hcju/isabelleps huggingface dataset:
lex_comp: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and is_scalar :: "'a itself \<Rightarrow> bool"   and lex_comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order"   and pot_comp :: "('a \<Rightarrow> 'a \<Rightarrow> order) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.nat_term_compare rep_nat_term splus less_eq less is_scalar lex_comp deg_comp pot_comp" shows "lex_comp = nat_term.lex_comp_aux rep_nat_term"
angle_actrans_def: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<langle>P\<rangle>_v \<equiv> \<not> _actrans (\<not> P) v"
nonintS_def: shows "nonintS \<equiv> \<forall>sl sl'. wffp sl \<and> wffp sl' \<and> hd sl = s0 \<and> hd sl' = s0 \<and> list_all2 f sl sl' \<longrightarrow> g (last sl) (last sl')"
sum_intro: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "\<forall>w. O w z = (O w x \<or> O w y)" shows "x \<oplus> y = z"
a: shows "\<not> is_final (steps0 (1, l, r) ap stpa) \<and> steps0 (1, l, r) ap (Suc stpa) = (0, l', r')"
T_partition3_eq: fixes x :: "'a"   and xs :: "'a list" shows "T_partition3 x xs = length xs + 1"
simps: fixes x :: "'a"   and y :: "'a" shows "neq\<cdot>x\<cdot>y = neg\<cdot>(eq\<cdot>x\<cdot>y)"
evals_final: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and es :: "char list exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "char list exp list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "finals es'"
fmdrop_set_null: fixes m :: "('a, 'b) fmap" shows "fmdrop_set {} m = m"
init: shows "(R ja, R j) \<notin> init_seg_of"
Union_disjoint: fixes C :: "'c set set"   and A :: "'c set" shows "(\<Union> C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
Seq_stepsnD: fixes c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int"   and n :: "nat"   and c' :: "com"   and t :: "char list \<Rightarrow> int" assumes "(c1;; c2, s) \<rightarrow>(n) (c', t)" shows "(\<exists>c1' m. c' = c1';; c2 \<and> (c1, s) \<rightarrow>(m) (c1', t) \<and> m \<le> n) \<or> (\<exists>s2 m1 m2. (c1, s) \<rightarrow>(m1) (SKIP, s2) \<and> (c2, s2) \<rightarrow>(m2) (c', t) \<and> m1 + m2 < n)"
openin_open_Int': fixes S :: "'b set"   and U :: "'b set" assumes "open S" shows "openin (top_of_set U) (S \<inter> U)"
prv_cnj_com: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (eqv (cnj \<phi> \<chi>) (cnj \<chi> \<phi>))"
le_Derives1_take: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and i :: "nat"   and j :: "nat"   and a :: "symbol list"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "i \<le> j"   and "CFG.Derives1 \<NN> \<TT> \<RR> a j r b" shows "take i a = take i b"
fix_ind_k: fixes P :: "'a \<Rightarrow> bool"   and k :: "nat"   and f :: "'a \<rightarrow> 'a" assumes "adm P"   and "\<forall>i<k. P (iterate i\<cdot>f\<cdot>\<bottom>)"   and "\<And>x. \<forall>i<k. P (iterate i\<cdot>f\<cdot>x) \<Longrightarrow> P (iterate k\<cdot>f\<cdot>x)" shows "P (fix\<cdot>f)"
closedI: fixes T :: "'b set set"   and s :: "'b set" assumes "carrier\<^bsub>T\<^esub> - s open\<^bsub>T\<^esub>" shows "s closed\<^bsub>T\<^esub>"
norm_form_uniq: fixes a :: "(bool \<times> 'c) list"   and b :: "(bool \<times> 'c) list"   and c :: "(bool \<times> 'c) list" assumes "cancels_to a b"   and "cancels_to a c"   and "canceled b"   and "canceled c" shows "b = c"
mk_disjoint_insert: fixes a :: "'e"   and A :: "'e set" assumes "a \<in> A" shows "\<exists>B. A = insert a B \<and> a \<notin> B"
