hcju/isabelleps huggingface dataset:
prems_ground: shows "{DA} \<union> set CAs \<subseteq> \<G>_Fset M"
sets_tree_sigma_eq: fixes M :: "'b measure" shows "sets (tree_sigma M) = sigma_sets (trees (space M)) (trees_cyl ` trees (sets M))"
mu_change: fixes j' :: "nat" assumes "j' \<le> j" shows "gs2.\<mu> i j' = gs1.\<mu> i j' - rat_of_int c * gs1.\<mu> j j'"
assms: fixes m :: "nat"   and n :: "nat" shows "\<bar>u (m + n) - u m - u n\<bar> \<le> C"
g: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and F :: "real filter" assumes "exp_asymp_equiv_linear f g" shows "\<forall>\<^sub>F x in F. g x = exp (f x)"
Stone_Weierstrass_polynomial_function_subspace: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and e :: "real"   and T :: "'b set" assumes "compact S"   and "continuous_on S f"   and "0 < e"   and "subspace T"   and "f ` S \<subseteq> T" obtains g :: "'a \<Rightarrow> 'b" where "polynomial_function g" and "g ` S \<subseteq> T" and "\<And>x. x \<in> S \<Longrightarrow> norm (f x - g x) < e"
retraction_if_Img_eq_Cod_1: fixes g :: "'s" assumes "arr g"   and "Img g = Cod g" shows "retraction g"
sets_into_space: fixes x :: "'a set" assumes "x \<in> tail_events A" shows "x \<subseteq> space M"
eq: fixes x :: "real" assumes "1 \<le> x" shows "\<psi> x * of_real (ln x) + dirichlet_prod' mangoldt \<psi> x = dirichlet_prod' moebius_mu G1 x"
assms: shows "x \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
n_comm_var: fixes x :: "'a"   and y :: "'a" shows "n (n x \<cdot> n y) = n (n y \<cdot> n x)"
deduct_if_closure_deduct: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M' :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" assumes "intruder_model arity Ana"   and "intruder_model.intruder_deduct arity public Ana M' t"   and "intruder_model.decomp_closure arity public Ana M M'" shows "intruder_model.intruder_deduct arity public Ana M t"
lebesgue_regular_inner: fixes S :: "'c set" assumes "S \<in> sets lebesgue" obtains K :: "'c set" and C :: "nat \<Rightarrow> 'c set" where "negligible K" and "\<And>n. compact (C n)" and "S = \<Union> (range C) \<union> K"
RID_def: fixes x :: "'c \<Longrightarrow> 'b"   and P :: "'b hrel" shows "RID x P = ((\<exists> $x \<bullet> \<exists> $x\<acute> \<bullet> P) \<and> $x\<acute> =\<^sub>u $x)"
prems: shows "h \<turnstile> get_component child \<rightarrow>\<^sub>r c"
downset_def: shows "ord.downset (\<lambda>x y. y \<le> x) = ord.downset_set (\<lambda>x y. y \<le> x) \<circ> (\<lambda>x. {x})"
size_image_mset: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" shows "size (f `# M) = size M"
congruence_inv: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex"   and H' :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0"   and "congruence M H = H'" shows "congruence (mat_inv M) H' = H"
subset_psubset_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq> B"   and "B \<subset> C" shows "A \<subset> C"
meta_spec: fixes P :: "'a \<Rightarrow> prop"   and x :: "'a" assumes "\<And>x. PROP P x" shows "PROP P x"
distinct_19: shows "IndexOfOp \<noteq> string_binop.LessOp"
BIT_lt0: fixes x :: "int"   and b :: "bool" shows "(x BIT b < 0) = (x < 0)"
rawpsubst_cnj: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubst (cnj \<phi>1 \<phi>2) txs = cnj (rawpsubst \<phi>1 txs) (rawpsubst \<phi>2 txs)"
lm130: fixes f :: "('a \<times> 'b) set"   and X :: "'a set" assumes "runiq f" shows "MiscTools.sum' f (X \<inter> Domain f) = sum (toFunction f) (X \<inter> Domain f)"
rename_image_Constrains: fixes h :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "bij h" shows "rename h ` (A Co B) = h ` A Co h ` B"
FG_suffix_G: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" assumes "w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<phi>)" shows "\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n G\<^sub>n \<phi>"
