hcju/isabelleps huggingface dataset:
eqv: fixes p :: "perm" assumes "p \<bullet> T = S"   and "p \<bullet> ba xs = ba ys"   and "supp p \<subseteq> ba xs \<union> ba ys" shows "p \<bullet> f xs T = f ys S"
intros: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "r a b" shows "Rangep r b"
intro: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and S :: "'c set" assumes "comm_ring R addition multiplication zero unit"   and "submonoid S R multiplication unit" shows "Comm_Ring.quotient_ring S R addition multiplication zero unit"
is_infI: fixes inf :: "'b"   and x :: "'b"   and y :: "'b" assumes "inf \<sqsubseteq> x"   and "inf \<sqsubseteq> y"   and "\<And>z. \<lbrakk>z \<sqsubseteq> x; z \<sqsubseteq> y\<rbrakk> \<Longrightarrow> z \<sqsubseteq> inf" shows "is_inf x y inf"
nn_integral_mono: fixes M :: "'c measure"   and u :: "'c \<Rightarrow> ennreal"   and v :: "'c \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
uniformly_continuous_on_dist: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "uniformly_continuous_on s f"   and "uniformly_continuous_on s g" shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
FW_2: shows "fw m n n n n i k = D m i k n"
negate_f: shows "\<forall>pts. matches \<gamma> (negate_f C pts) a p = matches \<gamma> (MatchNot (Match (C pts))) a p"
seq_lexord_antisym: fixes R :: "('b \<times> 'b) set"   and a :: "'b seq"   and b :: "'b seq" assumes "asym R"   and "(a, b) \<in> seq_lexord R" shows "(b, a) \<notin> seq_lexord R"
nat_pow_Suc: fixes n :: "nat"   and x :: "'a" shows "[Suc n] \<cdot> x = [n] \<cdot> x \<oplus> x"
oseq_assignTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and u :: "'e \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}\<lbrakk>u\<rbrakk> p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "a = \<tau>" and "\<sigma>' i = u (\<sigma> i)" and "q = p"
countable_acc: fixes X :: "'b set" assumes "countable X" shows "countable (K2.acc `` X)"
l_cancel_one': fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier H"   and "a \<in> carrier H" shows "(x = x \<otimes>\<^bsub>H\<^esub> a) = (a = \<one>\<^bsub>H\<^esub>)"
simps_3: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Inverse a) vs = approx_floatarith p a vs \<bind> inverse_aform_err p"
simps: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'d pmf"   and B :: "('e, 'e cset Monomorphic_Monad.id) nondetT" shows "cr_prob_ndi A p B = rel_set A (set_pmf p) (rcset (extract (run_nondet B)))"
in_language_bij: shows "bij (Collect \<circ> in_language)"
no_hw_le1: fixes h :: "complex \<Rightarrow> complex" assumes "h \<in> Y" shows "cmod (h w) \<le> 1"
ball_subset_open_map_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and r :: "real"   and a :: "'a" assumes "continuous_on (closure S) f"   and "open (f ` interior S)"   and "\<And>z. z \<in> frontier S \<Longrightarrow> r \<le> norm (f z - f a)"   and "bounded S"   and "a \<in> S"   and "0 < r" shows "ball (f a) r \<subseteq> f ` S"
norm_le_l1_cart: fixes x :: "(real, 'a) vec" shows "norm x \<le> (\<Sum>i\<in>UNIV. \<bar>x $ i\<bar>)"
all_Suc_split: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>i. P i) = (P 0 \<and> (\<forall>i. P (Suc i)))"
A_fv: fixes e :: "char list exp"   and A :: "char list set" assumes "\<A> e = \<lfloor>A\<rfloor>" shows "A \<subseteq> fv e"
Done: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and C :: "qtname" shows "G,A\<turnstile>{Normal (P\<leftarrow>\<diamondsuit> \<and>. initd C)} .Init C. {P}"
Ccard: shows "int (card C) \<le> int (card B) - int (card A)"
rep_eq: fixes x :: "ennreal set" shows "enn2ereal (Sup x) = sup 0 (Sup (enn2ereal ` x))"
