hcju/isabelleps huggingface dataset:
PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1_dir1: fixes ps :: "nat list" assumes "0 < length ps" shows "PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m 1 ps \<sqsubseteq> LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m (ps ! 0)"
tfr_ineq: shows "\<forall>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<longrightarrow> Q1 F X \<or> Q2 F X"
ind: assumes "is_sublist as ars" shows "is_sublist [x, y] bs"
abs_summable_on_comparison_test': fixes g :: "'c \<Rightarrow> real"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "g abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> g x" shows "f abs_summable_on A"
dom_lunit: fixes a :: "'c local.term set" assumes "ide a" shows "local.dom \<l>[a] = \<I> \<otimes> a"
construct_lhs_vector_clean_R: fixes p :: "real poly"   and i :: "nat"   and signs :: "rat list list"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "i < length signs" shows "of_rat (construct_lhs_vector_R p qs signs $ i) = of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec qs x = signs ! i})"
line_S_match_intersections: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2" assumes "p \<noteq> q"   and "r \<noteq> s"   and "p \<in> S"   and "q \<in> S"   and "r \<in> S"   and "s \<in> S"   and "proj2_set_Col {p, q, r, s}" shows "p = r \<and> q = s \<or> q = r \<and> p = s"
a: shows "UNIV = {PeerA, PeerB, PeerC}"
arr_cod: fixes f :: "'b" assumes "C'.arr f" shows "C'.arr (C'.cod f)"
nsets_mono: fixes A :: "'b set"   and B :: "'b set"   and n :: "nat" assumes "A \<subseteq> B" shows "[A]\<^bsup>n\<^esup> \<subseteq> [B]\<^bsup>n\<^esup>"
dvdg'_1: shows "gcd (a div gcd a b) (b div gcd a b) dvd a div gcd a b"
one_plus_x_powr_Taylor1_bigo: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and p :: "real" assumes "(f \<longlongrightarrow> 0) F" shows "(\<lambda>x. (1 + f x) powr p - 1) \<in> O[F](f)"
mono_set_mult: fixes H :: "'c set"   and H' :: "'c set"   and K :: "'c set"   and K' :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<subseteq> H'"   and "K \<subseteq> K'" shows "H <#>\<^bsub>G\<^esub> K \<subseteq> H' <#>\<^bsub>G\<^esub> K'"
map: shows "(y + x * xa) mod q = (y + x * ya) mod q"
entries_rbt_lookup: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t1 :: "('c, 'd) rbt"   and t2 :: "('c, 'd) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t1"   and "ord.rbt_sorted less t2" shows "(RBT_Impl.entries t1 = RBT_Impl.entries t2) = (ord.rbt_lookup less t1 = ord.rbt_lookup less t2)"
encode_operators_valid: fixes sv :: "sat_plan_variable"   and t :: "nat" assumes "sv \<in> atoms (encode_operators (\<phi> prob_with_noop abs_prob ) t)"   and "t < Suc h" shows "valid_state_var sv"
after_summary_union: fixes M :: "'t zmultiset"   and N :: "'t zmultiset"   and S :: "'sum antichain" shows "after_summary (M + N) S = after_summary M S + after_summary N S"
sim_val_def: fixes M :: "('b, 'c, 'd) KripkeStructure"   and M' :: "('b, 'c, 'e) KripkeStructure"   and f :: "'d \<Rightarrow> 'e" shows "sim_val M M' f \<equiv> \<forall>u\<in>worlds M. valuation M u = valuation M' (f u)"
absorb1: fixes a :: "'a"   and b :: "'a" assumes "a \<sqsubseteq> b" shows "ord.min (\<sqsubseteq>) a b = a"
jpTracesp_jpTraces_eq: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" shows "jpTracesp jp = (\<lambda>x. x \<in> jpTraces jp)"
Quadratic_Reciprocity_int: fixes p :: "int"   and q :: "int" assumes "prime (nat p)"   and "2 < p"   and "prime (nat q)"   and "2 < q"   and "p \<noteq> q" shows "Legendre p q * Legendre q p = (- 1) ^ nat ((p - 1) div 2 * ((q - 1) div 2))"
