hcju/isabelleps huggingface dataset:
sortedConc: fixes a :: "('a, 'b) Combinators"   and p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "FWNormalisationCore.sorted (a # p) l" shows "FWNormalisationCore.sorted p l"
hyps: fixes ivl :: "float interval" assumes "approx_tse_form' prec t f s ivl cmp"   and "x \<in>\<^sub>r ivl" shows "\<exists>ivl' ivly. x \<in>\<^sub>r ivl' \<and> ivl' \<le> ivl \<and> cmp ivly \<and> approx_tse prec 0 t (mid ivl') 1 f [Some ivl'] = Some ivly"
emeasure_notin_sets: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<notin> sets M" shows "emeasure M A = 0"
20: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) \<le> - (x \<sqinter> - - y)"
the_def: fixes option :: "'d option" shows "the option = (case option of Some x2 \<Rightarrow> x2)"
lincomb_isolate_2: fixes A :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a"   and v :: "'a vec" assumes "finite A"   and "A \<subseteq> carrier_vec n"   and "a \<in> A \<rightarrow> UNIV"   and "v \<in> A"   and "a v \<noteq> (0::'a)"   and "lincomb a A = 0\<^sub>v n" shows "v \<in> local.span (A - {v})"
scale_zero_right: fixes a :: "'a" shows "a \<cdot> (0::'a) = (0::'a)"
eqD_13: fixes y :: "cmsg"   and N :: "nat" assumes "eq y (cNumber N)" shows "y = cNumber N"
proj2_addpreRT: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip' :: "nat"   and npre :: "nat set" assumes "ip \<in> kD rt"   and "ip' \<in> kD rt" shows "\<pi>\<^sub>2 (the (the (addpreRT rt ip' npre) ip)) = \<pi>\<^sub>2 (the (rt ip))"
composition_closed: fixes a :: "('a set \<times> 'b) set"   and b :: "('a set \<times> 'b) set" assumes "a \<in> stk1.carrier_stalk"   and "b \<in> stk1.carrier_stalk" shows "stk1.mult_stalk a b \<in> stk1.carrier_stalk"
DJredlove: fixes p :: "fm"   and bs :: "real list" assumes "qfree p" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = Ifm bs (DJ redlove p) \<and> qfree (DJ redlove p)"
ok_add_reverse: fixes e :: "'b environment"   and a :: "char list"   and X :: "'b"   and b :: "char list"   and Y :: "'b" assumes "ok (e\<lparr>a:X\<rparr>\<lparr>b:Y\<rparr>)" shows "e\<lparr>b:Y\<rparr>\<lparr>a:X\<rparr> = e\<lparr>a:X\<rparr>\<lparr>b:Y\<rparr>"
hom_1_iff: fixes x :: "int" shows "(of_int x = (1::'b)) = (x = 1)"
LCons: fixes x :: "'d"   and xs :: "'d llist" assumes "x \<notin> lset xs"   and "ldistinct xs" shows "ldistinct (LCons x xs)"
dghm_HomDom: shows "\<FF>\<lparr>HomDom\<rparr> = op_dg \<AA>"
left_idem: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "semilattice f" shows "f a (f a b) = f a b"
assms: shows "\<exists>U. open U \<and> U \<inter> A = {x}"
true_eqvt: fixes pi :: "('a \<times> 'a) list" shows "pi \<bullet> True = True"
differentiable_on_polynomial_function: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "polynomial_function f" shows "f differentiable_on S"
dw_closedI: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; r a b\<rbrakk> \<Longrightarrow> b \<in> A" shows "relation.dw_closed r A"
poly2_monom: fixes a :: "'c poly"   and n :: "nat"   and x :: "'c"   and y :: "'c" shows "poly2 (monom a n) x y = poly a x * y ^ n"
unfold2: fixes s1 :: "'e \<Rightarrow> ('d, 'e, 'f) F1"   and s2 :: "'f \<Rightarrow> ('d, 'e, 'f) F2"   and z :: "'f" shows "dtor2 (unfold2 s1 s2 z) = F2map id (unfold1 s1 s2) (unfold2 s1 s2) (s2 z)"
simps_2: fixes xs :: "((int \<times> int) set set, int) Combinators list" shows "wellformed_policy2 (DenyAll # xs) = wellformed_policy2 xs"
