hcju/isabelleps huggingface dataset:
w\<^sub>2: shows "if a \<in> alphabet t\<^sub>2 then if b \<in> alphabet t\<^sub>2 then weight (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>2 a + freq t\<^sub>2 b = weight t\<^sub>2 + w\<^sub>a + w\<^sub>b else weight (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>2 a = weight t\<^sub>2 + w\<^sub>b else if b \<in> alphabet t\<^sub>2 then weight (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b) + freq t\<^sub>2 b = weight t\<^sub>2 + w\<^sub>a else weight (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b) = weight t\<^sub>2"
map_fun_apply: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'd"   and x :: "'b" shows "map_fun f g h x = g (h (f x))"
interior_of_complement: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of (topspace X - S) = topspace X - X closure_of S"
id_nec3_1: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" shows "[x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P \<^bold>\<equiv> \<^bold>\<box>x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P in v]"
c2p_mono_p2c: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 \<le> n"   and "\<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n \<le> n'" shows "n \<le> \<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n'"
ide_map\<^sub>0_obj: fixes a :: "'a" assumes "C.obj a" shows "D.ide (\<sigma>.map\<^sub>0 a)"
fzn: fixes n :: "nat" shows "f (z n) = z (Suc n)"
bit_eqI: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "\<And>n. power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<Longrightarrow> bit a n = bit b n" shows "a = b"
f1: shows "\<not> ((fst instr = load_store_type STH \<or> fst instr = load_store_type STHA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s1)))) \<noteq> 0) \<and> \<not> ((fst instr = load_store_type STH \<or> fst instr = load_store_type STHA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) \<noteq> 0)"
alw_holds_pred_stream_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" shows "alw (holds P) xs = pred_stream P xs"
has_lock_lock_lock_unlock_lock_id: fixes l :: "('b \<times> nat) option"   and t :: "'b" assumes "has_lock l t" shows "lock_lock (unlock_lock l) t = l"
r_def: shows "r = (MAX i\<in>{i \<in> {..<N}. coeff p i \<noteq> 0}. - fpxs_val (coeff p i) / (rat_of_nat N - rat_of_nat i))"
compactE_image: fixes S :: "'a set"   and C :: "'b set"   and f :: "'b \<Rightarrow> 'a set" assumes "compact S"   and "\<And>T. T \<in> C \<Longrightarrow> open (f T)"   and "S \<subseteq> \<Union> (f ` C)" obtains C' :: "'b set" where "C' \<subseteq> C" and "finite C'" and "S \<subseteq> \<Union> (f ` C')"
inf_sup_aci_6: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.lattice inf less_eq less sup" shows "sup (sup x y) z = sup x (sup y z)"
run_asset_iii_ex: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "pred_asset_iii n (run_owner_iii n s) (run_asset_iii n s)"
greaterThanAtMost_def: fixes l :: "'a"   and u :: "'a" shows "{l<..u} \<equiv> {l<..} \<inter> {..u}"
pr_algo_lts_length_bound: fixes p :: "pr_operation list"   and f' :: "nat \<times> nat \<Rightarrow> 'capacity"   and l' :: "nat \<Rightarrow> nat" assumes "((f'__, l'__), p, f', l') \<in> trcl pr_algo_lts" shows "length p \<le> pr_algo_len_bound"
select_convs_2: fixes empty :: "'b"   and lookup :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'e" shows "MapOps.lookup \<lparr>MapOps.empty = empty, lookup = lookup, update = update, \<dots> = more\<rparr> = lookup"
is_valid_problem_sas_plus_then_5: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "dom (\<Psi>\<^sub>G\<^sub>+) \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"
DOM_in_dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "monoidal_language.Dom C (free_monoidal_category.rep C f) \<in> partial_magma.dom (free_monoidal_category.comp C) f"
insert_eqvt: fixes p :: "perm"   and x :: "'a"   and A :: "'a set" shows "p \<bullet> insert x A = insert (p \<bullet> x) (p \<bullet> A)"