connectedin_connected_component_of: fixes X :: "'c topology"   and x :: "'c" shows "connectedin X (connected_component_of_set X x)"
forall_coeffs_conv: fixes P :: "'b \<Rightarrow> bool"   and p :: "'b poly" assumes "P (0::'b)" shows "(\<forall>n. P (coeff p n)) = (\<forall>c\<in>set (coeffs p). P c)"
mrbst_inter_diff_correct: fixes A :: "'a set"   and B :: "'a set"   and b :: "bool" assumes "finite A"   and "finite B" shows "random_bst A \<bind> (\<lambda>t1. random_bst B \<bind> mrbst_inter_diff b t1) = random_bst ((if b then (\<inter>) else (-)) A B)"
effect_to_assignments_iii_a: fixes v :: "'c"   and op :: "'c strips_operator"   and as :: "('c \<times> bool) list" assumes "v \<in> set (add_effects_of op)"   and "as = effect_to_assignments op" obtains a :: "'c \<times> bool" where "a \<in> set as" and "a = (v, True)"
HSem_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "HSem \<Gamma> = (\<Lambda> \<rho>. \<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<N>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>x\<^esub>)"
simps_107: fixes x111 :: "char list"   and x112 :: "ty"   and x113 :: "expr"   and x4 :: "val" shows "{x111:x112; x113} \<noteq> Val x4"
zero_minus_ennreal: fixes a :: "ennreal" shows "0 - a = 0"
steps: shows "P,E \<turnstile> \<langle>es'',s''\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>"
fg_2: shows "H = {f \<turnstile> \<C>, g \<turnstile> \<C>}"
assms_1: shows "u \<le>p r \<^sup>@ k"
758: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> x \<sqinter> (- y \<sqinter> - z) = x"
f_Exec_Stream_not_empty_conv: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list"   and c :: "'a" shows "(f_Exec_Comp_Stream trans_fun xs c \<noteq> []) = (xs \<noteq> [])"
b_e_type_return: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Return" shows "\<exists>ts_c ts''. ts = ts_c @ ts'' \<and> return \<C> = Some ts''"
bexI: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and A :: "'b set" assumes "P x"   and "x \<in> A" shows "\<exists>x\<in>A. P x"
simps_1: fixes p :: "'a" shows "Dioid_Models_New.last (p, []) = p"
Tietze: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and U :: "'a set"   and B :: "real" assumes "continuous_on S f"   and "closedin (top_of_set U) S"   and "0 \<le> B"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> B" obtains g :: "'a \<Rightarrow> 'b" where "continuous_on U g" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" and "\<And>x. x \<in> U \<Longrightarrow> norm (g x) \<le> B"
set_times_mono4: fixes a :: "'c"   and C :: "'c set"   and D :: "'c set" assumes "a \<in> C" shows "a *o D \<subseteq> D * C"
pivot_positions_1: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat nr nc"   and "pivot_fun A f nc" shows "set (pivot_positions A) = {(i, f i) |i. i < nr \<and> f i \<noteq> nc}"
word_sub_le: fixes y :: "'b word"   and x :: "'b word" assumes "y \<le> x" shows "x - y \<le> x"
kill_short_large_girth: fixes G :: "nat set \<times> nat set set"   and k :: "nat" assumes "finite (uverts G)" shows "enat k < girth (kill_short G k)"
Q_neg_distr: fixes x :: "'a H"   and y :: "'c H"   and L :: "'a \<Rightarrow> 'a \<times> 'c \<Rightarrow> bool"   and L' :: "'a \<times> 'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "Q x"   and "Q y"   and "rel_H (L OO L') x y" shows "(rel_H L OO eq_onp Q OO rel_H L') x y"
weak_ranking_aux_unique': fixes As :: "'a set list" assumes "\<Union> (set As) \<subseteq> carrier"   and "is_weak_ranking As"   and "of_weak_ranking As = restrict_relation (\<Union> (set As)) le" shows "As = weak_ranking_aux (\<Union> (set As))"