ringed_space_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" shows "ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
indep_F1: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs')"
index_transpose_mat_3: fixes A :: "'a mat" shows "dim_col A\<^sup>t = dim_row A"
d2_var: fixes p :: "'a"   and x :: "'a" assumes "test p" shows "- - (p \<sqinter> x) \<le> p"
l11_46: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "B \<noteq> C"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C \<or> Tarski_neutral_dimensionless.Obtuse Bet Cong A B C" shows "Tarski_neutral_dimensionless.Lt Bet Cong B A A C \<and> Tarski_neutral_dimensionless.Lt Bet Cong B C A C"
is_derivation_linear_on: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" assumes "is_derivation X p" shows "linear_on diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X"
enat_sum_le: fixes a :: "nat"   and b :: "nat"   and c :: "enat" assumes "enat (a + b) \<le> c" shows "enat b \<le> c"
tensor_tensor_2: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and h :: "'c local.term set" assumes "arr f"   and "arr g"   and "arr h" shows "f \<otimes>\<^sub>S g \<otimes>\<^sub>S h = \<F>C.mkarr (\<^bold>\<lfloor>\<F>C.rep f\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>\<F>C.rep g\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>\<F>C.rep h\<^bold>\<rfloor>)"
simps_25: fixes vc :: "'b"   and vd :: "bexp"   and ve :: "'b acom"   and vf :: "'b"   and v :: "char list"   and va :: "aexp"   and vb :: "'b" shows "({vc} WHILE vd DO ve {vf} \<sqsubseteq> v ::= va {vb}) = False"
mfinalE: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and x :: "'x"   and ln :: "'e \<Rightarrow>f nat" assumes "mfinal s"   and "thr s t = \<lfloor>(x, ln)\<rfloor>" obtains "final x" and "ln = no_wait_locks" and "wset s t = None"
arr_Par_comp_Par_id_Par_right: fixes \<alpha> :: "V"   and f :: "V"   and A :: "V" assumes "arr_Par \<alpha> f"   and "f\<lparr>ArrDom\<rparr> = A" shows "f \<circ>\<^sub>R\<^sub>e\<^sub>l id_Rel A = f"
sorted_sublist_def: fixes xs :: "'c list"   and lo :: "nat"   and hi :: "nat" shows "sorted_sublist xs lo hi = sorted_sublist_wrt (\<le>) xs lo hi"
rev_iterateoi_def: fixes S :: "'s" shows "reverse_iterateoi S \<equiv> it_to_it (rev_list_it S)"
Asm: fixes P :: "'d set"   and p :: "'e"   and Q :: "'d set"   and A :: "'d set"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and F :: "'f set" assumes "(P, p, Q, A) \<in> \<Theta>" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A"
content_ball_pos: fixes r :: "real"   and c :: "'b" assumes "0 < r" shows "0 < content (ball c r)"
gen_incl_Un: fixes P :: "'a set set"   and I :: "'a set" shows "gen P I \<subseteq> I \<union> \<Union> P"
trans': fixes si :: "('i, 'a) state"   and sj :: "('i, 'a) state"   and sk :: "('i, 'a) state" assumes "P si sj"   and "P sj sk" shows "P si sk"
assms_4: shows "\<forall>B\<in>P. W B \<le> real c"
square_eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a * a = b * b) = (a = b \<or> a = - b)"
expansion_factor_def: fixes p :: "proj2"   and J :: "cltn2" shows "expansion_factor p J \<equiv> (cart2_append1 p v* cltn2_rep J) $ 3"
rght_trpl_trd: fixes l :: "nat"   and st :: "nat"   and r :: "nat" shows "rght (trpl l st r) = r"
sets_restrict_restrict_space: fixes M :: "'c measure"   and A :: "'c set"   and B :: "'c set" shows "sets (restrict_space (restrict_space M A) B) = sets (restrict_space M (A \<inter> B))"
exeaxioms_of_def: fixes exetheory :: "exetheory" shows "exeaxioms_of exetheory = (case exetheory of ExeTheory x1 x2 \<Rightarrow> x2)"
