hcju/isabelleps huggingface dataset:
endofunctor_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'b" shows "endofunctor A F \<equiv> functor A A F"
has: shows "has_ell2_norm (\<lambda>i. x i + y i)"
simps: fixes \<I> :: "('c, 'd) \<I>"   and a :: "('c, 'd) resource" shows "lossless_resource \<I> a = (\<exists>res. a = res \<and> (\<forall>x. x \<in> outs_\<I> \<I> \<longrightarrow> lossless_spmf (run_resource res x)) \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> set_spmf (run_resource res x) \<longrightarrow> lossless_resource \<I> xb))"
assms_3: fixes z :: "'a" assumes "z \<in> B" shows "s0 \<bullet> (1::'a) < z \<bullet> (1::'a)"
cos_double: fixes x :: "'b" shows "cos ((2::'b) * x) = (cos x)\<^sup>2 - (sin x)\<^sup>2"
pole_theorem_analytic_open_superset: fixes g :: "complex \<Rightarrow> complex"   and S :: "complex set"   and T :: "complex set"   and a :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "g analytic_on S"   and "S \<subseteq> T"   and "open T"   and "\<And>z. z \<in> T - {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)) analytic_on S"
cltn2_rep_abs2: fixes A :: "((real, 3) vec, 3) vec" assumes "invertible A" shows "\<exists>k. k \<noteq> 0 \<and> cltn2_rep (cltn2_abs A) = k *\<^sub>R A"
state_split: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> 'e" assumes "\<tau> \<Turnstile> \<delta> X" shows "\<tau> \<Turnstile> X.oclIsNew() \<or> \<tau> \<Turnstile> X.oclIsDeleted() \<or> \<tau> \<Turnstile> X.oclIsMaintained() \<or> \<tau> \<Turnstile> X.oclIsAbsent()"
abs_eq: fixes x :: "('c \<times> nat) list" assumes "eq_onp (\<lambda>x. x \<in> Collect monom_inv) x x" shows "sum_var (Abs_monom x) = sum_var_list x"
permutation_insert_def: fixes i :: "'b"   and j :: "nat"   and p :: "'b \<Rightarrow> nat" shows "permutation_insert i j p \<equiv> insert_dom (insert_ran p j) i j"
simulation: fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts\<^sub>s\<^sub>b' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b' :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b' :: "nat \<Rightarrow> bool option"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "invariant ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b m\<^sub>s\<^sub>b"   and "safe_reach_direct safe_delayed (ts, m, \<S>)" shows "(\<exists>ts' \<S>'' m'. (ts, m, \<S>) \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>'') \<and> (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b') \<sim> (ts', m', \<S>'')) \<and> invariant ts\<^sub>s\<^sub>b' \<S>\<^sub>s\<^sub>b' m\<^sub>s\<^sub>b'"
e_nth_append_zeros: fixes b :: "nat"   and z :: "nat"   and i :: "nat" shows "e_nth (e_append_zeros b z) i = e_nth b i"
unit_disc_fix_conjugate_comp_moebius: fixes M :: "moebius" assumes "unit_disc_fix M" shows "unit_disc_fix_f (conjugate \<circ> moebius_pt M)"
Ord_add: fixes x :: "V"   and y :: "V" assumes "Ord x"   and "Ord y" shows "Ord (x + y)"
arr_cod_iff_arr: fixes f :: "'a" shows "Left.arr (Left.cod f) = Left.arr f"
word_le_exists': fixes x :: "'b word"   and y :: "'b word" assumes "x \<le> y" shows "\<exists>z. y = x + z \<and> uint x + uint z < 2 ^ LENGTH('b)"
surj_vimage_empty: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'c set" assumes "surj f" shows "(f -` A = {}) = (A = {})"
get_curr_win_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "fst (fst (get_curr_win () s1)) = fst (fst (get_curr_win () s2))"