finite_dimensionE': fixes K :: "'c set"   and E :: "'c set" assumes "S.finite_dimension K E" obtains n :: "nat" where "S.dimension n K E"
cindexE_roots_on_horizontal_border: fixes p :: "complex poly"   and q :: "complex poly"   and r :: "complex poly"   and a :: "complex"   and lb :: "real"   and ub :: "real"   and s :: "real" assumes "p = q * r"   and "lead_coeff r = 1"   and "\<forall>x\<in>proots r. Im x = Im a" shows "cindexE lb ub (\<lambda>t. Im ((poly p \<circ> linepath a (a + complex_of_real s)) t) / Re ((poly p \<circ> linepath a (a + complex_of_real s)) t)) = cindexE lb ub (\<lambda>t. Im ((poly q \<circ> linepath a (a + complex_of_real s)) t) / Re ((poly q \<circ> linepath a (a + complex_of_real s)) t))"
disjComm: fixes a :: "('b set set, 'c) Combinators"   and b :: "('b set set, 'd) Combinators" assumes "disjSD_2 a b" shows "disjSD_2 b a"
b_a_s1: shows "b - a = (b div m - a div m) * m + b mod m - a mod m"
cases: fixes x :: "'a list" obtains xa :: "'a" and xs :: "'a list" where "x = xa \<Colon> xs"   | "x = []"
int_nat_add: fixes n :: "int"   and m :: "int" assumes "0 < n"   and "0 < m" shows "nat (n - 1) + nat (m - 1) + Suc 0 = nat (n + m - 1)"
hcomp_emp: fixes P :: "interface" shows "(P \<otimes> \<epsilon>) = P"
Gcd_singleton: 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 Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<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" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Gcd {a} = normalize a"
pole_theorem_open: fixes g :: "complex \<Rightarrow> complex"   and S :: "complex set"   and a :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "g holomorphic_on S"   and "open S"   and "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" shows "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) holomorphic_on S"
preserves_reflects_arr: fixes f :: "'d \<times> 'c" shows "S.arr (Hom_DopxG.map f) = DopxC.arr f"
Ugp_def: fixes G :: "('e, 'f) Group_scheme" shows "Ugp G \<equiv> Group G \<and> carrier G = {\<one>\<^bsub>G\<^esub>}"
assigns_expr_simp1: fixes e :: "expr" shows "assigns \<langle>e\<rangle> = assignsE e"
plus_\<omega>_equals_\<omega>: fixes \<alpha> :: "V" assumes "\<alpha> \<in> elts \<omega>" shows "\<alpha> + \<omega> = \<omega>"
p_leI: fixes p' :: "real" assumes "(\<Sum>i<k. as ! i * bs ! i powr p') \<le> 1" shows "p \<le> p'"
integrable_join: fixes f :: "'c \<Rightarrow> real"   and N :: "'c measure"   and B :: "real"   and M :: "'c measure measure"   and B' :: "real" assumes "f \<in> borel_measurable N"   and "\<And>x. x \<in> space N \<Longrightarrow> \<bar>f x\<bar> \<le> B"   and "sets M = sets (subprob_algebra N)"   and "finite_measure M"   and "AE M' in M. emeasure M' (space M') \<le> ennreal B'" shows "integrable (join M) f"
map_add_restricted_self: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "m ++ m |` S = m"
ndist_C: fixes v :: "nat" assumes "v \<in> C" shows "ndist v = d"
Num_eq_subst_Var: fixes x :: "float"   and s :: "nat \<Rightarrow> nat"   and fa :: "floatarith" shows "(Num x = subst_floatarith (\<lambda>x. Var (s x)) fa) = (fa = Num x)"
Un_Diff_Int: fixes A :: "'a set"   and B :: "'a set" shows "A - B \<union> A \<inter> B = A"
circ_increasing: fixes x :: "'a" shows "x \<le> x\<^sup>\<star>"
i_drop_nth: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and x :: "nat" shows "(s \<Up> n) x = s (n + x)"
F_def: shows "(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. x ; (d::'a::refinement_lattice))"