in_set_conv_decomp_first: fixes x :: "'d"   and xs :: "'d list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
cases: fixes x :: "order" obtains "x = Lt"   | "x = Gt"   | "x = Eq"
euler_product_LIMSEQ: fixes f :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>n. norm (f n))"   and "multiplicative_function f" shows "(\<lambda>n. \<Prod>p\<le>n. if prime p then \<Sum>n. f (p ^ n) else (1::'a)) \<longlonglongrightarrow> (\<Sum>n. f n)"
cdf_is_right_cont: fixes a :: "real" shows "continuous (at_right a) (cdf M1)"
numer_cf3_pos: fixes x :: "real" assumes "- (464 / 10\<^sup>2) \<le> x" shows "0 < numer_cf3 x"
is_alignedI: fixes x :: "'b word"   and n :: "nat"   and k :: "'b word" assumes "x = 2 ^ n * k" shows "is_aligned x n"
normal_form_disjunct_with_flatten_pi_2_length_def: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set"   and N :: "'b ltln set" shows "normal_form_disjunct_with_flatten_pi_2_length \<phi> M N \<equiv> size \<phi>[N]\<^sub>\<Pi>\<^sub>2 + (\<Sum>\<psi>\<in>M. size (\<psi>[N]\<^sub>\<Sigma>\<^sub>1) + 2) + (\<Sum>\<psi>\<in>N. size (\<psi>[M]\<^sub>\<Pi>\<^sub>1) + 2)"
wset_find_path_def: fixes E :: "('b \<times> 'b) set"   and U0 :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "wset_find_path E U0 P \<equiv> ASSERT (finite U0) \<bind> (\<lambda>_. s_init U0 \<bind> (\<lambda>s0. WHILE\<^sub>T (\<lambda>(res, V, ws). res = None \<and> ws \<noteq> Map.empty) (\<lambda>(res, V, ws). ASSERT (ws \<noteq> Map.empty) \<bind> (\<lambda>_. SPEC (\<lambda>(u, p). ws u = Some p) \<bind> (\<lambda>(u, p). let ws = ws |` (- {u}) in if P u then RETURN (Some (rev p, u), V, ws) else ASSERT (finite (E `` {u})) \<bind> (\<lambda>_. ASSERT (dom ws \<subseteq> V) \<bind> (\<lambda>_. ws_update E u p V ws \<bind> (\<lambda>(V, ws). RETURN (None, V, ws))))))) s0 \<bind> (\<lambda>(res, uu_, uu_). RETURN res)))"
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'c \<times> 'd Filters.filter"   and x :: "('c, 'd) phi \<Rightarrow> 'c \<times> 'd Filters.filter" assumes "eq_onp (\<lambda>pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)) xa xa"   and "eq_onp (\<lambda>pf. \<forall>f. pf f \<in> triple.pairs (Rep_phi f)) x x" shows "(Abs_lifted_pair xa \<le> Abs_lifted_pair x) = (\<forall>f. triple.pairs_less_eq (xa f) (x f))"
new_element_no_shadow_root: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_shadow_root new_element_ptr \<rightarrow>\<^sub>r None"
gen_akiyama_tanigawa_0: fixes f :: "nat \<Rightarrow> real" shows "gen_akiyama_tanigawa f 0 = f"
length_trace_parallel_plan_strips_lt_length_plan_plus_one_then: fixes I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" assumes "length (trace_parallel_plan_strips I \<pi>) < length \<pi> + 1" shows "\<not> are_all_operators_applicable (execute_parallel_plan I (take (length (trace_parallel_plan_strips I \<pi>) - 1) \<pi>)) (\<pi> ! (length (trace_parallel_plan_strips I \<pi>) - 1)) \<or> \<not> are_all_operator_effects_consistent (\<pi> ! (length (trace_parallel_plan_strips I \<pi>) - 1))"
simps_1: fixes ns :: "'b list list" shows "listprepend [] ns = []"
equivpartpE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "equivpartp r x y" obtains "x = y"   | "r x y" and "r y x"
5: shows "direct_decomp (ideal F \<inter> P[X]) [ideal {f} \<inter> P[X], P']"
"case": fixes f :: "'d \<Rightarrow> 'f \<Rightarrow> 'c"   and x1 :: "'d"   and x2 :: "'f" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