safety_prop_INTER: fixes I :: "'c set"   and X :: "'c \<Rightarrow> 'd program set" assumes "\<And>i. i \<in> I \<Longrightarrow> safety_prop (X i)" shows "safety_prop (\<Inter> (X ` I))"
R: shows "\<nexists>E1 E2. closedin X E1 \<and> closedin X E2 \<and> topspace X \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {}"
euler_incr_fas_def: fixes X0 :: "floatarith list"   and h :: "floatarith"   and CX :: "floatarith list" shows "euler_incr_fas X0 h CX = map (euler_incr_fas_nth X0 h CX) [0..<length X0]"
gbinomial_int_pochhammer': fixes a :: "int"   and k :: "nat" shows "a gchoose k = pochhammer (a - int k + 1) k div fact k"
minimal_witness: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat" assumes "P k" obtains K :: "nat" where "K \<le> k" and "P K" and "\<And>k. k < K \<Longrightarrow> \<not> P k"
tminus_le_alt: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "a \<ominus> b = 0"
assms_1: shows "arr_mset arr1 l r = arr_mset arr2 l r"
comp_assoc: fixes h :: "'c"   and g :: "'c"   and f :: "'c" shows "(h \<cdot>\<^sub>C g) \<cdot>\<^sub>C f = h \<cdot>\<^sub>C g \<cdot>\<^sub>C f"
bit_last_iff: fixes w :: "'b word" shows "bit w (LENGTH('b) - Suc 0) = (sint w < 0)"
2: shows "\<guillemotleft>(\<tau> \<star> \<mu>) \<star> \<nu> : (local.dom \<tau> \<star> local.dom \<mu>) \<star> local.dom \<nu> \<Rightarrow> (cod \<tau> \<star> cod \<mu>) \<star> cod \<nu>\<guillemotright>"
odd_of_distinct_in_odd_partitions: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p \<in> {p. p partitions n \<and> (\<forall>i. p i \<le> 1)}" shows "odd_of_distinct p \<in> {p. p partitions n \<and> (\<forall>i. p i \<noteq> 0 \<longrightarrow> odd i)}"
zip_update: fixes xs :: "'d list"   and i :: "nat"   and x :: "'d"   and ys :: "'e list"   and y :: "'e" shows "zip (xs[i := x]) (ys[i := y]) = (zip xs ys)[i := (x, y)]"
fimage_is_fempty: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b fset" shows "(f |`| A = {||}) = (A = {||})"
abort_Bleast_def: fixes S :: "'g set"   and P :: "'g \<Rightarrow> bool" shows "abort_Bleast S P = (LEAST x. x \<in> S \<and> P x)"
set_unsat_simps_1: fixes I :: "'a list"   and s :: "('a, 'b) state" shows "\<B>\<^sub>i\<^sub>l (set_unsat I s) = \<B>\<^sub>i\<^sub>l s"
thetaParZOT_def: shows "thetaParZOT \<equiv> {(c1 | c2, d1 | d2) |c1 c2 d1 d2. c1 \<approx>01T d1 \<and> c2 \<approx>01T d2}"
nth_takefill: fixes m :: "nat"   and n :: "nat"   and fill :: "'b"   and l :: "'b list" assumes "m < n" shows "takefill fill n l ! m = (if m < length l then l ! m else fill)"
cases: fixes x :: "'a ifex list" obtains "x = []"   | v :: "'a" and uu_ :: "'a ifex" and uv_ :: "'a ifex" and r :: "'a ifex list" where "x = IF v uu_ uv_ # r__"   | r :: "'a ifex list" where "x = Trueif # ra__"   | r :: "'a ifex list" where "x = Falseif # r"
the_g: shows "(THE g. \<forall>a. f (cart_basis_Rn $ j) $ a = (\<Sum>x\<in>UNIV. g x * cart_basis_Rm $ x $ a)) = g"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (x \<le> y \<and> y \<le> x)"
deflation: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and y :: "'e" assumes "galois_connection G"   and "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y) \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y"
card_solovay_strassen_liars_composite: fixes n :: "nat" assumes "\<not> prime n"   and "2 < n"   and "odd n" shows "card {a \<in> {2..<n}. solovay_strassen_witness n a} < (n - 2) div 2"
eval_primfun_altdef: fixes f :: "primfun \<times> real"   and x :: "real" shows "eval_primfun f x = eval_primfun' (fst f) x powr snd f"