r_s_def: shows "r_s \<equiv> SOME rs. recfn 1 rs \<and> Partial_Recursive.total rs \<and> s = (\<lambda>x. eval rs [x])"
zero_le: fixes x :: "'h multiset" shows "{#} \<subseteq># x"
preserves_hseq: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (H \<mu>) \<and> D.arr (H \<nu>) \<and> src\<^sub>D (H \<mu>) = trg\<^sub>D (H \<nu>)"
Bot_map: fixes B :: "'f \<times> 'l" assumes "B \<in> Bot_FL" shows "\<G>_F_L B \<subseteq> Bot_G"
update_convs_2: fixes Map' :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and Pol :: "('c \<times> 'c) set"   and Map :: "'b \<Rightarrow> 'c"   and Doms :: "'c set"   and List :: "'b list"   and ListOp :: "'b list option"   and Set :: "'b set"   and more :: "'d" shows "Map_update Map' \<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> = \<lparr>Pol = Pol, Map = Map' Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr>"
bij_swap_compose_bij: fixes p :: "'b \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "bij p" shows "bij (Fun.swap a b id \<circ> p)"
mat_tensor_prod_2: fixes v :: "complex Matrix.mat"   and w :: "complex Matrix.mat" assumes "state 1 v"   and "state 1 w" shows "v \<Otimes> w = Matrix.mat 4 1 (\<lambda>(i, j). if i = 0 then v $$ (0, 0) * w $$ (0, 0) else if i = 3 then v $$ (1, 0) * w $$ (1, 0) else if i = 1 then v $$ (0, 0) * w $$ (1, 0) else v $$ (1, 0) * w $$ (0, 0))"
cover_of_subset_axioms: shows "cover_of_subset X U index cover"
ide_char': fixes a :: "('obj, 'obj) concrete_category.arr" shows "ide a = (arr a \<and> (local.dom a = a \<or> cod a = a))"
le_arcsin_iff: fixes x :: "real"   and y :: "real" assumes "- 1 \<le> x"   and "x \<le> 1"   and "- pi / 2 \<le> y"   and "y \<le> pi / 2" shows "(y \<le> arcsin x) = (sin y \<le> x)"
real_polynomial_function_diff: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "real_polynomial_function f"   and "real_polynomial_function g" shows "real_polynomial_function (\<lambda>x. f x - g x)"
i_take_commute: fixes b :: "nat"   and a :: "nat"   and s :: "nat \<Rightarrow> 'b" shows "s \<Down> a \<down> b = s \<Down> b \<down> a"
TensorDiag_preserves_Diag_1: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag t"   and "Diag u" shows "Diag (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"
combine_options_simps_3: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" shows "combine_options f (Some a) (Some b) = Some (f a b)"
A72_Acc: shows "Acc level1 sA72 = {sA82, sA93}"
Wens: fixes X :: "'b set"   and F :: "'b program"   and B :: "'b set"   and act :: "('b \<times> 'b) set" assumes "X \<in> wens_set F B"   and "act \<in> Acts F" shows "wens F act X \<in> wens_set F B"
monoseq_iff: fixes X :: "nat \<Rightarrow> 'a" shows "monoseq X = (incseq X \<or> decseq X)"
simps_53: fixes 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 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f5 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f6 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f7 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x41 :: "('d, 'e) Term.term"   and x42 :: "('d, 'e) Term.term" shows "(case insert\<langle>x41,x42\<rangle> of send\<langle>x\<rangle> \<Rightarrow> f1 x | receive\<langle>x\<rangle> \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle> \<Rightarrow> f3 x xa xb | insert\<langle>x,xa\<rangle> \<Rightarrow> f4 x xa | delete\<langle>x,xa\<rangle> \<Rightarrow> f5 x xa | \<langle>x: xa \<in> xb\<rangle> \<Rightarrow> f6 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa \<or>\<notin>: xb\<rangle> \<Rightarrow> f7 x xa xb) = f4 x41 x42"
eq_for_lvar_def: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and t :: "(nat \<times> linear_poly) list"   and v :: "nat" assumes "EqForLVar eq_idx_for_lvar" shows "EqForLVar.eq_for_lvar eq_idx_for_lvar t v \<equiv> t ! eq_idx_for_lvar t v"
antisym_converse: fixes r :: "('b \<times> 'b) set" shows "antisym (r\<inverse>) = antisym r"