Int_stable_Icc: shows "Int_stable (range (\<lambda>(a, b). {a..b}))"
initial_gso_cost_def: shows "initial_gso_cost = m * (m * (m + n + 2) + 2 * n + 1)"
ring_n_pd_pOp_K_pOp: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and x :: "'a"   and y :: "'a" assumes "distinct_pds K n P"   and "x \<in> carrier (O\<^bsub>K P n\<^esub>)"   and "y \<in> carrier (O\<^bsub>K P n\<^esub>)" shows "x \<plusminus>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub> y = x \<plusminus> y"
le_imp_max_x_impl'_ge: fixes v :: "nat list"   and y :: "nat list"   and i :: "nat"   and a :: "nat list"   and b :: "nat list" assumes "v \<le>\<^sub>v y"   and "i < length a" shows "max_x_impl' a b y i \<le> max_x_impl' a b v i"
dvd_neg_div: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.idom_divide times one plus zero minus uminus divide"   and "dvd.dvd times b a" shows "divide (uminus a) b = uminus (divide a b)"
homE_2: fixes CC :: "('b, 'c, 'd) category_scheme"   and A :: "'b"   and B :: "'b"   and f :: "'c" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>"   and "B \<in> Ob\<^bsub>CC\<^esub>"   and "f \<in> Hom\<^bsub>CC\<^esub> A B" shows "Cod\<^bsub>CC\<^esub> f = B"
phi: fixes n :: "nat" assumes "n < length cl" shows "mC_C Sbis (cl ! n) (dl ! n) s t (fst (PF n)) (snd (PF n))"
open_bijective_linear_image_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "linear f"   and "bij f" shows "open (f ` A) = open A"
PowD: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> Pow B" shows "A \<subseteq> B"
Rep_moebius_mat: fixes x :: "moebius_mat" shows "Rep_moebius_mat x \<in> {M. mat_det M \<noteq> 0}"
assms_2: shows "\<forall>P Q R. Pred (P, Q) \<and> Pred (Q, R) \<longrightarrow> Pred (P, R)"
is_open_def: fixes t :: "Core.term" shows "is_open t \<equiv> loose_bvar t 0"
irreducibleD: fixes p :: "'a"   and a :: "'a"   and b :: "'a" assumes "irreducible p"   and "p = a * b" shows "is_unit a \<or> is_unit b"
D: shows "distributed M lborel X (\<lambda>x. ennreal (normal_density \<mu> \<sigma> x))"
outs_\<I>_mono: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('a, 'b) \<I>" assumes "\<I> \<le> \<I>'" shows "outs_\<I> \<I> \<subseteq> outs_\<I> \<I>'"
square_rtrancl_reflcl_commute: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "square R S S\<^sup>*\<^sup>* R\<^sup>=\<^sup>=" shows "commute R\<^sup>*\<^sup>* S\<^sup>*\<^sup>*"
weakBisimWeakCoinductAux: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto><X> Q \<and> (Q, P) \<in> X" shows "P \<approx> Q"
frameCommutativity: fixes F :: "'b frame"   and G :: "'b frame" shows "F \<otimes>\<^sub>F G \<simeq>\<^sub>F G \<otimes>\<^sub>F F"
cancels_to_1_fold: fixes x1 :: "bool \<times> 'e"   and x2 :: "bool \<times> 'e"   and xs1 :: "(bool \<times> 'e) list"   and xs2 :: "(bool \<times> 'e) list" assumes "canceling x1 x2" shows "cancels_to_1 (xs1 @ x1 # x2 # xs2) (xs1 @ xs2)"
in_frontier_trans: fixes M :: "'b zmultiset"   and y :: "'b"   and x :: "'b" assumes "0 < zcount M y"   and "x \<in>\<^sub>A frontier M"   and "y \<le> x" shows "y \<in>\<^sub>A frontier M"
monom_subst_def: fixes \<sigma> :: "'d \<Rightarrow> ('b monom \<times> 'c) list"   and m :: "'d monom" shows "monom_subst \<sigma> m = monom_list_subst \<sigma> (monom_list m)"
reachable_supgraph: fixes x :: "'a"   and y :: "'a" assumes "E x y"   and "reachable x" shows "x \<rightarrow> y"
one_poly_def: shows "1 \<equiv> Abs_poly (\<lambda>n. of_bool (n = 0))"
subset_equiv_class: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and b :: "'a"   and a :: "'a" assumes "equiv A r"   and "r `` {b} \<subseteq> r `` {a}"   and "b \<in> A" shows "(a, b) \<in> r"
conversepD: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'a" assumes "r\<inverse>\<inverse> b a" shows "r a b"