simps_1: fixes uu :: "('a \<times> 'a) set"   and ys :: "'a list" shows "eval_sorter uu (Return ys) = ys"
I'_def: shows "I' \<equiv> Inr (positions_in_row (x @- sconst (z (Suc n))) 0) # I"
integral_sum: fixes I :: "'a set"   and M :: "'b measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "LINT x|M. (\<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
node_connectTE': fixes ii :: "nat"   and s :: "'c"   and R :: "nat set"   and i :: "nat"   and i' :: "nat"   and s' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(NodeS ii s R, connect(i, i'), s') \<in> node_sos S" obtains "i = ii" and "s' = NodeS ii s (insert i' R)"   | "i' = ii" and "s' = NodeS ii s (insert i R)"   | "s' = NodeS ii s R" and "ii \<noteq> i" and "ii \<noteq> i'"
enum_all_UNIV: fixes P :: "'a \<Rightarrow> bool" shows "enum_all P = Ball UNIV P"
simps_13: fixes f11 :: "'a"   and f12 :: "X \<Rightarrow> 'a \<Rightarrow> 'a"   and f13 :: "Y \<Rightarrow> 'b \<Rightarrow> 'a"   and f21 :: "X \<Rightarrow> 'a \<Rightarrow> 'b"   and f22 :: "Y \<Rightarrow> 'b \<Rightarrow> 'b"   and f23 :: "'b"   and x12 :: "X" shows "rec_X f11 f12 f13 f21 f22 f23 (X.B x12) = f12 x12 (rec_X f11 f12 f13 f21 f22 f23 x12)"
row_transpose: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d" shows "Square_Matrix.row (Square_Matrix.transpose M) i = col M i"
ex_map_conv: fixes ys :: "'e list"   and f :: "'d \<Rightarrow> 'e" shows "(\<exists>xs. ys = map f xs) = (\<forall>y\<in>set ys. \<exists>x. y = f x)"
backflows_split: shows "backflows (filternew_flows_state \<T>) \<union> (backflows (flows_state \<T>) - backflows (filternew_flows_state \<T>)) = backflows (flows_state \<T>)"
Suc: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "inverse (disc_rfr_proc r n w) * X n w = det_discount r n * X n w"
ide_mkIde: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>"   and "A \<subseteq> Collect (category.terminal S)"   and "|A| <o \<AA>" shows "partial_magma.ide S (set_category_given_img.mkIde S img \<AA> A)"
elims_2: fixes x :: "('c, 'd) graph"   and xa :: "'c"   and xb :: "('c \<times> 'd \<times> 'c) list"   and xc :: "'c" assumes "is_path_undir x xa xb xc" obtains G :: "('c, 'd) graph" and v :: "'c" and v' :: "'c" where "x = Ga__" and "xa = v__" and "xb = []" and "xc = v'__" and "v__ = v'__ \<and> v'__ \<in> nodes Ga__"   | G :: "('c, 'd) graph" and v :: "'c" and v1 :: "'c" and w :: "'d" and v2 :: "'c" and p :: "('c \<times> 'd \<times> 'c) list" and v' :: "'c" where "x = G" and "xa = v" and "xb = (v1, w, v2) # p" and "xc = v'" and "v = v1 \<and> ((v1, w, v2) \<in> edges G \<or> (v2, w, v1) \<in> edges G) \<and> is_path_undir G v2 p v'"
pr_weak'_def: fixes f :: "'f \<times> nat"   and g :: "'f \<times> nat" shows "kbo'.pr_weak' f g = (f \<notin> kbo'.lConsts \<and> pr_weak' f g \<or> g \<in> kbo'.lConsts)"
card_d: fixes k :: "nat" shows "card (d k) = Suc k"
l_set_child_nodes_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_child_nodes_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs \<equiv> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_set_child_nodes type_wf set_child_nodes set_child_nodes_locs \<and> l_set_child_nodes_get_child_nodes_axioms type_wf known_ptr get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs"
prems_2: shows "r \<in> redexes (Catch c\<^sub>1 c\<^sub>2)"
maxsimpchain_def: fixes xs :: "'a set list" shows "domain.gallery xs \<equiv> Ball (set xs) domain.chamber \<and> adjacentchain xs"