winding_number_paths_linear_eq: fixes g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path g"   and "path h"   and "pathstart h = pathstart g"   and "pathfinish h = pathfinish g"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)" shows "winding_number h z = winding_number g z"
a: shows "limsup (\<lambda>n. \<integral>\<^sup>+ x. ennreal (norm (f x)) + ennreal (norm (F n x)) \<partial>M) \<le> 2 * \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M"
encPf_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and enc :: "'c \<Rightarrow> 'b"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and P :: "'c"   and isTrue :: "'c \<Rightarrow> bool"   and Pf :: "'c" assumes "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf" shows "Minimal_Truth_Soundness_HBL1iff_Compl_Pf.encPf \<equiv> id"
sample_var_lookup_other: fixes x :: "'v"   and y :: "'v"   and f :: "int \<Rightarrow> int \<Rightarrow> 'm" assumes "x \<noteq> y" shows "bind (memo (sample_var X) y) (\<lambda>i. bind (lookup x) (f i)) = bind (lookup x) (\<lambda>j. bind (memo (sample_var X) y) (\<lambda>i. f i j))"
elims_2: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and x :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "('b \<times> 'c) list \<times> 'a"   and xb :: "('b \<times> 'd) list \<times> 'a" assumes "oalist_raw.prod_ord_raw rep_key_order x xa xb" obtains f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" and xs :: "('b \<times> 'c) list" and ox :: "'a" and ys :: "('b \<times> 'd) list \<times> 'a" where "x = f" and "xa = (xs, ox)" and "xb = ys" and "comparator.prod_ord_pair (key_compare (rep_key_order ox)) f xs (oalist_raw.sort_oalist_aux rep_key_order ox ys)"
compact_UN: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> compact (B x)" shows "compact (\<Union> (B ` A))"
Ext_foo_overrides_Base_foo: shows "tprg \<turnstile> (Ext, Ext_foo) overrides (Base, Base_foo)"
is_functor: shows "functor (\<cdot>\<^sub>J) (\<cdot>) A.map"
fw_Suc: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i' \<le> n"   and "j' \<le> n" shows "fw m n (Suc k) i' j' i j \<le> fw m n k n n i j"
assms_3: shows "r \<in> hyp2 \<union> S"
solve_depressed_quartic_complex: fixes x :: "complex"   and p :: "complex"   and q :: "complex"   and r :: "complex" shows "(x \<in> set (solve_depressed_quartic_complex p q r)) = (x ^ 4 + p * x\<^sup>2 + q * x + r = 0)"
split_list_first_prop_iff: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" shows "(\<exists>x\<in>set xs. P x) = (\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>y\<in>set ys. \<not> P y))"
p_ifce: shows "ipassmt (Iface (p_oiface p)) = Some p_ips \<and> p_dst p \<in> ipcidr_union_set (set p_ips)"
composeES_def: fixes ES1 :: "'a ES_rec"   and ES2 :: "'a ES_rec" shows "ES1 \<parallel> ES2 \<equiv> \<lparr>E_ES = E\<^bsub>ES1\<^esub> \<union> E\<^bsub>ES2\<^esub>, I_ES = I\<^bsub>ES1\<^esub> - O\<^bsub>ES2\<^esub> \<union> (I\<^bsub>ES2\<^esub> - O\<^bsub>ES1\<^esub>), O_ES = O\<^bsub>ES1\<^esub> - I\<^bsub>ES2\<^esub> \<union> (O\<^bsub>ES2\<^esub> - I\<^bsub>ES1\<^esub>), Tr_ES = {\<tau>. \<tau> \<upharpoonleft> E\<^bsub>ES1\<^esub> \<in> Tr\<^bsub>ES1\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES2\<^esub> \<in> Tr\<^bsub>ES2\<^esub> \<and> set \<tau> \<subseteq> E\<^bsub>ES1\<^esub> \<union> E\<^bsub>ES2\<^esub>}\<rparr>"