f19: shows "t' \<otimes> t \<otimes> (snd x \<otimes> snd x' \<otimes> (r \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (fst x \<otimes> fst x')) = t' \<otimes> t \<otimes> (snd x \<otimes> snd x') \<otimes> (r \<otimes> r') \<ominus> t' \<otimes> t \<otimes> (s \<otimes> s' \<otimes> (fst x \<otimes> fst x'))"
in_sorted_domain_raw_iff_lookup_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and k :: "'a"   and ko :: "'o" assumes "oalist_inv xs" shows "(k \<in> set (sorted_domain_raw ko xs)) = (lookup_raw xs k \<noteq> (0::'c))"
adjacent_def: fixes v :: "nat" shows "cf.adjacent v \<equiv> cf.incoming v \<union> cf.outgoing v"
quad_part_2_eq: fixes var :: "nat"   and L :: "real list"   and SQ :: "real"   and p :: "real mpoly"   and deg :: "nat"   and sq :: "real mpoly" assumes "var < length L"   and "0 \<le> SQ"   and "MPoly_Type.degree p var = deg"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) sq = SQ" shows "aEval (Eq p) (L[var := sqrt SQ]) = aEval (Eq (quadratic_part_2 var sq p)) (L[var := sqrt SQ])"
diff_closed: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and g :: "'c"   and h :: "'c" assumes "FGModule G smult V"   and "g \<in> V"   and "h \<in> V" shows "g - h \<in> V"
setdist_le_dist: fixes x :: "'b"   and s :: "'b set"   and y :: "'b"   and t :: "'b set" assumes "x \<in> s"   and "y \<in> t" shows "setdist s t \<le> dist x y"
knows_composition1_k: fixes m :: "Keys"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "kKS m \<notin> specKeysSecrets P"   and "kKS m \<notin> specKeysSecrets Q"   and "knows P [kE m]"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ" shows "knows PQ [kE m]"
d\<^sub>x': shows "profile_single d\<^sub>x"
P_equiv: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and u :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and t' :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and u' :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "P t u"   and "equiv_trans t t'"   and "equiv_utrans u u'" shows "P t' u'"
abc_append_frist_steps_halt_eq: fixes lm :: "nat list"   and A :: "abc_inst list"   and n :: "nat"   and lm' :: "nat list"   and B :: "abc_inst list" assumes "abc_steps_l (0, lm) A n = (length A, lm')" shows "\<exists>n'. abc_steps_l (0, lm) (A @ B) n' = (length A, lm')"
inv_closed: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "\<ominus> x \<in> carrier Zp"
string_of_word_single_atoi: fixes n :: "'a word" assumes "n < 10" shows "string_of_word_single True n = [char_of (48 + unat n)]"
complement_conv_sub: fixes y :: "'a"   and x :: "'a" shows "- (y \<sqinter> x) \<sqinter> id x \<le> - y"
simps_3: fixes hosts :: "'b set"   and flows_fix :: "('b \<times> 'b) set"   and flows_state :: "('b \<times> 'b) set"   and more :: "'c" shows "flows_state \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = flows_state"
fetch_t_wcode_14_Bk: shows "fetch t_wcode_main (14 + length t_twice div 2 + t_fourtimes_len) Bk = (L, Suc 0)"
unique_fun_value: fixes f :: "ZF"   and x :: "ZF" assumes "isFun f"   and "Elem x (HOLZF.Domain f)" shows "\<exists>!y. Elem (Opair x y) f"
oexp_0_right: fixes \<alpha> :: "V" shows "\<alpha> \<up> 0 = 1"
eval_fm_PfP: fixes e :: "(name, hf) finfun"   and y :: "tm" shows "eval_fm e (PfP y) = Pf \<lbrakk>y\<rbrakk>e"
Prim_Invar2_ctd_loc_axioms: shows "Prim_Invar2_ctd_loc w g r (Q' Q \<pi> u) (\<pi>' Q \<pi> u)"
one_matrix_mult_left: fixes A :: "'b matrix"   and n :: "nat" assumes "nrows A \<le> n" shows "one_matrix n * A = A"
hcomp_def: fixes g :: "'c"   and f :: "'c" shows "C'.hcomp g f = (if C'.arr f \<and> C'.arr g \<and> trg\<^sub>C f = src\<^sub>C g then g \<star>\<^sub>C f else C'.null)"