of_bl_drop': fixes lend :: "nat"   and bl :: "bool list" assumes "lend = length bl - LENGTH('a)" shows "of_bl (drop lend bl) = of_bl bl"
ecard_def: fixes A :: "'c set" shows "ecard A = (if finite A then enat (card A) else \<infinity>)"
not_ARR_empty: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "free_monoidal_category C" shows "\<not> free_monoidal_category.ARR C {}"
a: shows "(\<lambda>i. Sigma_Algebra.measure M (f i)) sums Sigma_Algebra.measure M (\<Union> (range f))"
simps_1: fixes t :: "'b btree" shows "height_up\<^sub>i (T\<^sub>i t) = height t"
noExceptionals_def: fixes g :: "graph"   and V :: "nat set" shows "noExceptionals g V \<equiv> \<forall>v\<in>V. \<not> exceptionalVertex g v"
minus1_div2: fixes n :: "nat" shows "(n - Suc 0) div 2 = (if odd n then n div 2 else n div 2 - 1)"
aligned_add_mask_less_eq: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word" assumes "is_aligned x n"   and "is_aligned y n"   and "n < LENGTH('b)" shows "(x + mask n < y) = (x < y)"
nat_group: fixes g :: "nat \<Rightarrow> 'b"   and k :: "nat"   and n :: "nat" shows "(\<Sum>m<n. sum g {m * k..<m * k + k}) = sum g {..<n * k}"
apex_def: shows "src_f.cod.apex \<equiv> C.dom src_f.cod.leg0"
\<G>\<F>_semantics': fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> \<phi> w = subformulas\<^sub>\<mu> \<phi> \<inter> {\<psi>. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)}"
prv_prfOf: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "prv \<phi> = (\<exists>prf\<in>proof. prfOf prf \<phi>)"
map_gals_2: fixes Cs :: "'b set list" assumes "SimplicialComplex.min_maxsimpchain W (C # Cs)" shows "pgallery (g \<Turnstile> (C # Cs))"
TLA1: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "|~ P \<and> Unchanged f \<longrightarrow> \<circle>P" shows "\<turnstile> \<box>P = (P \<and> \<box>[P \<longrightarrow> \<circle>P]_f)"
tmod4: fixes mn :: "char list"   and s :: "'b state"   and ck :: "bool"   and env :: "v sem_env"   and specs :: "spec list option"   and ds :: "dec list" assumes "[mn] \<in> defined_mods s" shows "evaluate_top ck env s (Tmod mn specs ds) (s, Rerr (Rabort Rtype_error))"
ang_vec_opposite_opposite': fixes z1 :: "complex"   and z :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> z"   and "z2 \<noteq> z" shows "\<angle> (z - z1) (z - z2) = \<angle> (z1 - z) (z2 - z)"
run_env_def: fixes envT :: "('a, 'b) envT" shows "run_env envT = (case envT of EnvT x \<Rightarrow> x)"
and_foldl_sem_conv: fixes \<Gamma> :: "('d, 'e, 'f) formula list"   and \<nu> :: "(real, 'f) vec \<times> (real, 'f) vec"   and I :: "('d, 'e, 'f) interp" assumes "\<And>\<phi>. List.member \<Gamma> \<phi> \<Longrightarrow> \<nu> \<in> fml_sem I \<phi>" shows "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma> TT)"
iwlsFSbSwTR_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.iwlsFSbSwTR Delta MOD \<equiv> FixSyn.igWlsAllDisj Delta MOD \<and> FixSyn.igWlsAbsIsInBar Delta MOD \<and> FixSyn.igConsIPresIGWlsSTR Delta MOD \<and> FixSyn.igSubstAllIPresIGWlsAllSTR Delta MOD \<and> igFreshClsSTR MOD \<and> FixSyn.igSubstClsSTR Delta MOD \<and> igAbsRenSTR MOD"
interleave_empty: fixes a :: "'b list"   and b :: "'b list" shows "([] \<in> a \<otimes> b) = (a = [] \<and> b = [])"
preserves_hseq: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "B.arr \<mu> \<and> B.arr \<nu> \<and> src\<^sub>B \<mu> = trg\<^sub>B \<nu>" shows "C.arr (F \<mu>) \<and> C.arr (F \<nu>) \<and> src\<^sub>C (F \<mu>) = trg\<^sub>C (F \<nu>)"
assms: fixes k :: "nat" shows "\<exists>v\<le>w. enat k < llength v \<and> lproject A v \<le> x"
s_def: shows "s = local.punit.monom_mult (- (1::'b) / (punit.lc p * punit.lc q)) (0::'a) (punit.tail p * punit.tail q)"
positive_is_normal: fixes A :: "complex mat" assumes "positive A" shows "A * adjoint A = adjoint A * A"