alignUp_not_aligned_eq: fixes a :: "'b word"   and n :: "nat" assumes "\<not> is_aligned a n"   and "n < LENGTH('b)" shows "alignUp a n = (a div 2 ^ n + 1) * 2 ^ n"
mu_nu: fixes i :: "nat"   and j :: "nat" assumes "seq (i + i) = seq i"   and "mu \<le> j" shows "seq (j + i) = seq j"
elims_1: fixes x :: "'a list"   and y :: "bool" assumes "incr x = y" obtains "x = []" and "y"   | uu_ :: "'a" where "x = [uu_]" and "y"   | xa :: "'a" and ya :: "'a" and xs :: "'a list" where "x = xa # ya # xs" and "y = (xa < ya \<and> incr (ya # xs))"
main: fixes pad :: "'a vec \<Rightarrow> 'a vec"   and unpad :: "'a vec \<Rightarrow> 'a vec" assumes "\<And>A a. \<lbrakk>A \<subseteq> U; finite A\<rbrakk> \<Longrightarrow> pad (N.lincomb a A) = NM.lincomb (a \<circ> unpad) (pad ` A)"   and "\<And>v. unpad (pad v) = v"   and "pad ` U \<subseteq> carrier_vec (n + m)" shows "pad ` N.span U = NM.span (pad ` U)"
ab_diff_conv_add_uminus: fixes a :: "'c"   and b :: "'c" shows "a - b = a + - b"
qelim: fixes qe :: "fm \<Rightarrow> fm"   and vs :: "'b list"   and p :: "fm"   and bs :: "'b list" assumes "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm vs bs (qe p) = Ifm vs bs (E p)" shows "qfree (qelim p qe) \<and> Ifm vs bs (qelim p qe) = Ifm vs bs p"
induct: fixes x :: "nat"   and F :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "x \<in> NatMultiples F"   and "\<And>k. k \<in> F \<Longrightarrow> P k"   and "\<And>k m. \<lbrakk>k \<in> F; m \<in> NatMultiples F; P m\<rbrakk> \<Longrightarrow> P (k * m)" shows "P x"
elims_3: fixes x :: "'b extT"   and xa :: "'b extT" assumes "\<not> x < xa" obtains x :: "'b extT" and y :: "'b extT" where "xa__ = x" and "xa = y" and "\<not> (x \<le> y \<and> \<not> y \<le> x)"
fmeasurableI_null_sets: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> null_sets M" shows "A \<in> fmeasurable M"
hd_rotate_conv_nth: fixes xs :: "'a list"   and n :: "nat" assumes "xs \<noteq> []" shows "hd (rotate n xs) = xs ! (n mod length xs)"
zero_divisor_iff_eq_0: fixes a :: "'a" shows "zero_divisor a = (a = (0::'a))"
bet2__out: 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 B' :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "A \<noteq> B'"   and "Bet A B C"   and "Bet A B' C" shows "Tarski_neutral_dimensionless.Out Bet A B B'"
pcs_subset_2: fixes es :: "nat exp list"   and pc :: "nat"   and d :: "nat" shows "pcs (compxEs\<^sub>2 es pc d) \<subseteq> {pc..<pc + length (compEs\<^sub>2 es)}"
natural_isomorphism_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" shows "natural_isomorphism A B F G \<tau> \<equiv> natural_transformation A B F G \<tau> \<and> natural_isomorphism_axioms A B \<tau>"
lt_set_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "lt_set F = lt ` (F - {0})"
MGFn_Fin: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and c1 :: "stmt"   and c2 :: "stmt" assumes "wf_prog G"   and "G,A\<turnstile>{=:n} In1r c1\<succ> {G\<rightarrow>}"   and "G,A\<turnstile>{=:n} In1r c2\<succ> {G\<rightarrow>}" shows "G,A\<turnstile>{=:n} In1r (c1 Finally c2)\<succ> {G\<rightarrow>}"
closed_Times: fixes S :: "'c set"   and T :: "'d set" assumes "closed S"   and "closed T" shows "closed (S \<times> T)"
simplex_like_pow: fixes A :: "'b set" assumes "finite A" shows "order.simplex_like (Pow A)"