sym_reachableI_in_awalk: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c"   and w1 :: "'c"   and w2 :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "w1 \<in> set (pre_digraph.awalk_verts G u p)"   and "w2 \<in> set (pre_digraph.awalk_verts G u p)" shows "w1 \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> w2"
err_semilat_JType_esl: fixes wf_mb :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" assumes "wf_prog wf_mb P" shows "err_semilat (SemiType.esl P)"
supi_def: fixes x :: "'a"   and y :: "'a" shows "supi x y = Supi {x, y}"
hyps: fixes p :: "'a poly" assumes "degree p < degree pa__" shows "Q p"
prob_eq_1: fixes M :: "'b measure"   and A :: "'b set" assumes "prob_space M"   and "A \<in> sets M" shows "(Sigma_Algebra.measure M A = 1) = (AE x in M. x \<in> A)"
p'_stop_zero: shows "p' t_stop = (if a = 0 then v else 0)"
map_map_rexp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and r :: "'d rexp" shows "map_rexp f (map_rexp g r) = map_rexp (\<lambda>r. f (g r)) r"
Abs_downset_inject: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> range \<down>"   and "y \<in> range \<down>" shows "(Abs_downset x = Abs_downset y) = (x = y)"
cod_cod: fixes f :: "'a" assumes "Right.arr f" shows "Right.cod (Right.cod f) = Right.cod f"
fds_ln_const: fixes l :: "'b"   and c :: "'b" shows "fds_ln l (fds_const c) = fds_const l"
wf_lex_prod: fixes ra :: "('a \<times> 'a) set"   and rb :: "('b \<times> 'b) set" assumes "wf ra"   and "wf rb" shows "wf (ra <*lex*> rb)"
prv_dsj_cases: fixes \<phi>1 :: "fm"   and \<phi>2 :: "fm"   and \<chi> :: "fm" assumes "\<phi>1 \<in> UNIV"   and "\<phi>2 \<in> UNIV"   and "\<chi> \<in> UNIV"   and "{} \<turnstile> \<phi>1 OR \<phi>2"   and "{} \<turnstile> \<phi>1 IMP \<chi>"   and "{} \<turnstile> \<phi>2 IMP \<chi>" shows "{} \<turnstile> \<chi>"
diverg: fixes t :: "nat" assumes "t\<^sub>0 \<le> t" shows "local.prefixes t j \<up>"
in_square_lemma: fixes x :: "nat"   and D :: "nat"   and y :: "nat"   and C :: "nat" assumes "x < D"   and "y < C" shows "x * C + y < D * C"
proj_iff: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "equivP P \<phi>"   and "P x"   and "P y" shows "(proj \<phi> x = proj \<phi> y) = \<phi> x y"
f_expand_take_mult_Suc: fixes n :: "nat"   and xs :: "'b message_af list"   and i :: "nat"   and k :: "nat" assumes "n < length xs"   and "i < k" shows "xs \<odot> k \<down> (n * k + Suc i) = xs \<down> n \<odot> k @ xs ! n # NoMsg\<^bsup>i\<^esup>"
simps_2: fixes p :: "real mpoly" shows "aNeg (Eq p) = Neq p"
minus_eq: fixes G :: "('h, 'i) ring_scheme"   and x :: "'h"   and y :: "'h" assumes "abelian_group G" shows "x \<ominus>\<^bsub>G\<^esub> y = x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> y"
constant_rep_point_not_0: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and design_rep_number :: "int"   and x :: "'b" assumes "constant_rep_design point_set block_collection design_rep_number"   and "x \<in> point_set" shows "block_collection rep x \<noteq> 0"
det_identical_rows: fixes i :: "'c"   and j :: "'c"   and A :: "(('d, 'c) vec, 'c) vec" assumes "i \<noteq> j"   and "Finite_Cartesian_Product.row i A = Finite_Cartesian_Product.row j A" shows "Determinants.det A = (0::'d)"
simps_2: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and a :: "'b" shows "applied_rule C a [] = \<bottom>"
M3: fixes p :: "'a" assumes "\<I> p" shows "\<exists>q r. q \<parallel> p \<and> p \<parallel> r"