compact_attains_sup: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.linorder_topology less_eq less open"   and "topological_space.compact open S"   and "S \<noteq> {}" shows "\<exists>s\<in>S. \<forall>t\<in>S. less_eq t s"
parallel_reduction_nonredexE: fixes A :: "'a trm"   and B :: "'a trm"   and X :: "'a trm" assumes "App A B >> X"   and "\<And>x T A'. A \<noteq> Fn x T A'" shows "\<exists>A' B'. A >> A' \<and> B >> B' \<and> X = App A' B'"
ord_\<Gamma>_def: shows "gr.ord_\<Gamma> = {uu_. \<exists>CAs DA AAs As E. uu_ = Infer (mset CAs) DA E \<and> gr.ord_resolve CAs DA AAs As E}"
not_lfinite_lprefix_conv_eq: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lfinite xs" shows "xs \<sqsubseteq> ys = (xs = ys)"
support_eq: fixes x :: "char list"   and E :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> nat" shows "support (\<lambda>l s. l x = E l s) \<subseteq> supportE E \<union> {x}"
merge_t_inclusion: fixes x :: "nat"   and n :: "nat"   and y :: "nat"   and j :: "nat"   and i :: "nat" assumes "x \<le> n"   and "(\<exists>j'. token_run x n \<noteq> token_run y n \<and> y \<le> n \<and> state_rank (token_run y n) n = Some j') \<or> y = Suc n"   and "token_run x (Suc n) = token_run y (Suc n)"   and "token_run x (Suc n) \<notin> F"   and "state_rank (token_run x n) n = Some j"   and "j < i" shows "n \<in> merge_t i"
inverse_negative_iff_negative: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse a) zero = less a zero"
distinct_remdups_id: fixes xs :: "'c list" assumes "distinct xs" shows "remdups xs = xs"
foldr_crypt_size: fixes KS :: "key list"   and X :: "msg" shows "size (foldr Crypt KS X) = size X + length KS"
funpow_cycle: fixes k :: "nat"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and l :: "nat" assumes "(f ^^ k) x = x" shows "(f ^^ l) x = (f ^^ (l mod k)) x"
InitConfig_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('d, 'c, 'a) Combined.configuration" assumes "dataflow_topology summary results_in" shows "dataflow_topology.InitConfig c = ((\<forall>p. init c p = False) \<and> order.init_config (exchange_config c) \<and> (\<forall>p loc t. zcount (c_pts (prop_config c p) loc) t = zcount (c_glob (exchange_config c) p) (loc, t)) \<and> (\<forall>w. local.init_config (prop_config c w)))"
intro: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1"   and "vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2" shows "vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
fcomp_def: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'e" shows "fcomp f g = (\<lambda>x. g (f x))"
prv_imp_scnj_insert: fixes F :: "'fmla set"   and \<phi> :: "'fmla" assumes "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla" shows "prv (imp (scnj (insert \<phi> F)) (cnj \<phi> (scnj F)))"
unpadl_padl: fixes m :: "nat"   and v :: "'b vec" shows "unpadl m (0\<^sub>v m @\<^sub>v v) = v"
inject_2: fixes x21 :: "char list"   and x22 :: "'a list"   and y21 :: "char list"   and y22 :: "'a list" shows "(Neg x21 x22 = Neg y21 y22) = (x21 = y21 \<and> x22 = y22)"
that: fixes s' :: "dB" assumes "term_brackets vs t = Some s'"   and "term_bracket v s' = Some t'" shows "thesis"
head_\<omega>_eq_0_iff: fixes m :: "hmultiset" shows "(head_\<omega> m = 0) = (m = 0)"
is_extensional: fixes f :: "'a1 \<times> 'a2" assumes "\<not> A1xA2.arr f" shows "F f = B.null"
attach_compose: fixes conv1 :: "('y, 'z, 'ac, 'ad) converter"   and conv2 :: "('ac, 'ad, 'aa, 'ab) converter"   and res :: "('aa, 'ab) resource" shows "conv1 \<odot> conv2 \<rhd> res = conv1 \<rhd> conv2 \<rhd> res"
axioms_1: fixes \<Omega> :: "'b set"   and M :: "'b set set" assumes "semiring_of_sets \<Omega> M" shows "subset_class \<Omega> M"
rg_succ2_def: fixes am :: "nat \<Rightarrow> nat list"   and cf :: "nat \<times> nat \<Rightarrow> int"   and u :: "nat" shows "rg_succ2 am cf u \<equiv> ps_get_op am u \<bind> monadic_filter_rev (\<lambda>v. cf_get cf (u, v) \<bind> (\<lambda>x. return (0 < x)))"
