hcju/isabelleps huggingface dataset:
eval_cases_2: 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 n :: "nat"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" assumes "(M, l, s, n, h, v) \<in> Exec" obtains os :: "Val list" and s :: "(Var \<times> Val) list" where "sa__ = (v # os, s, h)" and "na__ = Suc 0" and "get_ins M l = Some vreturn"   | n :: "nat" and ll :: "int" and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" and m :: "nat" where "na__ = Suc (max n m)" and "(M, l, sa__, n, ll, t) \<in> Step" and "(M, ll, t, m, h, v) \<in> Exec"
intros_4: fixes a1 :: "aexp"   and s :: "char list \<Rightarrow> val"   and i1 :: "int"   and a2 :: "aexp"   and i2 :: "int" assumes "taval a1 s (Iv i1)"   and "taval a2 s (Iv i2)" shows "tbval (Less a1 a2) s (i1 < i2)"
IH: shows "\<forall>m<n. \<forall>\<Gamma>. ( \<Gamma> \<Rightarrow>* Em, m) \<in> derivable R* \<longrightarrow> ( \<Gamma> \<Rightarrow>* C, m) \<in> derivable R*"
Un_absorb: fixes A :: "'e set" shows "A \<union> A = A"
dickson_lessD1: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "dickson_less d m s t" shows "d s \<le> m"
assms_2: shows "n \<noteq> m \<and> dominates g n m"
exp_cf7_def: shows "exp_cf7 \<equiv> \<lambda>x. numer_cf7 x / numer_cf7 (- x)"
lprefix_LNil: fixes xs :: "'c llist" shows "lprefix xs LNil = lnull xs"
Lindelof: fixes \<F> :: "'a set set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> open S" obtains \<F>' :: "'a set set" where "\<F>' \<subseteq> \<F>" and "countable \<F>'" and "\<Union> \<F>' = \<Union> \<F>"
to_fract_power_le: fixes a :: "'b fract"   and b :: "'b fract"   and e :: "nat"   and f :: "nat" assumes "a * b ^ e \<in> range to_fract"   and "a \<in> range to_fract"   and "f \<le> e" shows "a * b ^ f \<in> range to_fract"
push_pres_Labeling: fixes e :: "nat \<times> nat" assumes "push_precond f l e" shows "Labeling c s t (push_effect f e) l"
almost_full_on_mon: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> P x y = Q (h x) (h y)"   and "bij_betw h A B"   and "almost_full_on Q B" shows "almost_full_on P A"
arr_iff_in_hom: fixes f :: "'a" shows "Right_a.arr f = Right_a.in_hom f (Right_a.dom f) (Right_a.cod f)"
run_shift_elim: fixes execute :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and enabled :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'c list"   and s :: "'c stream"   and p :: "'d" assumes "transition_system.run execute enabled (r @- s) p" obtains "transition_system.path execute enabled r p" and "transition_system.run execute enabled s (fold execute r p)"
Abs_lst1_fcb: fixes x :: "'b"   and T :: "'a"   and y :: "'b"   and S :: "'a"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "[[atom x]]lst. T = [[atom y]]lst. S"   and "\<lbrakk>x \<noteq> y; atom y \<sharp> T; atom x \<sharp> (y \<leftrightarrow> x) \<bullet> T\<rbrakk> \<Longrightarrow> atom x \<sharp> f x T"   and "\<lbrakk>x \<noteq> y; atom y \<sharp> T; atom x \<sharp> (y \<leftrightarrow> x) \<bullet> T\<rbrakk> \<Longrightarrow> atom y \<sharp> f x T"   and "\<lbrakk>S = (x \<leftrightarrow> y) \<bullet> T; x \<noteq> y; atom y \<sharp> T; atom x \<sharp> S\<rbrakk> \<Longrightarrow> (x \<leftrightarrow> y) \<bullet> f x T = f y S" shows "f x T = f y S"
mod_in_carrier: fixes a :: "int" shows "a mod int p \<in> carrier R"
pp_of_term_sminus: fixes v :: "'t"   and t :: "'a" shows "pp_of_term (v \<ominus> t) = pp_of_term v - t"
iMin_singleton: fixes a :: "'a" shows "iMin {a} = a"
assms_4: fixes i :: "'a" assumes "i \<in> I" shows "f i = x"
inf_in_red_inf: fixes N :: "'f set" shows "Inf_between UNIV (Red_F N) \<subseteq> Red_I N"