distinct_disc_12: fixes event :: "('a, 'b, 'c) event" assumes "isSnapshot event" shows "\<not> isSend event"
or_sem: fixes I :: "('b, 'c, 'a) interp"   and \<phi> :: "('b, 'c, 'a) formula"   and \<psi> :: "('b, 'c, 'a) formula" shows "fml_sem I (\<phi> || \<psi>) = fml_sem I \<phi> \<union> fml_sem I \<psi>"
swap_simps_3: fixes a :: "name"   and c :: "name"   and b :: "name" assumes "a \<noteq> c"   and "b \<noteq> c" shows "[(a, b)] \<bullet> c = c"
zmod_numeral_Bit0: fixes v :: "num"   and w :: "num" shows "numeral (num.Bit0 v) mod numeral (num.Bit0 w) = 2 * (numeral v mod numeral w)"
permute_fset_2: fixes p :: "perm"   and x :: "'a"   and S :: "'a fset" shows "p \<bullet> finsert x S = finsert (p \<bullet> x) (p \<bullet> S)"
product_lists_in_set_nth: fixes xs :: "'c list"   and ys :: "'c list list" assumes "xs \<in> set (product_lists ys)" shows "\<forall>i<length ys. xs ! i \<in> set (ys ! i)"
cong_3_swap: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'" shows "Tarski_neutral_dimensionless.Cong3 Cong B A C B' A' C'"
Diff_insert2: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "A - insert a B = A - {a} - B"
obtain_elem_frontier: fixes M :: "'a zmultiset"   and t :: "'a" assumes "0 < zcount M t" obtains s :: "'a" where "s \<le> t \<and> s \<in>\<^sub>A frontier M"
card_equiv_class_restricted_same_size: fixes A :: "'a set"   and R :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and k :: "nat" assumes "equiv A R"   and "P respects R"   and "\<And>F. F \<in> {x \<in> A. P x} // R \<Longrightarrow> card F = k" shows "card {x \<in> A. P x} = k * card ({x \<in> A. P x} // R)"
that: fixes \<alpha> :: "nat \<Rightarrow> int" assumes "f \<bullet> \<alpha> = \<zero>"   and "\<alpha> \<in> carrier Zp"   and "val_Zp (f' \<bullet> a) < val_Zp (a \<ominus> \<alpha>)"   and "val_Zp (a \<ominus> \<alpha>) = val_Zp (local.divide (f \<bullet> a) (f' \<bullet> a))"   and "val_Zp (f' \<bullet> \<alpha>) = val_Zp (f' \<bullet> a)" shows "thesis"
inverse_matrix_def: fixes A :: "'b matrix"   and X :: "'b matrix" shows "inverse_matrix A X \<equiv> right_inverse_matrix A X \<and> left_inverse_matrix A X"
simps: fixes s :: "name"   and x :: "tm"   and x' :: "tm"   and k :: "name" assumes "atom s \<sharp> (x, x', k)"   and "atom k \<sharp> (x, x')" shows "HRP x x' = SyntaxN.Ex s (SyntaxN.Ex k (SeqHRP x x' (Var s) (Var k)))"
ga_gen_def: fixes gc :: "('c, 'ls, 'gs) pid_global_config" shows "ga_gen gc \<equiv> {(pid, c, a, c'). cstep c a c' \<and> pid_valid gc pid \<and> c = cmd_of_pid gc pid \<and> en (fs_of_pid gc pid) a}"
rel_def: shows "rel \<equiv> \<Union> (rels ` {pointsTo. wf {(l, l'). pointsTo l' = Link l}})"
ordinal_utility_left: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and u :: "'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "ordinal_utility carrier relation u"   and "x \<succeq>[relation] y" shows "u y \<le> u x"
inf_from_subs: shows "no_labels.Inf_between (fst ` active_subset N) {C0} \<subseteq> no_labels.Red_I_\<G> (fst ` (N \<union> {(C0, active)} \<union> M))"
anti_deduction: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A IMP B" shows "insert A H \<turnstile> B"
with_bot_iff: fixes x :: "'a"   and A :: "'a set" shows "(Some x \<in> A\<^sub>\<bottom>) = (x \<in> A)"
phis'_def: shows "phis' \<equiv> \<lambda>g (n, v, m, def). if m = n \<and> n \<in> phiDefNodes g v \<and> v \<in> vars g \<and> def = PhiDef then Some (map (\<lambda>m. lookupDef g m v) (predecessors g n)) else None"