landau_symbols_5: shows "landau_symbol bigtheta bigtheta bigtheta"
inter_restrict: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "finite A" shows "sum g (A \<inter> B) = (\<Sum>x\<in>A. if x \<in> B then g x else (0::'b))"
interp: shows "\<forall>x i. (Functions I i has_derivative FunctionFrechet I i x) (at x)"
matrix_sum_remove: fixes j :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and d :: "nat"   and f' :: "nat \<Rightarrow> 'a mat" assumes "j < n"   and "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. f' k = (if k = j then 0\<^sub>m d d else f k)" shows "matrix_sum d f n = f j + matrix_sum d f' n"
is_up: fixes c :: "'b word \<Rightarrow> 'c word" shows "is_up c = (LENGTH('b) \<le> LENGTH('c))"
xor_negative_int_iff: fixes k :: "int"   and l :: "int" shows "(k XOR l < 0) = ((k < 0) \<noteq> (l < 0))"
integral_completion: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "integral\<^sup>L (completion M) f = integral\<^sup>L M f"
sig_red_zero_nonzero: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red_zero sing_reg F r"   and "rep_list r \<noteq> 0"   and "sing_reg = (\<preceq>\<^sub>t) \<or> sing_reg = (\<prec>\<^sub>t)" shows "is_sig_red sing_reg (=) F r"
equivalence_pseudofunctor_axioms: shows "equivalence_pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>"
order_le_less: fixes x :: "'d"   and y :: "'d" shows "(x \<le> y) = (x < y \<or> x = y)"
mem_closed_if_AE_lebesgue_open: fixes S :: "'c set"   and C :: "'c set"   and x :: "'c" assumes "open S"   and "closed C"   and "AE x\<in>S in lebesgue. x \<in> C"   and "x \<in> S" shows "x \<in> C"
assms_3: shows "\<not> stack c\<^sub>2 \<lesssim> stack c\<^sub>4"
tendsto_null_power: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and n :: "nat" assumes "(f \<longlongrightarrow> (0::'c)) F"   and "0 < n" shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> (0::'c)) F"
hausdorff_distance_triangle: fixes B :: "'b set"   and A :: "'b set"   and C :: "'b set" assumes "B \<noteq> {}"   and "bounded B" shows "hausdorff_distance A C \<le> hausdorff_distance A B + hausdorff_distance B C"
contour_integral_div: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and c :: "complex" assumes "f contour_integrable_on g" shows "contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
symE: fixes r :: "('a \<times> 'a) set"   and b :: "'a"   and a :: "'a" assumes "sym r"   and "(b, a) \<in> r" obtains "(a, b) \<in> r"
Notin_def: shows "Notin \<equiv> \<lambda>\<delta> D. list_ex (\<lambda>f. f \<cdot>\<^sub>p \<delta> \<circ>\<^sub>s \<I> \<notin> set D \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>)"
bidel_arc_same_face_tail_conn_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" shows "bidel_arc_same_face_tail_conn G M a \<equiv> bidel_arc_same_face G M a \<and> bidel_arc_tail_conn G M a"
continuous_map_diff: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_map X euclidean f"   and "continuous_map X euclidean g" shows "continuous_map X euclidean (\<lambda>x. f x - g x)"
mult_ge_one: fixes a :: "'a"   and b :: "'a" assumes "(1::'a) \<le> a"   and "(1::'a) \<le> b" shows "(1::'a) \<le> a * b"
g: fixes i :: "'s"   and j :: "'t" assumes "R i j" shows "R (g j) j"
simps_1: fixes ys :: "[:'r:]" shows "[::] :@ ys = ys"
red2_in_Hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b monoidal_language.term"   and b :: "'b monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Ide C a"   and "monoidal_language.Diag C a"   and "monoidal_language.Ide C b"   and "monoidal_language.Diag C b" shows "monoidal_language.red2 C a b \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = monoidal_language.Tensor a b \<and> monoidal_language.Cod C t = monoidal_language.Diagonalize C (monoidal_language.Tensor a b)}"