ufact: shows "unique_factorization_m f (lead_coeff f, mset hs)"
hom_uminus: fixes x :: "'b poly"   and i :: "nat" shows "coeff (- x) i = - coeff x i"
c4_1: shows "(fst \<gamma>1''', map (Pair True) (snd \<gamma>1''')) \<in> conv ars"
exhaustive_fusion_code: fixes g :: "('a, 'd) generator"   and s :: "'d" shows "exhaustive_fusion g s = (list.has_next g s \<and> (let (x, s') = list.next g s in \<not> proper_interval None (Some x) \<and> exhaustive_above_fusion g x s'))"
orderE: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" obtains "a = a \<sqinter> b"
intros: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and c :: "'b"   and is' :: "nat list"   and "is" :: "nat list"   and i :: "nat"   and ip :: "('b, 'a) antecedent" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "c \<in> set conclusions"   and "is' \<in> (#) 0 ` it_paths (Abstract_Task.globalize freshenLC subst_renameLCs anyP antecedent [fidx (Abstract_Task.conc_forms conclusions) c, 0] (freshenLC v_away) (Abstract_Task.to_it freshenLC renameLCs lconsts subst subst_lconsts anyP antecedent consequent rules assumptions conclusions (Solved_Task.ts freshenLC renameLCs lconsts subst subst_lconsts antecedent consequent rules assumptions c)))"   and "prefix (is @ [i]) is'"   and "ip = Solved_Task.in_port_at freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions (c, is) i" shows "Solved_Task.scope' freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions (c, is) ip (c, is')"
pair_def: fixes d :: "('fun, 'a) Term.term \<times> ('fun, 'a) Term.term" shows "pair d \<equiv> case d of (t, t') \<Rightarrow> Fun Pair [t, t']"
mset_rev: fixes xs :: "'b list" shows "mset (rev xs) = mset xs"
emeasure_return: fixes A :: "'c set"   and M :: "'c measure"   and x :: "'c" assumes "A \<in> sets M" shows "emeasure (return M x) A = indicator A x"
assms: shows "\<phi>\<^sub>O \<Psi> op\<^sub>1 = \<phi>\<^sub>O \<Psi> op\<^sub>2"
hom_dvd_iff: fixes p :: "'a"   and q :: "'a" shows "(hom p dvd hom q) = (p dvd q)"
recurrent_subset_meas_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.recurrent_subset_infty T A \<in> sets M"
Next_not_member_subset: fixes S' :: "'a set"   and S :: "'a set" assumes "finite S'"   and "S \<subseteq> S'" shows "Next S' \<notin> S"
liftAll2_def: fixes phi :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and inp1 :: "'h \<Rightarrow> 'f option"   and inp2 :: "'h \<Rightarrow> 'g option" shows "liftAll2 phi inp1 inp2 \<equiv> \<forall>i v1 v2. inp1 i = Some v1 \<and> inp2 i = Some v2 \<longrightarrow> phi v1 v2"
temp: shows "\<forall>a\<in>A. F' a \<subseteq> Field r \<and> |F a| =o |F' a| \<and> |F' a| <o r"
separation_normal: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T"   and "S \<inter> T = {}" obtains U :: "'a set" and V :: "'a set" where "open U" and "open V" and "S \<subseteq> U" and "T \<subseteq> V" and "U \<inter> V = {}"
real_sqrt_sum_squares_ge2: fixes y :: "real"   and x :: "real" shows "y \<le> sqrt (x\<^sup>2 + y\<^sup>2)"
in_arcs_imp_in_arcs_ends: fixes e :: "'b" assumes "e \<in> arcs G" shows "tail G e \<rightarrow> head G e"
PartialMeetContraction_axioms: shows "PartialMeetContraction Cn \<gamma>"
sum_of_squares: fixes n :: "nat" shows "real (sum power2 {..n}) = (2 * real n ^ 3 + 3 * (real n)\<^sup>2 + real n) / 6"
pderiv_add: fixes p :: "'a poly"   and q :: "'a poly" shows "pderiv (p + q) = pderiv p + pderiv q"
prems: fixes D' :: "char list" assumes "P \<turnstile> C1 \<preceq>\<^sup>* D'" shows "D' \<noteq> C"