Act_tau_eq_iff: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and x1 :: "('d, 'e, 'c) formula"   and \<alpha> :: "'c"   and x2 :: "('d, 'e, 'c) formula" assumes "weak_nominal_ts satisfies transition \<tau>" shows "(Act \<tau> x1 = Act \<alpha> x2) = (\<alpha> = \<tau> \<and> x2 = x1)"
emv_subst_imp_num_unique_vars_le: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "emv s t"   and "s = t \<cdot> \<sigma>" shows "num_unique_vars s \<le> num_unique_vars t"
simps: fixes l :: "('b \<times> color) tree"   and x :: "'b"   and r :: "('b \<times> color) tree" shows "joinL l x r = (if bheight r \<le> bheight l then R l x r else case r of \<langle>l', (x', Red), r'\<rangle> \<Rightarrow> R (joinL l x l') x' r' | \<langle>l', (x', Black), r'\<rangle> \<Rightarrow> baliL (joinL l x l') x' r')"
assms_1: shows "\<PP>\<^sub>\<infinity> \<phi> \<G> w"
prems_3: shows "\<forall>b\<in>B. f b \<noteq> 0"
shift_def: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "shift f n = (\<lambda>p. f (p + n))"
inf_prefix_le2: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix (xs \<sqinter> ys) ys"
assms_3: shows "L \<in> carrier A \<rightarrow> carrier B"
vdiff_alt_def: fixes w :: "nat list"   and v :: "nat list" assumes "length w = length v" shows "w -\<^sub>v v = map2 (-) w v"
ss_rewrite: fixes e' :: "nat"   and e :: "nat"   and a :: "'a"   and h :: "'a"   and z :: "nat"   and z' :: "nat" assumes "e' < e"   and "e < order \<G>"   and "a \<in> carrier \<G>"   and "h \<in> carrier \<G>"   and "a \<otimes> h [^] e = \<^bold>g [^] z"   and "a \<otimes> h [^] e' = \<^bold>g [^] z'" shows "h = \<^bold>g [^] ((int z - int z') * Number_Theory_Aux.inverse (e - e') (order \<G>) mod int (order \<G>))"
g_def: shows "g = the_inv_into (vertices G') (\<lambda>x. if x \<in> m ` C then Inl (the_inv_into C m x) else Inr x)"
flows_edges: fixes \<T> :: "'b stateful_policy"   and G :: "'b graph"   and M :: "'b SecurityInvariant_configured list" assumes "stateful_policy_compliance \<T> G M" shows "flows_fix \<T> \<subseteq> edges G"
proj6_def: shows "\<pi>\<^sub>6 \<equiv> \<lambda>(uu_, uu_, uu_, uu_, nhip). nhip"
ij_le3_2: shows "j < prod_list (nths (dims B) (- I))"
mayl: shows "\<forall>l\<in>collect_locks' las. may_lock (ls $ l) t"
intro: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a" assumes "\<And>x. Dmonoid.unit prod one x" shows "Dgrp_axioms prod one"
abs_summable_on_normI: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "f abs_summable_on A" shows "(\<lambda>x. norm (f x)) abs_summable_on A"
ifbb_def: shows "bb\<^sup>-\<^sub>\<F> = op\<^sub>K \<circ> fb\<^sup>-\<^sub>\<F>"
closure_inclusion: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "closure_of A \<subseteq> closure_of B"
smooth_on_open_subsetsI: fixes S :: "'b set"   and k :: "enat"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. x \<in> T \<and> open T \<and> k-smooth_on T f" shows "k-smooth_on S f"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_all_front_tickfree_traces2: fixes s :: "'a event list" assumes "front_tickFree s" shows "s \<in> T (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV)"
integral_by_parts: assumes "a \<le> b"   and "continuous_on {a..b} f"   and "continuous_on {a..b} g"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (F has_real_derivative f x) (at x within {a..b})"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (G has_real_derivative g x) (at x within {a..b})" shows "LBINT x. F x * g x * indicat_real {a..b} x = F b * G b - F a * G a - (LBINT x. f x * G x * indicat_real {a..b} x)"