inv_solve_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = inv b \<otimes> c) = (c = b \<otimes> a)"
selectlike_equal_cocond_corule: fixes \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set"   and X :: "(variable \<Rightarrow> real) set"   and Y :: "(variable \<Rightarrow> real) set" assumes "\<And>\<mu>. Uvariation \<mu> \<nu> V \<Longrightarrow> (\<mu> \<in> X) = (\<mu> \<in> Y)" shows "selectlike X \<nu> (- V) = selectlike Y \<nu> (- V)"
NFOR_def: shows "NFOR = nFOR \<circ> sorted_list_of_set"
rem_implicit_pres_wf_operator: fixes op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "consistent_pres_op op"   and "ast_problem.wf_operator prob op" shows "ast_problem.wf_operator (rem_implicit_pres_ops prob) (rem_implicit_pres op)"
mk_mop_rl1: fixes mf :: "'e \<Rightarrow> 'f nres"   and P :: "'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "\<And>x. mf x \<equiv> ASSERT (P x) \<bind> (\<lambda>_. RETURN (f x))" shows "(RETURN \<circ> f, mf) \<in> Id \<rightarrow> \<langle>Id\<rangle>nres_rel"
ownership_distinct_axioms: shows "ownership_distinct ts\<^sub>s\<^sub>b"
FactGroup_universal: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and N :: "'c set" assumes "h \<in> hom G H"   and "N \<lhd> G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G; N #>\<^bsub>G\<^esub> x = N #>\<^bsub>G\<^esub> y\<rbrakk> \<Longrightarrow> h x = h y" obtains g :: "'c set \<Rightarrow> 'd" where "g \<in> hom (G Mod N) H" and "\<And>x. x \<in> carrier G \<Longrightarrow> g (N #>\<^bsub>G\<^esub> x) = h x"
nth_list_update: fixes i :: "nat"   and xs :: "'c list"   and x :: "'c"   and j :: "nat" assumes "i < length xs" shows "xs[i := x] ! j = (if i = j then x else xs ! j)"
g2: shows "map (\<lambda>x. (x, f x)) (filter (\<lambda>y. f x \<noteq> f y) xs) = filter (\<lambda>y. f x \<noteq> snd y) (map (\<lambda>x. (x, f x)) xs)"
cases: fixes a :: "fm" assumes "a \<in> boolean_axioms" obtains A :: "fm" where "a = A__ IMP A__"   | A :: "fm" and B :: "fm" where "a = Aa__ IMP Aa__ OR B__"   | A :: "fm" where "a = Ab__ OR Ab__ IMP Ab__"   | A :: "fm" and B :: "fm" and C :: "fm" where "a = Ac__ OR Ba__ OR C__ IMP (Ac__ OR Ba__) OR C__"   | C :: "fm" and A :: "fm" and B :: "fm" where "a = C OR A IMP (C IMP B) IMP A OR B"
simps_1: shows "are_all_operators_non_interfering [] = True"
push_bit_of_1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n one = power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n"
id_o: fixes g :: "'k \<Rightarrow> 'l" shows "id \<circ> g = g"
\<Delta>_in_hom_2: shows "\<guillemotleft>(\<mu> \<star> r\<^sub>0) \<cdot> \<rho> : r\<^sub>1 \<Rightarrow> s \<star> r\<^sub>0\<guillemotright>"
product_category_axioms: shows "product_category (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) (\<cdot>\<^sub>A)"
fundchamber_W_image_inj_on: shows "inj_on (\<lambda>w. w `\<rightarrow> C0) \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"
ZSkip: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "\<turnstile>\<^sub>Z {\<lambda>s. eSuc (P s)} SKIP {P}"
lower_on_lit: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list"   and L :: "'a literal"   and x :: "'a" assumes "lower_on \<sigma> \<eta> (vars_of_lit L)"   and "(Var x \<lhd> \<sigma>, Var x \<lhd> \<eta>) \<in> trm_ord"   and "x \<in> vars_of_lit L" shows "(subst_lit L \<sigma>, subst_lit L \<eta>) \<in> lit_ord"