AlphaNegNeg: fixes A :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<stileturn> A # G" shows "\<stileturn> Neg (Neg A) # G"
LSQ_def: shows "LSQ = cbox (case (real_of_int l - \<delta>, snd p) of (x, y) \<Rightarrow> (x, real_of_int y)) (case (l, real_of_int (snd p) + \<delta>) of (x, y) \<Rightarrow> (real_of_int x, y))"
K: fixes i :: "nat" shows "K i = prod_emb I M (J i) (X i)"
tensor_P_testN_sum: shows "matrix_sum d (\<lambda>k. adjoint (tensor_P (testN k) (1\<^sub>m K)) * tensor_P post (1\<^sub>m K) * tensor_P (testN k) (1\<^sub>m K)) N = tensor_P post (1\<^sub>m K)"
indS_int: shows "(indicat_real S has_integral Sigma_Algebra.measure lebesgue S) (cbox a b)"
L_bins_lower_bound_card: fixes P :: "'a set set"   and P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "bp P"   and "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 (S \<union> L)"   and "\<forall>B\<in>P\<^sub>1 \<union> wrap B\<^sub>1. B \<inter> {u \<in> U. real c / 2 < w u} \<noteq> {}"   and "S \<subseteq> {u \<in> U. w u \<le> real c / 2} \<and> L \<subseteq> {u \<in> U. real c / 2 < w u}" shows "card (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> {{v} |v. v \<in> L}) \<le> card P"
Abs_riemann_sphere_inverse: fixes y :: "real \<times> real \<times> real" assumes "y \<in> unit_sphere" shows "Rep_riemann_sphere (Abs_riemann_sphere y) = y"
simps: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and "structure" :: "'f set"   and sat :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('g \<times> bool \<times> 'g) set"   and Ord :: "('h \<times> 'h) set"   and \<sigma> :: "'g \<times> 'f \<Rightarrow> 'h"   and a1 :: "bool stream"   and a2 :: "'g stream"   and a3 :: "('e \<times> 'd) stream" assumes "Infinite_Soundness eff rules structure sat \<delta> Ord \<sigma>" shows "Infinite_Soundness.follow \<delta> a1 a2 a3 = (\<exists>M' Ms s' steps M b s r bs. a1 = b ## bs \<and> a2 = M ## Ms \<and> a3 = (s, r) ## steps \<and> M' = shd Ms \<and> s' = fst (shd steps) \<and> (M, b, M') \<in> \<delta> s r s' \<and> Infinite_Soundness.follow \<delta> bs Ms steps)"
bind_fv_remove_var_from_fv: fixes idn :: "variable"   and \<tau> :: "typ"   and t :: "Core.term" shows "fv (bind_fv (idn, \<tau>) t) = fv t - {(idn, \<tau>)}"
that: fixes cms\<^sub>2' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2' :: "'Var \<Rightarrow> 'Val"   and mems' :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list" assumes "map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> (cms\<^sub>2, mem\<^sub>2) \<rightarrow> (cms\<^sub>2', mem\<^sub>2') \<and> makes_compatible (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2', mem\<^sub>2') mems'" shows "thesis"
not_psubset_empty: fixes A :: "'b set" shows "\<not> A \<subset> {}"
lm096: fixes pair :: "'a \<times> 'b set"   and N :: "'a set"   and G :: "'b set"   and bids :: "'a \<times> 'b set \<Rightarrow> nat" assumes "pair \<in> N \<times> (Pow G - {{}})" shows "summedBidVector (real \<circ> bids) N G pair = real (summedBidVector bids N G pair)"
refl: fixes t :: "('c, 'd) tm" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b t"
cont: shows "continuous_map (subtopology X (S - U)) (subtopology X S) id"
fromFull: fixes \<sigma> :: "'tp"   and a :: "univ" shows "intT \<sigma> (fromFull \<sigma> a)"
priSK_symK: fixes X :: "agent"   and n :: "nat" shows "priSK X \<noteq> symK n"