adjoint_functors_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" shows "adjoint_functors C D F G = (\<exists>\<phi> \<psi>. meta_adjunction C D F G \<phi> \<psi>)"
commute: fixes a :: "'d"   and b :: "'d" shows "a + b = b + a"
dirvec_minus: fixes x :: "'a \<times> 'a" shows "dirvec x = snd x - fst x"
in_traverse_fst: fixes t :: "('c, 'd) tree" shows "map fst (in_traverse_pairs t) = in_traverse t"
empty_Collect_eq: fixes P :: "'b::type \<Rightarrow> bool" shows "({} = Collect (P::'b::type \<Rightarrow> bool)) = (\<forall>x::'b::type. \<not> P x)"
set: fixes x1 :: "16 word"   and x2 :: "'b set"   and x3 :: "'a" shows "set_flow_entry_match (OFEntry x1 x2 x3) = {x3}"
that: fixes n :: "nat" assumes "\<And>m. n \<le> m \<Longrightarrow> (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n" shows "thesis"
intros_2: fixes c1 :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c2 :: "('c, 'd) conf"   and ls :: "('c, 'd) label list"   and c3 :: "('c, 'd) conf" assumes "se c1 l c2"   and "se_star c2 ls c3" shows "se_star c1 (l # ls) c3"
insert_xlate: fixes e :: "('c, 'd) BsSkewElem"   and q :: "('c, 'd) BsSkewBinomialTree list" shows "map bsmapt (local.insert e q) = SkewBinomialHeapStruc.insert e (eprio e) (map bsmapt q)"
Eats_EQ_Zero_EH_1: fixes t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (Eats t u EQ Zero) H \<turnstile> A"
starprime: shows "starprime = {p. 1 < p \<and> (\<forall>m. m dvd p \<longrightarrow> m = 1 \<or> m = p)}"
reachability_closed_under_execution_step: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "reachable s"   and "revision_step r s s'" shows "reachable s'"
corec_2: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g21 :: "'a \<Rightarrow> 'b"   and q22 :: "'a \<Rightarrow> bool"   and g221 :: "'a \<Rightarrow> 'b llist"   and g222 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "corec_llist p g21 q22 g221 g222 a = LCons (g21 a) (if q22 a then g221 a else corec_llist p g21 q22 g221 g222 (g222 a))"
ord_dual: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "\<partial> y \<le> \<partial> x"
make\<kappa>_inject: fixes x :: "\<nu> option"   and y :: "\<nu> option" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(make\<kappa> x = make\<kappa> y) = (x = y)"
assms_1: shows "ms_exp_gt 0 (ms_aux_hd_exp xs)"
in_components_nonempty: fixes c :: "'c set"   and s :: "'c set" assumes "c \<in> components s" shows "c \<noteq> {}"
sq_mtx_inv_unique: fixes A :: "'b sq_mtx"   and B :: "'b sq_mtx" assumes "A * B = 1"   and "B * A = 1" shows "A\<^sup>-\<^sup>1 = B"
2: shows "compose (set_dom f) (\<lambda>x\<in>set_cod f. x) (set_func f) = set_func f"
fresh_def: fixes xs :: "'b"   and x :: "'c"   and X :: "('d, 'e, 'b, 'c, 'f) qTerm set" shows "fresh xs x X \<equiv> qFresh xs x (pick X)"
mult_gcd_left: fixes c :: "'a"   and a :: "'a"   and b :: "'a" shows "c * gcd a b = unit_factor c * gcd (c * a) (c * b)"
simps: fixes i :: "'a"   and c :: "constraint" shows "i_constraint_to_qdelta_constraint (i, c) = map (Pair i) (constraint_to_qdelta_constraint c)"
invrules_5: shows "S \<subseteq> {u \<in> U. w u \<le> real c / 2} \<and> L \<subseteq> {u \<in> U. real c / 2 < w u}"
triv_finprod_iff_compl_fam_Pi: fixes I :: "'d set"   and S :: "'d \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (S i) G" shows "compl_fam S I = (\<forall>f\<in>Pi I S. finprod G f I = \<one> \<longrightarrow> (\<forall>i\<in>I. f i = \<one>))"