supp_fun_eqvt: fixes f :: "'a" assumes "eqvt f" shows "supp f = {}"
sq_mtx_col_def: shows "\<c>\<o>\<l> \<equiv> map_fun id (map_fun to_vec id) column"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "blinfun_cblinfun_eq x xa = ((*\<^sub>v) x = (*\<^sub>V) xa)"
insertCI: fixes a :: "'a"   and B :: "'a set"   and b :: "'a" assumes "a \<notin> B \<Longrightarrow> a = b" shows "s2p (insert b B) a"
simps_4: fixes m1 :: "'b match_expr"   and m2 :: "'b match_expr" shows "normalized_nnf_match (MatchAnd m1 m2) = (normalized_nnf_match m1 \<and> normalized_nnf_match m2)"
axioms_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "set_valued_functor C S F" shows "category C"
SigmaI: fixes a :: "'e"   and A :: "'e set"   and b :: "'f"   and B :: "'e \<Rightarrow> 'f set" assumes "a \<in> A"   and "b \<in> B a" shows "(a, b) \<in> Sigma A B"
i: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "vc C Q" shows "\<turnstile>\<^sub>2 {pre C Q} strip C {Q}"
ln_complex_def: shows "ln \<equiv> \<lambda>z. THE w. exp w = z \<and> - pi < Im w \<and> Im w \<le> pi"
xor_self: fixes x :: "'a" shows "x \<oplus> x = \<zero>"
allVars_def: fixes g :: "'g" shows "allVars g \<equiv> \<Union>n\<in>set (\<alpha>n g). allDefs g n \<union> allUses g n"
sg7: shows "cycleLength (nC i) = cycleLength (nC j)"
alpha_qFresh_qSwap_compose: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and x :: "'var" assumes "qGood X"   and "qFresh zs y X"   and "qFresh zs z X" shows "X #[[y \<and> x]]_zs #[[z \<and> y]]_zs #= X #[[z \<and> x]]_zs"
continuous_at_Sup_mono: fixes f :: "'b \<Rightarrow> 'd"   and S :: "'b set" assumes "mono f"   and "continuous (at_left (Sup S)) f"   and "S \<noteq> {}"   and "bdd_above S" shows "f (Sup S) = Sup (f ` S)"
\<Phi>o_preserves_ide: fixes a :: "'s" assumes "S.ide a" shows "S'.ide (\<Phi>o a)"
that: fixes t' :: "'state" assumes "(d, t) \<rightarrow>*t t'"   and "s' \<approx> t'" shows "thesis"
Cons_2: fixes ub :: "real" assumes "\<forall>p\<in>set (p # ps). \<forall>x. poly p x = 0 \<longrightarrow> x < ub"   and "\<forall>x\<ge>ub. \<forall>p\<in>set (p # ps). sgn (poly p x) = sgn_pos_inf p"   and "a < ub" shows "thesis"
seq_char: fixes g :: "'d"   and f :: "'d" shows "D'.seq g f = (D'.arr f \<and> D'.arr g \<and> D.seq g f)"
not_empty_card_gr0_conv: fixes A :: "'a set" assumes "finite A" shows "(A \<noteq> {}) = (0 < card A)"
set_cases: fixes e :: "'c"   and a :: "'c option" assumes "e \<in> set_option a" obtains "a = Some e"
gen_model_world_refl: fixes w :: "'b"   and M :: "('c, 'd, 'b) KripkeStructure" assumes "w \<in> worlds M" shows "w \<in> worlds (gen_model M w)"
invar_imp_correct: fixes res :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option" assumes "dinvar ({}, res)" shows "is_shortest_path_map v0 res"
impr_def: fixes a :: "'b G"   and b :: "'b G" shows "a r\<rightarrow> b \<equiv> Abs_G ((u - Rep_G a + Rep_G b) \<sqinter> u)"
geq: fixes x :: "('a \<times> 'b) \<times> real" assumes "x \<in> C" shows "g x = (fst \<circ> f) x"
col_2: fixes j :: "nat" assumes "\<not> j < nc1"   and "j < nc1 + nc2" shows "col (four_block_mat A2 B2 C2 D2) j = col B2 (j - nc1) @\<^sub>v col D2 (j - nc1)"
coboundedI2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c \<le> b" shows "c \<le> max a b"
zero_less_mult_pos2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero (times b a)"   and "less zero a" shows "less zero b"
Union_Un_distrib: fixes A :: "'d set set"   and B :: "'d set set" shows "\<Union> (A \<union> B) = \<Union> A \<union> \<Union> B"
simps_2: fixes v :: "'d \<Rightarrow> 'g"   and Vc :: "'g set \<times> ('d, 'e, 'f) com"   and vs :: "('g set \<times> ('d, 'e, 'f) com) list" shows "switch v (Vc # vs) = Cond {s. v s \<in> fst Vc} (snd Vc) (switch v vs)"
abs_eq: fixes x :: "complex" shows "of_complex_hcoords x = Abs_complex_homo_coords (of_complex_cvec x)"