valid_path_reversepath: fixes g :: "real \<Rightarrow> 'a" shows "valid_path (reversepath g) = valid_path g"
indRelEQ_def: shows "indRelEQ \<equiv> _Collect (x, y) (indRelEQp x y)"
Fun_comp: fixes g :: "'s"   and f :: "'s" assumes "S.seq g f" shows "S.Fun (g \<cdot>\<^sub>S f) = restrict (S.Fun g \<circ> S.Fun f) (S.Dom f)"
assms_3: assumes "\<phi> \<pi>l" shows "\<chi>"
length_concat_negGorbit_list: fixes smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and gs :: "'c list"   and T :: "'d \<Rightarrow> 'a"   and as :: "'d list" shows "length (concat (aezfun_scalar_mult.negGorbit_list smult gs T as)) = length gs * length as"
assms: shows "\<forall>x\<in>{0..}. \<forall>y\<in>{0..}. x * y div y = x"
min_absorb1: fixes x :: "'d"   and y :: "'d" assumes "x \<le> y" shows "min x y = x"
PiE_insert_eq: fixes x :: "'a"   and S :: "'a set"   and T :: "'a \<Rightarrow> 'b set" shows "Pi\<^sub>E (insert x S) T = (\<lambda>(y, g). g(x := y)) ` (T x \<times> Pi\<^sub>E S T)"
continuous_on_id: fixes s :: "'d set" shows "continuous_on s (\<lambda>x. x)"
succss_closed_dfs: fixes xs :: "'a list" assumes "list_all isNode xs" shows "nodeAbs ` succss (set_of (dfs empt xs)) \<subseteq> nodeAbs ` set_of (dfs empt xs)"
assms_1: shows "wa_precise_refine WC WB \<alpha>1"
skelBinp_def: fixes binp :: "'d \<Rightarrow> ('e, 'f, 'g, 'h, 'i) qAbs set option" shows "skelBinp binp = qSkelBinp (pickBinp binp)"
ereal_diff_eq_0_iff: fixes a :: "ereal"   and b :: "ereal" assumes "\<bar>a\<bar> = \<infinity> \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity>" shows "(a - b = 0) = (a = b)"
acc_lengths_interact_injective: fixes as' :: "'a list list"   and bs' :: "'a list list"   and as :: "'a list list"   and bs :: "'a list list"   and a :: "nat"   and b :: "nat" assumes "interact as' bs' = interact as bs"   and "acc_lengths a as' = acc_lengths a as"   and "acc_lengths b bs' = acc_lengths b bs" shows "as' = as \<and> bs' = bs"
finite_prime_divisors: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "finite {p. prime p \<and> p dvd x}"
nl42: fixes l' :: "real"   and r' :: "real"   and ll :: "real"   and r :: "real" assumes "1 \<le> l'"   and "1 \<le> r'"   and "1 \<le> ll"   and "1 \<le> r" shows "4 * (l' + r') * (l' + ll) * (r' + r) \<le> (l' + ll + r' + r) ^ 3"
l14: shows "2 * e * b * d - 2 * a * e\<^sup>2 - 2 * d * f \<in> radical_sqrt"
"3_4": shows "is_heap (T v2 l2 r2)"
dot_square_norm: fixes x :: "'a" shows "x \<bullet> x = (norm x)\<^sup>2"
resolve_eq_Bin_ord_resolve: fixes C :: "'a literal list"   and D :: "'a literal list" shows "mset ` set (resolve C D) = Bin_ord_resolve (mset C) (mset D)"
measure_UNION_le: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" shows "Sigma_Algebra.measure M (\<Union> (F ` I)) \<le> (\<Sum>i\<in>I. Sigma_Algebra.measure M (F i))"
hyps_1: shows "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f"
token_stays_in_sink: fixes q :: "'b"   and x :: "nat"   and n :: "nat"   and m :: "nat" assumes "sink q"   and "token_run x n = q" shows "token_run x (n + m) = q"
gtr_empty_conf_s_1: fixes w :: "'d"   and c' :: "'e multiset"   and S :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'e \<times> 'e multiset) set" shows "({#}, w, c') \<notin> gtr S"
lm151: fixes f :: "('a \<times> 'b) set"   and g :: "('a \<times> 'b) set" assumes "f \<subseteq> g"   and "runiq g"   and "Domain f = Domain g" shows "g \<subseteq> f"
compute_monom_mult_poly_mapping: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and c :: "'d"   and t :: "'b"   and xs :: "('a, 'd) fmap" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.monom_mult pair_of_term term_of_pair c t (Pm_fmap xs) = Pm_fmap (if c = (0::'d) then fmempty else term_powerprod.shift_map_keys pair_of_term term_of_pair t ((*) c) xs)"