simps_2: fixes uv :: "'a \<Rightarrow> value option" shows "gval (Bc False) uv = false"
k_1: shows "local.F (Suc local.k) = 0"
is_binqueue_min_Min_prios: fixes l :: "nat"   and xs :: "('c, 'd) bintree option list" assumes "is_binqueue l xs"   and "normalized xs"   and "xs \<noteq> []" shows "Binomial_Queue.min xs = Some (Min (set (prios xs)))"
interpret_floatarith_Sin: fixes x :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Sin x) vs = sin (interpret_floatarith x vs)"
openproc_parq_axioms_def: fixes qp :: "('a, 'b seq_action) automaton" shows "openproc_parq_axioms qp \<equiv> init qp \<noteq> {}"
ln_neg_is_const: fixes x :: "real" assumes "x \<le> 0" shows "ln x = (THE x. False)"
hIm_HComplex: fixes x :: "real star"   and y :: "real star" shows "hIm (HComplex x y) = y"
len_empty_on_subview1: fixes v :: "view"   and ts :: "traffic"   and c :: "cars"   and v1 :: "view"   and v2 :: "view" shows "\<parallel>len v ts c\<parallel> = 0 \<and> v=v1\<parallel>v2 \<longrightarrow> \<parallel>len v1 ts c\<parallel> = 0"
suffixes_alt_def: fixes xs :: "'a list" shows "suffixes xs = rev (prefix (length xs) (\<lambda>i. drop i xs))"
R_def: shows "R = (\<lambda>a. cbox (of_int_vec a) (of_int_vec (a + 1)))"
PEE_strict_appI: fixes xs :: "nat \<Rightarrow> ValD"   and fs :: "nat \<Rightarrow> ValD \<rightarrow> ValD" assumes "xs \<in> unlr PE.delta"   and "\<And>xs. xs \<in> unlr PE.delta \<Longrightarrow> (\<lambda>j. fs j\<cdot>(xs j)) \<in> unlr PE.delta" shows "(\<lambda>j. strictify\<cdot>(fs j)\<cdot>(xs j)) \<in> unlr PE.delta"
assms_2: shows "t \<le> v * v\<^sup>T \<sqinter> - - g"
card_UNIV_option_def: shows "card_UNIV_class.card_UNIV = Phantom('a option) (let c = of_phantom card_UNIV_class.card_UNIV in if c \<noteq> 0 then Suc c else 0)"
multiset_sum_monotone_f': fixes CC :: "('b \<times> 'c) multiset"   and DD :: "('b \<times> 'c) multiset"   and f :: "'b \<Rightarrow> nat" assumes "CC \<subset># DD" shows "(\<Sum>(C, i)\<in>#CC. Suc (f C)) < (\<Sum>(C, i)\<in>#DD. Suc (f C))"
cones_map_ide: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and \<chi> :: "'ag \<Rightarrow> 'ah"   and a :: "'ah" assumes "diagram J C D"   and "\<chi> \<in> {\<chi>. Limit.cone J C D a \<chi>}" shows "(\<lambda>\<chi>\<in>{\<chi>. Limit.cone J C D (partial_magma.cod C a) \<chi>}. \<lambda>j. if partial_magma.arr J j then C (\<chi> j) a else partial_magma.null C) \<chi> = \<chi>"
union_disjoint: fixes A :: "'e set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'f" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "prod g (A \<union> B) = prod g A * prod g B"
col_obtuse__bet: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C"   and "Obtuse A B C" shows "Bet A B C"
inorder_del: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (RBT_Set.del x t) = del_list x (Tree2.inorder t)"
set_eq_sym: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set"   and B :: "'a set" assumes "equivalence S"   and "A \<subseteq> carrier S"   and "B \<subseteq> carrier S"   and "A {.=}\<^bsub>S\<^esub> B" shows "B {.=}\<^bsub>S\<^esub> A"
simps_4: fixes r :: "real" shows "ereal r + - \<infinity> = - \<infinity>"
gfp_mono: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "\<And>Z. f Z \<le> g Z" shows "gfp f \<le> gfp g"
Kruskal_Impl_axioms: shows "Kruskal_Impl E V vertices joins forest connected weight \<alpha> endpoints getEdges getEdges_impl superE endpoints_impl"
Op2_eq_b_least2_at_point: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" shows "loc_Op2 f x y = b_least2 f x y"