nths_append: fixes l :: "'a list"   and l' :: "'a list"   and A :: "nat set" shows "nths (l @ l') A = nths l A @ nths l' {j. j + length l \<in> A}"
nonzero_closed: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> nonzero Zp" shows "a \<in> carrier Zp"
in_language_L_size: fixes idx :: "'n"   and I :: "'i set"   and w :: "'x list"   and x :: "'x" assumes "in_language (L idx I) w"   and "x \<in> set w" shows "size x = idx"
da_a: fixes x :: "'a" shows "d (a x) = a x"
insert3_d_def2: fixes x :: "'a"   and t :: "'a tree"   and n :: "nat"   and dl :: "nat" shows "insert3_d x (t, n, dl) = (let t' = ins3 (n + dl) 0 x t in case t' of Same2 \<Rightarrow> (t, n, dl) | Bal2 t' \<Rightarrow> (t', n + 1, dl))"
Ball_set: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
compl_one: shows "NOT (- (1::'b)) = (0::'b)"
obj_terminalE: fixes \<CC> :: "V"   and t :: "V" assumes "obj_terminal \<CC> t" obtains "t \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" and "\<And>a. a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr> \<Longrightarrow> Ex1 (is_arr \<CC> a t)"
H_values_right_bottom: fixes i :: "nat"   and j :: "nat" assumes "i = 1 \<and> j = 1" shows "H $$ (i, j) = complex_of_real (- 1 / sqrt 2)"
neq_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<noteq> y) = (lt x y \<or> lt y x)"
prems_2: shows "\<forall>x\<in>A. f x \<in> A \<and> f x \<noteq> x \<and> f (f x) = x"
su: shows "set_mset \<Gamma>'' \<subseteq> \<Gamma>"
ins: shows "(ea, eb) \<in> set (routing_ipassmt_wi tbl)"
butlast_neq: shows "butlast (prio_emb_pos prio (prio_emb_step prio t s) s) \<noteq> butlast (prio_emb_pos prio t s)"
card_of_Sigma_ordLeq_infinite_Field: fixes r :: "('a \<times> 'a) set"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "infinite (Field r)"   and "Card_order r"   and "|I| \<le>o r"   and "\<forall>i\<in>I. |A i| \<le>o r" shows "|Sigma I A| \<le>o r"
newpkt_def: shows "newpkt \<equiv> \<lambda>(d, dip). Newpkt d dip"
pmdl_f4_red_aux: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "set (f4_red_aux bs ps) \<subseteq> pmdl (args_to_set ([], bs, ps))"
eq: shows "poly ([a, 1::'a] %^ Suc n *** p) = poly ([a, 1::'a] %^ n *** ([a, 1::'a] *** p))"
bvxorxor_elim_help: fixes x :: "nat"   and a :: "bit list"   and b :: "bit list" assumes "x \<le> length a"   and "length a = length b" shows "bv_prepend x \<zero> (drop x (bvxor (bv_prepend x \<zero> (drop x (bvxor a b))) b)) = bv_prepend x \<zero> (drop x a)"
ltl_prop_implies_min_dnf: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<phi> \<longrightarrow>\<^sub>P \<psi> = (\<forall>A\<in>min_dnf \<phi>. \<exists>B\<in>min_dnf \<psi>. B |\<subseteq>| A)"
swapnorm_def: fixes n :: "nat"   and m :: "nat"   and t :: "'b poly" shows "swapnorm n m t = polynate (swap n m t)"
FOREACH_def: fixes S :: "'e set" shows "FOREACH S \<equiv> FOREACH\<^sub>C S (\<lambda>_. True)"
inj_on_iff_eq_card: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i" assumes "finite A" shows "inj_on f A = (card (f ` A) = card A)"
dvd_div_div_eq_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "c \<noteq> zero"   and "dvd.dvd times a b"   and "dvd.dvd times c d" shows "(divide b a = divide d c) = (times b c = times a d)"
dense_bot_closed: shows "bot \<le> bot \<sqinter> bot"