eint_ord_code_5: fixes n :: "int" shows "(\<infinity> \<le> eint n) = False"
ide_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.ide (src\<^sub>B \<mu>)"
2: shows "\<guillemotleft>S.UP \<eta> : S.UP (src f) \<Rightarrow>\<^sub>S S.UP (g \<star> f)\<guillemotright>"
unif_rtrancl_empty_imp_unifiable: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "(E, {#}) \<in> unif\<^sup>*" shows "unifiable (set_mset E)"
regexp_ewp: fixes t :: "'a rexp" shows "\<exists>t'. t \<sim> ro t +\<^sub>r t' \<and> ro t' = 0\<^sub>r"
\<Phi>_def: fixes p :: "(nat \<times> int) list"   and x :: "nat \<Rightarrow> real" shows "\<Phi> p x = (\<Prod>d<length p. \<phi> (p ! d) (x d))"
Xij_def: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" shows "Xij xs i j = prod_list (map (X xs) [i..<j])"
ws0_1: shows "set ws0 \<subseteq> carrier_vec n"
cases: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "char list exp list"   and a3 :: "ty list" assumes "P,a1,h \<turnstile> a2 [:] a3" obtains E :: "char list \<Rightarrow> ty option" where "a1 = Ea__" and "a2 = []" and "a3 = []"   | E :: "char list \<Rightarrow> ty option" and e :: "char list exp" and T :: "ty" and es :: "char list exp list" and Ts :: "ty list" where "a1 = E" and "a2 = e # es" and "a3 = T # Ts" and "P,E,h \<turnstile> e : T" and "P,E,h \<turnstile> es [:] Ts"
trading_strategy_adapted: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and Mkt :: "('a, 'c) discrete_market" assumes "trading_strategy p"   and "support_adapt Mkt p" shows "borel_adapt_stoch_proc F (val_process Mkt p)"
cut: fixes p :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list"   and r :: "('a, 'b) fm"   and q :: "('a, 'b) fm" assumes "p # ps \<turnstile> r"   and "q # ps \<turnstile> p" shows "q # ps \<turnstile> r"
wf_iff_wfp_on: fixes p :: "('a \<times> 'a) set" shows "wf p = wfp_on (\<lambda>a b. (a, b) \<in> p) UNIV"
eq2: shows "(if f i < j then f i else Suc (f i)) = (if f i' < j then f i' else Suc (f i'))"
truncate_down_nonneg_mono: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "0 \<le> x"   and "x \<le> y" shows "truncate_down prec x \<le> truncate_down prec y"
weak_get_via_put: fixes x :: "'c \<Longrightarrow> 'd"   and s :: "'d" assumes "mwb_lens x"   and "s \<in> \<S>\<^bsub>x\<^esub>" shows "get\<^bsub>x\<^esub> s = (THE v. put\<^bsub>x\<^esub> s v = s)"
components_are_iso: fixes a :: "'b" assumes "ide a" shows "local.iso (\<rr> a)"
Bit_eq_0_iff: fixes w :: "int"   and b :: "bool" shows "(w BIT b = 0) = (w = 0 \<and> \<not> b)"
Ints_nonzero_abs_ge1: fixes x :: "'b" assumes "x \<in> \<int>"   and "x \<noteq> (0::'b)" shows "(1::'b) \<le> \<bar>x\<bar>"
test_not: fixes test :: "'c \<Rightarrow> 'd"   and not :: "'d \<Rightarrow> 'd"   and p :: "'c" assumes "drat test not" shows "test (- p) = not (test p)"
integrable_component: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and y :: "'d" assumes "f integrable_on A" shows "(\<lambda>x. f x \<bullet> y) integrable_on A"
runFw_scheme_def: fixes s :: "'a word"   and d :: "'a word"   and c :: "'b parts_connection_scheme"   and p :: "('a, 'c) simple_packet_scheme"   and rs :: "'a simple_rule list" shows "runFw_scheme s d c p rs = simple_fw rs (p\<lparr>p_iiface := pc_iiface c, p_oiface := pc_oiface c, p_src := s, p_dst := d, p_proto := pc_proto c, p_sport := pc_sport c, p_dport := pc_dport c\<rparr>)"
sum_permutations_inverse: fixes f :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a"   and S :: "'b set" shows "sum f {p. p permutes S} = (\<Sum>p | p permutes S. f (inv p))"