older_seniors_older: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and y :: "nat"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "y \<in> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n" shows "y < x"
ident: fixes D :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and e :: "'b"   and x :: "'b" assumes "ACeD D f e"   and "x \<in> D" shows "f x e = x"
distinct_take: fixes xs :: "'a list"   and i :: "nat" assumes "distinct xs" shows "distinct (take i xs)"
cases: fixes x :: "'fmla list" obtains "x = []"   | \<phi> :: "'fmla" and \<phi>s :: "'fmla list" where "x = \<phi> # \<phi>s"
simps_4: fixes uv :: "'d list"   and uw :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and F' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>uv\<langle>\<or>\<noteq>: uw \<or>\<notin>: F'\<rangle> = set F'"
sumlist_snoc: fixes xs :: "'a vec list"   and x :: "'a vec" assumes "set xs \<subseteq> carrier_vec n"   and "x \<in> carrier_vec n" shows "M.sumlist (xs @ [x]) = M.sumlist xs + x"
Hwit1_def: fixes b :: "'q"   and c :: "'r" shows "Hwit1 b c = wit1_G wit_F1 (wit_F3 b c)"
one_step_equiv: fixes N1 :: "('f \<times> 'l) set"   and N2 :: "('f \<times> 'l) set" assumes "N1 \<leadsto>GC N2" shows "N1 \<rhd>L N2"
mem_in_image3: fixes b :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "b \<in> f ` A" shows "\<exists>a\<in>A. b = f a"
count_inject: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(count x = count y) = (x = y)"
components_trdsp_subset: fixes bs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and p :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and q :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" shows "component_of_term ` keys (trdsp bs (p, q)) \<subseteq> component_of_term ` Keys (insert (fst p) (insert (fst q) (set bs)))"
pim: fixes T :: "'a set" assumes "T \<subseteq> C"   and "\<phi> T" shows "\<psi> (p ` T)"
star2: fixes x :: "'a list set" shows "p_star (p_one \<union> x) = p_star x"
11: shows "rawpsubstT (substT r (Var v) x) (zip (map Var vs) xs) = substT (rawpsubstT r (zip (map Var vs) xs)) (Var v) x"
assms_1: shows "z \<cdot> w \<le> w \<cdot> z \<cdot> w"
col_ket_vec: fixes M :: "complex Matrix.mat" assumes "dim_col M = 1" shows "|Matrix.col M 0\<rangle> = M"
bex_empty: fixes P :: "'c \<Rightarrow> bool" shows "Bex {} P = False"
is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F.map G.map local.map"
p_1: shows "poly (of_int_poly p) (\<i> * complex_of_real pi) = 0"
zero_power_eq: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero n = zero.when zero one (n = 0)"
l_identity: fixes w :: "i"   and x :: "'a"   and y :: "'a"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "[x \<^bold>= y in w]"   and "[\<phi> x in w]" shows "[\<phi> y in w]"
assms: shows "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) F"
representation_sum: fixes basis :: "'b set"   and I :: "'c set"   and v :: "'c \<Rightarrow> 'b" assumes "cindependent basis"   and "\<And>i. i \<in> I \<Longrightarrow> v i \<in> cspan basis" shows "crepresentation basis (sum v I) = (\<lambda>b. \<Sum>i\<in>I. crepresentation basis (v i) b)"
asprod_def: fixes m :: "int"   and x :: "ant" shows "m *\<^sub>a x \<equiv> if x = \<infinity> then if 0 < m then \<infinity> else if m < 0 then - \<infinity> else if m = 0 then 0 else undefined else if x = - \<infinity> then if 0 < m then - \<infinity> else if m < 0 then \<infinity> else if m = 0 then 0 else undefined else ant m * x"