strips_state_to_state_dom_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<times> 'd \<Rightarrow> bool option" assumes "is_valid_problem_sas_plus \<Psi>" shows "dom (\<phi>\<^sub>S\<inverse> \<Psi> s) = (\<Union>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). {uu_. \<exists>a. uu_ = v \<and> a \<in> \<R>\<^sub>+ \<Psi> v \<and> s (v, a) = Some True})"
mem_existence_ivl_subset: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "t \<in> T"
Cons_1: assumes "distinct (map fst xs)" shows "lookup_pair (sort_oalist xs) = lookup_dflt xs"
build_eq: fixes a\<^sub>1 :: "'a"   and w\<^sub>1 :: "nat \<Rightarrow> 'a"   and a\<^sub>2 :: "'a"   and w\<^sub>2 :: "nat \<Rightarrow> 'a" shows "(a\<^sub>1 ## w\<^sub>1 = a\<^sub>2 ## w\<^sub>2) = (a\<^sub>1 = a\<^sub>2 \<and> w\<^sub>1 = w\<^sub>2)"
MakeNT_dom: fixes f :: "('d, 'g, 'e, 'h, 'f, 'i) NatTrans" shows "NTCatDom f = NTCatDom (MakeNT f)"
induct: fixes x :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>"   and xa :: "('d, 'e, 'f) gpv"   and P :: "('d, 'e, 'f) gpv \<Rightarrow> bool" assumes "x \<in> sub_gpvs \<I> xa"   and "\<And>out c gpv input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out; x = c input\<rbrakk> \<Longrightarrow> P gpv"   and "\<And>out c gpv input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out; x \<in> sub_gpvs \<I> (c input); P (c input)\<rbrakk> \<Longrightarrow> P gpv" shows "P xa"
out_subseqs1_set_as_out_subseqs2_set: fixes i :: "nat"   and v :: "'b"   and prevs :: "(nat \<times> 'b) list" shows "out_subseqs1_set ((i, v) # prevs) = {f head v} \<union> out_subseqs1_set prevs \<union> out_subseqs2_set v [0..<i]"
t0_space: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
subring_props_5: fixes h :: "'a" assumes "h \<in> K" shows "\<ominus> h \<in> K"
sta_p: shows "\<forall>p<length ss. stable r step ss p"
less_eq_char_def: fixes c :: "char"   and d :: "char" shows "less_eq_char c d = (of_char c \<le> of_char d)"
advantage_def: fixes \<A> :: "(bool, 'a, 'a) gpv" shows "advantage \<A> = \<bar>spmf (prp_game \<A>) True - spmf (prf_game \<A>) True\<bar>"
basis_in_max_indep_in: fixes \<E> :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> insert x__ X"   and "X_insert.basis_in \<E> X"   and "x \<in> \<E> - X" shows "\<not> X_insert.indep_in \<E> (insert x X)"
vector_derivative_within_cbox: fixes a :: "real"   and b :: "real"   and x :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "'a" assumes "a < b"   and "x \<in> cbox a b"   and "(f has_vector_derivative f') (at x within cbox a b)" shows "vector_derivative f (at x within cbox a b) = f'"
img_lub_def: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and Lub :: "'c set \<Rightarrow> 'd" shows "img_lub f g Lub = (\<lambda>A. g (Lub (f ` A)))"
borel_measurable_implies_simple_function_sequence': fixes u :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "u \<in> borel_measurable M" obtains f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal" where "\<And>i. simple_function M (f i)" and "incseq f" and "\<And>i x. f i x < top" and "\<And>x. (SUP i. f i x) = u x"
eval_quot_fm_ignore: fixes e :: "(name, hf) finfun"   and A :: "fm"   and e' :: "(name, hf) finfun" shows "\<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e'"
"3_1": shows "real (count_derangements (Suc n)) = fact (Suc n) * (\<Sum>k = 0..Suc n. (- 1) ^ k / fact k)"
nu_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "has_greatest_fixpoint f" shows "\<nu> f = f (\<nu> f)"
fsps_def: fixes R :: "('j, 'k) Ring_scheme"   and M :: "('h, 'j, 'l) Module_scheme"   and N :: "('i, 'j, 'm) Module_scheme"   and f :: "'h \<Rightarrow> 'i"   and H :: "'h set" shows "fsps R M N f H = {(A, g) |A g. fsp R M N f H A g}"
Exit_Use_empty: fixes V :: "'c" assumes "V \<in> Use Exit" shows "False"
slice_Nil: fixes "begin" :: "nat"   and "end" :: "nat" shows "Misc.slice begin end [] = []"
stable_on_allocation: fixes ds :: "'d set"   and X :: "'x set" assumes "stable_on ds X" shows "inj_on Xd X"