sort_append_Cons_swap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and x :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq (\<lambda>x. x) (ys @ x # xs) = linorder.sort_key less_eq (\<lambda>x. x) (ys @ xs @ [x])"
full_rank_imp_is_basis2: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "rank A = ncols A" shows "vec.independent (columns A) \<and> vec.span (columns A) = col_space A \<and> card (columns A) = ncols A"
lm022: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> partitionValuedUniverse"
lsc_at_open: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "lsc_at x0 f = (\<forall>S. open S \<and> f x0 \<in> S \<longrightarrow> (\<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>x'\<in>T. f x' \<le> f x0 \<longrightarrow> f x' \<in> S)))"
finite: shows "integral\<^sup>N (count_space B) g \<noteq> \<top>"
is_type_ArrayD: fixes P :: "'b prog"   and T :: "ty" assumes "is_type P (T\<lfloor>\<rceil>)" shows "is_type P T"
sublinear_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "sublinear t = (\<forall>a b c P Q s. sound P \<and> sound Q \<and> 0 \<le> a \<and> 0 \<le> b \<and> 0 \<le> c \<longrightarrow> a * t P s + b * t Q s \<ominus> c \<le> t (\<lambda>s'. a * P s' + b * Q s' \<ominus> c) s)"
fib_def: shows "fib \<equiv> fix\<cdot>fib_body"
\<R>_union: shows "\<Union> \<R>\<^sub>\<beta> = V"
lfinite_llength_enat: fixes xs :: "'c llist" assumes "lfinite xs" shows "\<exists>n. llength xs = enat n"
LI_completeness_single: fixes S\<^sub>1 :: "('fun, 'var) strand_step list"   and \<theta>\<^sub>1 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"   and "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>1,\<theta>\<^sub>1\<rangle>"   and "\<not> simple S\<^sub>1" shows "\<exists>S\<^sub>2 \<theta>\<^sub>2. (S\<^sub>1, \<theta>\<^sub>1) \<leadsto> (S\<^sub>2, \<theta>\<^sub>2) \<and> \<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>2,\<theta>\<^sub>2\<rangle>"
IH_1: fixes x :: "perm" shows "(x \<bullet> t1 =\<^sub>\<alpha> t2) = (p \<bullet> x \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2)"
vbrelation_vsingleton: fixes a :: "V"   and b :: "V" shows "vbrelation (set {\<langle>a, b\<rangle>})"
case_3: fixes f1 :: "'a"   and f2 :: "bool \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "real \<Rightarrow> 'a"   and f5 :: "val \<Rightarrow> val \<Rightarrow> 'a"   and x3 :: "int" shows "(case IntVal x3 of UnitVal \<Rightarrow> f1 | BoolVal x \<Rightarrow> f2 x | IntVal x \<Rightarrow> f3 x | RealVal x \<Rightarrow> f4 x | <|x, xa|> \<Rightarrow> f5 x xa) = f3 x3"
r_init: fixes s :: "'e"   and T :: "('e, 'f) TS_scheme" assumes "s \<in> init T" shows "s \<in> reach T"
rtrancl_into_rtrancl: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>*\<^sup>* a c"
inline_map_gpv: fixes callee :: "'i \<Rightarrow> 'l \<Rightarrow> ('m \<times> 'i, 'j, 'k) gpv"   and f :: "'n \<Rightarrow> 'h"   and g :: "'o \<Rightarrow> 'l"   and gpv :: "('n, 'o, 'm) gpv"   and s :: "'i" shows "inline callee (map_gpv f g gpv) s = map_gpv (apfst f) id (inline (\<lambda>s x. callee s (g x)) gpv s)"
IH_1: assumes "\<And>i t e. \<lbrakk>lowest_tops [i, t, e] = None; i = Trueif\<rbrakk> \<Longrightarrow> P i t e"   and "\<And>i t e. \<lbrakk>lowest_tops [i, t, e] = None; i = Falseif\<rbrakk> \<Longrightarrow> P i t e"   and "\<And>i t e a. \<lbrakk>\<And>val. P (restrict_top i a val) (restrict_top t a val) (restrict_top e a val); lowest_tops [i, t, e] = Some a\<rbrakk> \<Longrightarrow> P i t e" shows "P (restrict_top i a True) (restrict_top t a True) (restrict_top e a True)"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and