fBallI: fixes A :: "'b fset"   and P :: "'b \<Rightarrow> bool" assumes "\<And>x. x |\<in>| A \<Longrightarrow> P x" shows "fBall A P"
W3b: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a r\<rightarrow> b) r\<rightarrow> (b r\<rightarrow> c) l\<rightarrow> a r\<rightarrow> c = (1::'a)"
assms_3: shows "(a, w, c) \<notin> edges H"
p'_2: fixes x :: "real" shows "(p has_vector_derivative p' x) (at x)"
neq_bi_alpha1ialpha4mi: fixes q :: "'b"   and i :: "'b" assumes "(q, i) \<in> b\<inverse>"   and "(q, i) \<in> e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>" shows "False"
intro: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "irreflexive A less"   and "transitive A less" shows "strict_ordered_set A less"
mapl_F_cong: fixes x :: "('p, 'q, 'r, 's, 't, 'u, 'v, 'w, 'x, 'y) F"   and l1 :: "'p \<Rightarrow> 'z"   and l1' :: "'p \<Rightarrow> 'z"   and l2 :: "'q \<Rightarrow> 'aa"   and l2' :: "'q \<Rightarrow> 'aa"   and l3 :: "'r \<Rightarrow> 'ab"   and l3' :: "'r \<Rightarrow> 'ab" assumes "\<And>z. z \<in> set1_F x \<Longrightarrow> l1 z = l1' z"   and "\<And>z. z \<in> set2_F x \<Longrightarrow> l2 z = l2' z"   and "\<And>z. z \<in> set3_F x \<Longrightarrow> l3 z = l3' z" shows "mapl_F l1 l2 l3 x = mapl_F l1' l2' l3' x"
elims: fixes x :: "('a \<times> 'a) set"   and xa :: "('a \<times> 'a) set"   and xb :: "('a \<times> 'a) set"   and xc :: "('a \<times> 'a) set"   and xd :: "SN_rel_ext_type"   and y :: "('a \<times> 'a) set" assumes "SN_rel_ext_step x xa xb xc xd = y" obtains P :: "('a \<times> 'a) set" and Pw :: "('a \<times> 'a) set" and R :: "('a \<times> 'a) set" and Rw :: "('a \<times> 'a) set" where "x = P__" and "xa = Pw__" and "xb = R__" and "xc = Rw__" and "xd = top_s" and "y = P__"   | P :: "('a \<times> 'a) set" and Pw :: "('a \<times> 'a) set" and R :: "('a \<times> 'a) set" and Rw :: "('a \<times> 'a) set" where "x = Pa__" and "xa = Pwa__" and "xb = Ra__" and "xc = Rwa__" and "xd = top_ns" and "y = Pwa__"   | P :: "('a \<times> 'a) set" and Pw :: "('a \<times> 'a) set" and R :: "('a \<times> 'a) set" and Rw :: "('a \<times> 'a) set" where "x = Pb__" and "xa = Pwb__" and "xb = Rb__" and "xc = Rwb__" and "xd = normal_s" and "y = Rb__"   | P :: "('a \<times> 'a) set" and Pw :: "('a \<times> 'a) set" and R :: "('a \<times> 'a) set" and Rw :: "('a \<times> 'a) set" where "x = P" and "xa = Pw" and "xb = R" and "xc = Rw" and "xd = normal_ns" and "y = Rw"
lt_right_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Lt Bet Cong A B C D" shows "Tarski_neutral_dimensionless.Lt Bet Cong A B D C"
from_mregex_eq: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list" assumes "safe_regex m g r"   and "to_mregex r = (mr, \<phi>s)" shows "from_mregex mr \<phi>s = r"
arrI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "Cop.in_hom f a b" shows "Cop.arr f"
RV_def: shows "RV = {(V', V''). (V', V'') \<in> RV' \<and> set (PPV V') \<subseteq> set (PPV V) \<and> set (PPV V'') \<subseteq> set (PPV V)}"
kstar_inductl: fixes h :: "'a \<Rightarrow> 'b set"   and f :: "'a \<Rightarrow> 'a set"   and g :: "'a \<Rightarrow> 'b set" assumes "h \<squnion> f \<circ>\<^sub>K g \<le> g" shows "kstar f \<circ>\<^sub>K h \<le> g"
commute: fixes a :: "'c"   and b :: "'c" shows "a \<sqinter> b = b \<sqinter> a"
crel_vs_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and v :: "'d"   and s :: "('b, 'e) state" shows "crel_vs R v s \<equiv> \<forall>M. cmem M \<and> P M \<longrightarrow> (case run_state s M of (v', M') \<Rightarrow> R v v' \<and> cmem M' \<and> P M')"
pred_def: fixes x :: "hf" shows "pred x \<equiv> THE y. succ y = x \<or> x = 0 \<and> y = 0"
