close_Var j xT t)) = open_Var j zV (close_Var j xT (open_Var i yU t))"
final_is_s_t_len_t: fixes t :: "('a, 'b, 'c) event list" assumes "trace init t final" shows "final = S t (length t)"
minus_eq_not_plus_1: 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 uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "uminus a = plus (not a) one"
convol_expand_snd: fixes f :: "'d \<Rightarrow> 'e \<times> 'f"   and g :: "'d \<Rightarrow> 'e" assumes "fst \<circ> f = g" shows "<g , snd \<circ> f> = f"
fdia_rel_im1: fixes R :: "('a \<times> 'a) set"   and P :: "'a set" shows "s2r (R\<inverse> `` P) = relfdia R (s2r P)"
\<gamma>'_def: shows "\<gamma>' = (\<lambda>t. - \<i> * (\<gamma> t - z) + z)"
simps_2: fixes f :: "'d"   and ss1 :: "('d, 'e) Term.term list"   and C :: "('d, 'e) ctxt"   and ss2 :: "('d, 'e) Term.term list"   and s :: "('d, 'e) Term.term" shows "(More f ss1 C ss2)\<langle>s\<rangle> = Fun f (ss1 @ C\<langle>s\<rangle> # ss2)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_nil: shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t [] = []"
open_map_iff_lower_hemicontinuous_preimage: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and T :: "'a set" assumes "f ` S \<subseteq> T" shows "(\<forall>U. openin (top_of_set S) U \<longrightarrow> openin (top_of_set T) (f ` U)) = (\<forall>U. closedin (top_of_set S) U \<longrightarrow> closedin (top_of_set T) {y \<in> T. {x \<in> S. f x = y} \<subseteq> U})"
lift_Use_High: fixes V :: "'g"   and H :: "'g set"   and E :: "'h"   and Use :: "'h \<Rightarrow> 'g set"   and X :: "'h"   and L :: "'g set" assumes "V \<in> H" shows "(Node E, V) \<in> lift_Use_set Use E X H L"
S5Reply: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCReply crCh rmCh rst p \<and> $S5 rmhist p \<and> unchanged (e p, c p, m p, rmhist!p) \<longrightarrow> S6 rmhist p$"
g_def: fixes n :: "nat"   and y :: "'v" shows "g n y = enn2real (weight \<Gamma> y * indicator (Y n) y)"
ptypings_induct: fixes x4 :: "(char list \<times> pat) list"   and x5 :: "(char list \<times> POPLmarkRecord.type) list"   and x6 :: "binding list"   and P2 :: "(char list \<times> pat) list \<Rightarrow> (char list \<times> POPLmarkRecord.type) list \<Rightarrow> binding list \<Rightarrow> bool" assumes "\<turnstile> x4 [:] x5 \<Rightarrow> x6"   and "P2 [] [] []"   and "\<And>p T \<Delta>\<^sub>1 fps fTs \<Delta>\<^sub>2 l. \<lbrakk>\<turnstile> p : T \<Rightarrow> \<Delta>\<^sub>1; \<turnstile> fps [:] fTs \<Rightarrow> \<Delta>\<^sub>2; P2 fps fTs \<Delta>\<^sub>2; fps\<langle>l\<rangle>\<^sub>? = \<bottom>\<rbrakk> \<Longrightarrow> P2 ((l, p) \<Colon> fps) ((l, T) \<Colon> fTs) (\<up>\<^sub>e \<parallel>\<Delta>\<^sub>1\<parallel> 0 \<Delta>\<^sub>2 @ \<Delta>\<^sub>1)" shows "P2 x4 x5 x6"