P4: shows "X Midpoint C C' \<and> Cong A C A C'"
finite_dimensional_vector_space_with_axioms_def: fixes plus\<^sub>V\<^sub>S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S :: "'d"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set" shows "finite_dimensional_vector_space_with_axioms plus\<^sub>V\<^sub>S zero\<^sub>V\<^sub>S scale basis \<equiv> finite basis \<and> (with (0::'e) zero\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale : \<guillemotleft>independent\<guillemotright> basis) \<and> (with zero\<^sub>V\<^sub>S plus\<^sub>V\<^sub>S scale : \<guillemotleft>span\<guillemotright> basis) = UNIV"
ltl_prop_implies_def: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<phi> \<longrightarrow>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> \<longrightarrow> \<A> \<Turnstile>\<^sub>P \<psi>"
scaleR_fun_def: fixes r :: "real"   and f :: "'c \<Rightarrow> 'd" shows "r *\<^sub>R f = (\<lambda>x. r *\<^sub>R f x)"
Nil2_notin_lex: fixes xs :: "'g list"   and r :: "('g \<times> 'g) set" shows "(xs, []) \<notin> lex r"
effective_matrix_Tensor_elements: fixes i :: "nat"   and M1 :: "'a list list"   and M2 :: "'a list list"   and j :: "nat" assumes "i < row_length M1 * row_length M2"   and "j < length M1 * length M2"   and "mat (row_length M1) (length M1) M1"   and "mat (row_length M2) (length M2) M2" shows "(M1 \<otimes> M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)"
conv_complement_1: fixes x :: "'a" shows "- x\<^sup>T \<squnion> (- x)\<^sup>T = (- x)\<^sup>T"
has_fields_types: fixes P :: "'a prog"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list"   and FD :: "String.literal \<times> String.literal"   and T :: "ty"   and fm :: "fmod"   and wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "P \<turnstile> C has_fields FDTs"   and "(FD, T, fm) \<in> set FDTs"   and "wf_prog wf_md P" shows "is_type P T"
\<delta>_prod_Un_2: fixes \<delta>1 :: "('a, 'c) ta_rule set"   and \<delta>2 :: "('b, 'c) ta_rule set"   and \<delta>2' :: "('b, 'c) ta_rule set" shows "\<delta>_prod \<delta>1 (\<delta>2 \<union> \<delta>2') = \<delta>_prod \<delta>1 \<delta>2 \<union> \<delta>_prod \<delta>1 \<delta>2'"
spr_abs_refl: fixes t :: "'s Trace" assumes "t \<in> SPR.jkbpC" shows "tLast t \<in> spr_abs t"
singular_relcycle: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_relcycle p X S c = (singular_chain p X c \<and> singular_chain (p - 1) (subtopology X S) (chain_boundary p c))"
comp_eq_id_dest: fixes a :: "'f \<Rightarrow> 'e"   and b :: "'d \<Rightarrow> 'f"   and c :: "'d \<Rightarrow> 'e"   and v :: "'d" assumes "a \<circ> b = id \<circ> c" shows "a (b v) = c v"
ics_SDG_path_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n'' :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' ns' n'"   and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''" shows "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ ns') n'"
semiring_simprules_1: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
bb_def: shows "bb \<equiv> \<lambda>n. real_of_int (b n)"
plus_closed': shows "\<forall>x\<in>U. \<forall>y\<in>U. x +\<^sub>o\<^sub>w y \<in> U"
onorm_le_matrix_component: fixes A :: "((real, 'b) vec, 'a) vec"   and B :: "real" assumes "\<And>i j. \<bar>A $ i $ j\<bar> \<le> B" shows "onorm ((*v) A) \<le> real CARD('a) * real CARD('b) * B"
exhaust: fixes y :: "univ" obtains x :: "nat" where "y = UU x"