Taylor_has_integral: fixes p :: "nat"   and Df :: "nat \<Rightarrow> real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real" assumes "0 < p"   and "Df 0 = f"   and "\<And>m t. \<lbrakk>m < p; a \<le> t; t \<le> b\<rbrakk> \<Longrightarrow> (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})"   and "a \<le> b" shows "((\<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x) has_integral f b - (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}"
dBox_test: fixes p :: "'b upred"   and \<Phi> :: "'b upred" shows "\<^bold>[?[p]\<^bold>]\<Phi> = (p \<Rightarrow> \<Phi>)"
transitive: fixes Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi"   and Rel :: "(pi \<times> pi) set"   and Rel'' :: "(pi \<times> pi) set"   and P :: "pi" assumes "Q \<leadsto><Rel'> R"   and "eqvt Rel"   and "eqvt Rel''"   and "Rel O Rel' \<subseteq> Rel''"   and "\<And>S T. (S, T) \<in> Rel \<Longrightarrow> S \<leadsto><Rel> T"   and "(P, Q) \<in> Rel" shows "P \<leadsto><Rel''> R"
lub_dual_glb: fixes S :: "'b set"   and cl :: "'b potype" shows "lub S cl = glb S (dual cl)"
C_def: shows "C = ffUnion (fmdom |`| constructors |`| fmran C_info)"
simps_1: fixes x :: "floatarith"   and a :: "floatarith"   and b :: "floatarith"   and f :: "form"   and vs :: "real list" shows "interpret_form (Bound x a b f) vs = (interpret_floatarith x vs \<in> {interpret_floatarith a vs..interpret_floatarith b vs} \<longrightarrow> interpret_form f vs)"
is_close_no_incr_boundvars: fixes t :: "Core.term"   and inc :: "nat" assumes "is_closed t" shows "incr_boundvars inc t = t"
union_mset_add_mset_right: fixes A :: "'b multiset"   and a :: "'b"   and B :: "'b multiset" shows "A + add_mset a B = add_mset a (A + B)"
filter_tab_dom_subset: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd option" shows "dom (filter_tab C t) \<subseteq> dom t"
antisym: fixes b :: "'c"   and a :: "'c" assumes "le_of_comp compare b a"   and "le_of_comp compare a b" shows "a = b"
suc_least: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b, a) \<in> r" shows "(suc B, a) \<in> r"
cinner_ket_same: fixes i :: "'c" shows "ket i \<bullet>\<^sub>C ket i = 1"
inf_Inf: 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 bot :: "'c"   and top :: "'c"   and A :: "'c set"   and x :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "inf x (Inf A) = Inf (inf x ` A)"
cod_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "H.VVV.cod f = (if H.VVV.arr f then H.VxVxV.cod f else H.VxVxV.null)"
columnvector_def: fixes v :: "('a, 'c) vec" shows "columnvector v = (\<chi>i j. v $ i)"
parse_text_def: shows "parse_text = many ((\<noteq>) CHR ''<'') \<bind> (\<lambda>ts. let text = trim ts in if text = [] then Parser_Monad.return None else Parser_Monad.return (Some (rev (trim (rev text)))))"
int_subsets: fixes A1 :: "'c set"   and A :: "'c set"   and B1 :: "'c set"   and B :: "'c set" assumes "A1 \<subseteq> A"   and "B1 \<subseteq> B" shows "A1 \<inter> B1 \<subseteq> A \<inter> B"
product_run_project_snd: fixes \<delta>\<^sub>1 :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and \<delta>\<^sub>2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>1 :: "'c"   and q\<^sub>2 :: "'a"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "project_transition_snd (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w i) = run\<^sub>t \<delta>\<^sub>2 q\<^sub>2 w i"
trm_prm_unit_inaction: fixes a :: "'b"   and X :: "'b trm"   and b :: "'b" assumes "a \<notin> fvs X"   and "b \<notin> fvs X" shows "[a \<leftrightarrow> b] \<cdot> X = X"