preserves_reflects_arr: fixes f :: "'t" shows "S.arr (\<Psi> f) = S'.arr f"
tickFree_def: fixes s :: "'b event list" shows "tickFree s = (tick \<notin> set s)"
PSuf_def: fixes s :: "'a list" shows "PSuf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
runit_in_hom_2: fixes f :: "'d" assumes "D.ide f" shows "\<guillemotleft>\<r>\<^sub>D[f] : f \<star>\<^sub>D src\<^sub>D f \<Rightarrow>\<^sub>D f\<guillemotright>"
assms: shows "\<forall>x\<in>vars p. v x = v' x"
adds_triv_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus a (plus a b)"
Nge0_def: shows "0\<le>\<^sub>N = (\<lambda>(a, b). 0 \<le> a)"
kripke_def: fixes M :: "('e, 'f, 'g) KripkeStructure" shows "kripke M \<equiv> \<forall>a. relations M a \<subseteq> worlds M \<times> worlds M"
vars_snd_rcv_strand2: fixes S :: "('d, 'c) strand_step list" shows "wfrestrictedvars\<^sub>s\<^sub>t S = \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S)) \<union> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S)) \<union> \<Union> (set (map (fv\<^sub>e\<^sub>q assign) S))"
induct: fixes P :: "nat bdd \<Rightarrow> nat bdd \<Rightarrow> bool list list \<Rightarrow> bool"   and a0 :: "nat bdd"   and a1 :: "nat bdd"   and a2 :: "bool list list" assumes "\<And>i j T. P (Leaf i) (Leaf j) T"   and "\<And>l r i T. \<lbrakk>P l (Leaf i) T; P r (Leaf i) T\<rbrakk> \<Longrightarrow> P (Branch l r) (Leaf i) T"   and "\<And>i l r T. \<lbrakk>P (Leaf i) l T; P (Leaf i) r T\<rbrakk> \<Longrightarrow> P (Leaf i) (Branch l r) T"   and "\<And>l1 r1 l2 r2 T. \<lbrakk>P l1 l2 T; P r1 r2 T\<rbrakk> \<Longrightarrow> P (Branch l1 r1) (Branch l2 r2) T" shows "P a0 a1 a2"
connected_imp_perfect_aff_dim: fixes S :: "'b set"   and a :: "'b" assumes "connected S"   and "aff_dim S \<noteq> 0"   and "a \<in> S" shows "a islimpt S"
P_mod_erH: fixes P :: "'a set"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and g :: "'a"   and v :: "'a"   and t :: "'a" assumes "Idomain S"   and "ideal S P"   and "PolynRg R' (S /\<^sub>r P) Y"   and "g \<in> carrier R"   and "v \<in> carrier R"   and "t \<in> P" shows "erH R S X R' (S /\<^sub>r P) Y (pj S P) g = erH R S X R' (S /\<^sub>r P) Y (pj S P) (g \<plusminus> t \<cdot>\<^sub>r v)"
intro: fixes f :: "'a \<Rightarrow> 'b" assumes "injective f"   and "surjective f" shows "bijective f"
steps: shows "\<Gamma> \<turnstile> cfg\<^sub>1 \<rightarrow>\<^sup>* cfg\<^sub>2"
assms_1: shows "Functor F : AA \<longrightarrow> BB"
join_and_local_commute: fixes s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>1 :: "'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 \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau>' :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and r'' :: "'r"   and s\<^sub>2' :: "'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 ls :: "('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr"   and \<sigma>' :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and v :: "('r, 'l, 'v) val" assumes "s\<^sub>2 = (s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r'' := None)"   and "s\<^sub>2' = s\<^sub>1(r' \<mapsto> ls)"   and "r \<noteq> r'"   and "r' \<noteq> r''"   and "revision_step r' s\<^sub>2 ((s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r'' := None)(r' \<mapsto> ls))"   and "s\<^sub>2' r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r''))])"   and "s\<^sub>2' r'' = Some (\<sigma>', \<tau>', VE v)" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
inj: shows "\<forall>a b. QB (f a) \<and> QA b \<and> f a = f b \<longrightarrow> a = b"