WHILEI_lfp_conv: fixes m :: "'b nres"   and I :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b nres" shows "m \<bind> WHILE\<^bsup>I\<^esup> c f = lfp (WHILEI_lfp_body I m c f) \<bind> (\<lambda>s. ASSERT (I s) \<bind> (\<lambda>_. ASSUME (\<not> c s) \<bind> (\<lambda>_. RETURN s)))"
preserves_unity: shows "D.isomorphic \<I>\<^sub>D (F \<I>\<^sub>C)"
extends_to_elementary_category_with_pullbacks: shows "elementary_category_with_pullbacks (\<cdot>) prj0 prj1"
inverse_of_nat_le: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m"   and "n \<noteq> 0" shows "(1::'b) / of_nat m \<le> (1::'b) / of_nat n"
fds_ind_True: shows "fds_ind (\<lambda>_. True) = fds_zeta"
gen_division_def: fixes s :: "'a set"   and S :: "'a set set" shows "gen_division s S \<equiv> finite S \<and> \<Union> S = s \<and> pairwise (\<lambda>X Y. negligible (X \<inter> Y)) S"
th2: shows "(\<Sum>f\<in>{p. p permutes {0..<n}}. det (mat\<^sub>r n n (\<lambda>i. A $$ (i, f i) \<cdot>\<^sub>v row B (f i)))) = det A * det B"
dual_equality: fixes x :: "'b"   and y :: "'b" shows "(dual x = dual y) = (x = y)"
thenOrdering_simps_2: fixes y :: "Ordering" shows "thenOrdering\<cdot>EQ\<cdot>y = y"
le_plus_1: fixes s :: "'sum"   and s' :: "'sum" shows "s \<le> followed_by s s'"
axioms_1: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "valid_ownership \<S> ts" shows "outstanding_non_volatile_refs_owned_or_read_only \<S> ts"
dom_simp_4: shows "partial_magma.dom (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) parallel_pair.j1 = quiver.Zero"
abs_triangle_ineq: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (plus a b)) (plus (abs a) (abs b))"
E_ss_VxV: fixes c :: "nat \<times> nat \<Rightarrow> 'b" shows "Graph.E c \<subseteq> Graph.V c \<times> Graph.V c"
degree_Poly_berlekamp_le: fixes i :: "nat"   and u :: "'b mod_ring poly" assumes "i < degree u" shows "degree (Poly (list_of_vec (row (berlekamp_mat u) i))) < degree u"
bij: shows "bij_betw (\<lambda>x. poly q1 x / poly q2 x) A B"
basic_just_sp_def: fixes G :: "('a, 'b) pre_digraph"   and dist :: "'a \<Rightarrow> ereal"   and c :: "'b \<Rightarrow> real"   and s :: "'a"   and num :: "'a \<Rightarrow> enat" shows "basic_just_sp G dist c s num \<equiv> basic_sp G dist c s \<and> basic_just_sp_axioms G dist c s num"
basis_indep: fixes X :: "'a set" assumes "\<E>.basis X" shows "indep_in carrier X"
satPB_Un: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi>1 :: "(('fsym, 'tp) efsym, 'psym) lit list set"   and \<Phi>2 :: "(('fsym, 'tp) efsym, 'psym) lit list set" shows "satPB \<xi> (\<Phi>1 \<union> \<Phi>2) = (satPB \<xi> \<Phi>1 \<and> satPB \<xi> \<Phi>2)"
inv_iso_block_img: shows "image_mset ((`) (inv_into \<V> \<pi>)) \<B>' = \<B>"
cardSuc_least_aux: fixes r :: "('c \<times> 'c) set"   and r' :: "('c set \<times> 'c set) set" assumes "Card_order r"   and "Card_order r'"   and "r <o r'" shows "cardSuc r \<le>o r'"
\<R>_typed_step: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma> {c\<^sub>1} \<Gamma>'"   and "mds_consistent mds \<Gamma>"   and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"   and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>" shows "\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>1, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>"
continuous_on_snd: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b \<times> 'c" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. snd (f x))"
indep_def: fixes M :: "'a measure"   and A :: "'a set"   and B :: "'a set" assumes "prob_space M" shows "prob_space.indep M A B = (Sigma_Algebra.measure M (A \<inter> B) = Sigma_Algebra.measure M A * Sigma_Algebra.measure M B)"
sgn_1_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(sgn a = one) = less zero a"
Inf_between_alt: fixes N :: "'a literal multiset set"   and M :: "'a literal multiset set" shows "F.Inf_between N M = {\<iota> \<in> F_Inf. \<iota> \<in> F.Inf_from (N \<union> M) \<and> set (prems_of \<iota>) \<inter> M \<noteq> {}}"