fps_div_fps_X_nth: fixes f :: "'a fps"   and k :: "nat" shows "fps_nth (f / fps_X) k = fps_nth f (Suc k)"
leg1_in_hom_1: shows "\<guillemotleft>g : src\<^sub>C f \<rightarrow>\<^sub>C trg\<^sub>C r\<guillemotright>"
disc_21: fixes x3 :: "'b formula" shows "is_Neg (Neg x3)"
valid_eqvt: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and x :: "('d, 'b, 'c) formula"   and p :: "perm" assumes "nominal_ts satisfies transition"   and "nominal_ts.valid satisfies transition P x" shows "nominal_ts.valid satisfies transition (p \<bullet> P) (p \<bullet> x)"
scc_of_empty_conv: fixes u :: "'a" shows "(scc_of u = {}) = (u \<notin> verts G)"
w1: shows "w1 \<in> set (awalk_verts u p)"
v': shows "vars_term_ms (SCF (Fun f (ss1 @ t # ss2))) \<subseteq># vars_term_ms (SCF (Fun f (ss1 @ s # ss2)))"
assms_1: shows "OAlist_tc_lex_ord f xs ys = aux"
safe_assignment_def: fixes X :: "nat set"   and \<phi> :: "Formula.formula" shows "safe_assignment X \<phi> = (case \<phi> of formula.Eq (trm.Var x) (trm.Var y) \<Rightarrow> (x \<notin> X) = (y \<in> X) | formula.Eq (trm.Var x) (trm.Const event_data) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Const event_data) \<subseteq> X | formula.Eq (trm.Var x) (trm.Plus trm1 trm2) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Plus trm1 trm2) \<subseteq> X | formula.Eq (trm.Var x) (trm.Minus trm1 trm2) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Minus trm1 trm2) \<subseteq> X | formula.Eq (trm.Var x) (trm.UMinus trm) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.UMinus trm) \<subseteq> X | formula.Eq (trm.Var x) (trm.Mult trm1 trm2) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Mult trm1 trm2) \<subseteq> X | formula.Eq (trm.Var x) (trm.Div trm1 trm2) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Div trm1 trm2) \<subseteq> X | formula.Eq (trm.Var x) (trm.Mod trm1 trm2) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Mod trm1 trm2) \<subseteq> X | formula.Eq (trm.Var x) (trm.F2i trm) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.F2i trm) \<subseteq> X | formula.Eq (trm.Var x) (trm.I2f trm) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.I2f trm) \<subseteq> X | formula.Eq (trm.Const event_data) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Const event_data) \<subseteq> X | formula.Eq (trm.Const event_data) _ \<Rightarrow> False | formula.Eq (trm.Plus trm1 trm2) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Plus trm1 trm2) \<subseteq> X | formula.Eq (trm.Plus trm1 trm2) _ \<Rightarrow> False | formula.Eq (trm.Minus trm1 trm2) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Minus trm1 trm2) \<subseteq> X | formula.Eq (trm.Minus trm1 trm2) _ \<Rightarrow> False | formula.Eq (trm.UMinus trm) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.UMinus trm) \<subseteq> X | formula.Eq (trm.UMinus trm) _ \<Rightarrow> False | formula.Eq (trm.Mult trm1 trm2) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Mult trm1 trm2) \<subseteq> X | formula.Eq (trm.Mult trm1 trm2) _ \<Rightarrow> False | formula.Eq (trm.Div trm1 trm2) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Div trm1 trm2) \<subseteq> X | formula.Eq (trm.Div trm1 trm2) _ \<Rightarrow> False | formula.Eq (trm.Mod trm1 trm2) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.Mod trm1 trm2) \<subseteq> X | formula.Eq (trm.Mod trm1 trm2) _ \<Rightarrow> False | formula.Eq (trm.F2i trm) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.F2i trm) \<subseteq> X | formula.Eq (trm.F2i trm) _ \<Rightarrow> False | formula.Eq (trm.I2f trm) (trm.Var x) \<Rightarrow> x \<notin> X \<and> fv_trm (trm.I2f trm) \<subseteq> X | formula.Eq (trm.I2f trm) _ \<Rightarrow> False | _ \<Rightarrow> False)"