strand_step :: "('d, 'e) strand_step" shows "P (case strand_step of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = ((\<forall>x1. strand_step = send\<langle>x1\<rangle>\<^sub>s\<^sub>t \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. strand_step = receive\<langle>x2\<rangle>\<^sub>s\<^sub>t \<longrightarrow> P (f2 x2)) \<and> (\<forall>x31 x32 x33. strand_step = \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t \<longrightarrow> P (f3 x31 x32 x33)) \<and> (\<forall>x41 x42. strand_step = \<forall>x41\<langle>\<or>\<noteq>: x42\<rangle>\<^sub>s\<^sub>t \<longrightarrow> P (f4 x41 x42)))"
of_rat_mult: fixes a :: "rat"   and b :: "rat" shows "of_rat (a * b) = of_rat a * of_rat b"
Inf_comp: fixes X :: "'a set"   and z :: "'a" shows "\<Sqinter>X * z = (\<Sqinter>x\<in>X. x * z)"
absorb1: fixes b :: "'t"   and a :: "'t" assumes "b \<preceq>\<^sub>t a" shows "ord_term_lin.max a b = a"
Diff_insert: fixes A :: "'e set"   and a :: "'e"   and B :: "'e set" shows "A - insert a B = A - B - {a}"
assms_1: shows "A \<Longrightarrow>\<^sub>A A' * true"
wf_execI: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" assumes "is_write_seen P E ws"   and "thread_start_actions_ok E" shows "P \<turnstile> (E, ws) \<surd>"
sup_set_def: fixes A :: "'e set"   and B :: "'e set" shows "A \<union> B = Collect (sup (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
less_eq_llist_def: fixes xs :: "'a llist"   and ys :: "'a llist" shows "(xs \<le> ys) = lprefix xs ys"
eval_cases_6: fixes P :: "(char list list \<times> char list exp) prog"   and V :: "char list"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>V:=e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains v :: "val" and h :: "heap" and l :: "sfields" and sh :: "sheap" where "e'a__ = unit" and "s' = (h, l(V \<mapsto> v), sh)" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>Val v,(h, l, sh)\<rangle>"   | e' :: "char list exp" where "e'a__ = throw e'" and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e',s'\<rangle>"
more_arith_simps_11: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * b * c = a * (b * c)"
uncurry_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "??.Misc.uncurry f \<equiv> \<lambda>(a, b). f a b"
smult_r_null: fixes a :: "'a" assumes "a \<in> carrier K" shows "a \<odot>\<^bsub>V\<^esub> \<zero>\<^bsub>V\<^esub> = \<zero>\<^bsub>V\<^esub>"
infl: shows "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"
FORWEAK_LIST_def: fixes Xi :: "'d list"   and di :: "'c nres"   and fi :: "'d \<Rightarrow> 'c nres"   and ci :: "'c \<Rightarrow> 'c \<Rightarrow> 'c nres" shows "FORWEAK_LIST Xi di fi ci \<equiv> if Xi = [] then di else fi (hd Xi) \<bind> nfoldli (tl Xi) (\<lambda>_. True) (\<lambda>x b. fi x \<bind> ci b)"
CTerm_def: fixes vf :: "bool"   and t :: "hf" shows "CTerm vf t \<equiv> \<exists>s k. SeqCTerm vf s k t"
discharge_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and u :: "nat" shows "discharge f l n u \<equiv> assert (u \<in> V - {s, t}) \<bind> (\<lambda>_. while\<^sub>T (\<lambda>(f, l, n). excess f u \<noteq> (0::'capacity)) (\<lambda>(f, l, n). (select v. v \<in> n u) \<bind> (\<lambda>v. case v of None \<Rightarrow> relabel f l u \<bind> (\<lambda>l. return (f, l, n(u := adjacent_nodes u))) | Some v \<Rightarrow> assert (v \<in> V \<and> (u, v) \<in> E \<union> E\<inverse>) \<bind> (\<lambda>_. if (u, v) \<in> cfE_of f \<and> l u = l v + 1 then push f l (u, v) \<bind> (\<lambda>f. return (f, l, n)) else assert ((u, v) \<notin> adm_edges f l) \<bind> (\<lambda>_. return (f, l, n(u := n u - {v})))))) (f, l, n))"