op_dghm_dghm_id: fixes \<CC> :: "V" shows "op_dghm (dghm_id \<CC>) = dghm_id (op_dg \<CC>)"
that: fixes l' :: "'action list"   and w' :: "'action list"   and w\<^sub>1' :: "'action list"   and w\<^sub>2' :: "'action list"   and u' :: "'action stream" assumes "reduced_run q (v\<^sub>1 @ [a]) v\<^sub>2 l' (w @ w') (w\<^sub>1 @ w\<^sub>1') (w\<^sub>2 @ w\<^sub>2') u'" shows "thesis"
disc_15: fixes x51 :: "nat"   and x52 :: "'d"   and x53 :: "'d" shows "\<not> isRecv (RecvMarker x51 x52 x53)"
paths_le_mono: fixes k :: "nat" shows "mono (paths_le k)"
rel_gpv''_colossless_gpvD2: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'k) gpv"   and \<I>' :: "('i, 'k) \<I>"   and \<I> :: "('h, 'j) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "colossless_gpv \<I>' gpv'"   and "rel_\<I> C R \<I> \<I>'" shows "colossless_gpv \<I> gpv"
less_not_ipdk: fixes j :: "nat" assumes "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"   and "j < (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k))" shows "ipd (\<pi> j) \<noteq> ipd (\<pi> k)"
\<Z>_Limit_\<alpha>\<omega>: shows "Limit (\<alpha> + \<omega>)"
sumPairsInverse: fixes P :: "('d \<times> 'c) set"   and f :: "'c \<Rightarrow> 'e" assumes "runiq (P\<inverse>)" shows "sum (f \<circ> snd) P = sum f (Range P)"
prod_eq_def: fixes eqa :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and eqb :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x1 :: "'e \<times> 'g"   and x2 :: "'f \<times> 'h" shows "prod_eq eqa eqb x1 x2 \<equiv> case x1 of (a1, b1) \<Rightarrow> case x2 of (a2, b2) \<Rightarrow> eqa a1 a2 \<and> eqb b1 b2"
similar_term1: fixes A :: "'b set"   and a :: "'b"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "'a" assumes "ideal R A"   and "a \<in> A"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> carrier M"   and "m \<in> f ` {j. j \<le> n}" shows "\<exists>t\<in>{j. j \<le> n} \<rightarrow> A. \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s f j) n \<plusminus> a \<cdot>\<^sub>s m = \<Sigma>\<^sub>e M (\<lambda>j. t j \<cdot>\<^sub>s f j) n"
add_commute: fixes a :: "'a list set"   and b :: "'a list set" shows "a \<union> b = b \<union> a"
\<beta>: shows "\<beta> < \<omega> \<up> \<delta>"
d: shows "(if k = 0 then ([], p) else if root_unity k dvd p then map_prod ((#) k) id (decompose_prod_root_unity_main (p div root_unity k) k) else decompose_prod_root_unity_main p (k - 1)) = (ks', g)"
qmpt_power: fixes n :: "nat" shows "qmpt M (T ^^ n)"
the1_equality: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "\<exists>!x. P x"   and "P a" shows "(THE x. P x) = a"
fix_def: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set" shows "fix f A \<equiv> {x \<in> A. f x = x}"
purge_Nil: fixes G :: "'U set" shows "purge G [] = []"
even_odd_interpolate_compose: fixes f :: "'d \<Rightarrow> 'c"   and u :: "nat \<Rightarrow> 'd"   and v :: "nat \<Rightarrow> 'd" shows "even_odd_interpolate (f \<circ> u) (f \<circ> v) = f \<circ> even_odd_interpolate u v"
error_free_raise_if1: fixes a :: "abopt"   and s :: "st"   and p :: "bool"   and x :: "xname" assumes "error_free (a, s)" shows "error_free (raise_if p x a, s)"
hoare3o_valid_alt: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<Turnstile>\<^sub>3\<^sub>' {P} c { Q}" shows "\<exists>k>0. \<forall>ps n. P (ps, n div k) \<longrightarrow> (\<exists>ps' ps'' m e e'. (c, ps) \<Rightarrow>\<^sub>A m \<Down> ps' + ps'' \<and> ps' ## ps'' \<and> n = e + e' + m \<and> Q (ps', e div k))"