inter_add_left2: fixes x :: "'a"   and N :: "'a multiset"   and M :: "'a multiset" assumes "x \<in># N" shows "(x, M) \<inter># N = x, M \<inter># (N - (x, {#}))"
eucl_truncate_up_prod_def: fixes p :: "nat"   and a :: "'a \<times> 'b" shows "eucl_truncate_up p a = (eucl_truncate_up p (fst a), eucl_truncate_up p (snd a))"
dom_def: fixes f :: "'c" shows "C.dom f = (if C.domains f \<noteq> {} then SOME a. a \<in> C.domains f else C.null)"
signed_of_nat: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus (word_of_nat n) = ring_1.of_int one plus zero minus (signed_take_bit (LENGTH('c) - Suc 0) (int n))"
octo_eq_iff: fixes x :: "octo"   and y :: "octo" shows "(x = y) = (Ree x = Ree y \<and> Im1 x = Im1 y \<and> Im2 x = Im2 y \<and> Im3 x = Im3 y \<and> Im4 x = Im4 y \<and> Im5 x = Im5 y \<and> Im6 x = Im6 y \<and> Im7 x = Im7 y)"
one: fixes n :: "nat" shows "ind (is_nth_power n) 1 = (1::'b)"
aff_dim_def: fixes V :: "'a set" shows "aff_dim V = (SOME d. \<exists>B. affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = d + 1)"
Neg_Imp_E: fixes B :: "fm"   and A :: "fm"   and H :: "fm set"   and C :: "fm" assumes "insert (Neg B) (insert A H) \<turnstile> C" shows "insert (Neg (A IMP B)) H \<turnstile> C"
trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "codomain.greater_eq b a"   and "codomain.greater_eq c b" shows "codomain.greater_eq c a"
pointbased_def: fixes f :: "'c set \<Rightarrow> 'd set" shows "pointbased f = (\<exists>F. f = pointbase F)"
shortest_path_pos_cost_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 "shortest_path_pos_cost G dist c s num \<equiv> basic_just_sp G dist c s num \<and> shortest_path_pos_cost_axioms G dist c s num"
signed_take_bit_negative_iff: fixes n :: "nat"   and k :: "int" shows "(signed_take_bit n k < 0) = bit k n"
cases: fixes a :: "nat"   and F :: "nat set" assumes "a \<in> NatMultiples F" obtains k :: "nat" where "a = ka__" and "ka__ \<in> F"   | k :: "nat" and m :: "nat" where "a = k * m" and "k \<in> F" and "m \<in> NatMultiples F"
map2_add_mset_map: fixes AAs' :: "'a multiset list"   and n :: "nat"   and As' :: "'a list"   and \<eta> :: "'s" assumes "length AAs' = n"   and "length As' = n" shows "Map2.map2 add_mset (As' \<cdot>al \<eta>) (AAs' \<cdot>aml \<eta>) = Map2.map2 add_mset As' AAs' \<cdot>aml \<eta>"
FullSpec_impl_Spec: shows "\<turnstile> FullSpec \<longrightarrow> Spec inp out"
Accessible_language: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.language (dfa.Accessible_dfa M) = dfa.language M"
divisors_nat_def: fixes n :: "nat" shows "divisors_nat n \<equiv> if n = 0 then [] else remdups_adj (sort (map prod_list (subseqs (prime_factorization_nat n))))"
distinct_23: fixes x31 :: "('a, 'b, 'c) cmd"   and x32 :: "('a, 'b, 'c) cmd"   and x41 :: "'a"   and x42 :: "('a, 'b, 'c) cmd" shows "x31 ;; x32 \<noteq> While x41 x42"
remove: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "x \<in> A" shows "F g A = g x \<^bold>* F g (A - {x})"
rep_eq: fixes x :: "'c monom" shows "sum_var x = sum_var_list (Rep_monom x)"
subspaceD: fixes U :: "'b set"   and V :: "'b set"   and x :: "'b" assumes "Subspace.subspace U V"   and "x \<in> U" shows "x \<in> V"
x: shows "\<bar>x\<bar> \<noteq> \<infinity>"
section_is_mono: fixes g :: "'a" assumes "S.section g" shows "S.mono g"
terminationLess_def: fixes F0 :: "Literal list list"   and decisionVars :: "nat set" shows "terminationLess F0 decisionVars \<equiv> {(stateA, stateB). (stateA, stateB) \<in> lexLessState F0 decisionVars \<or> (stateA, stateB) \<in> boolLessState \<or> (stateA, stateB) \<in> multLessState \<or> (stateA, stateB) \<in> learnLessState}"
order_finite_set_obtain_foundation: fixes M :: "'a set"   and t :: "'a" assumes "finite M"   and "t \<in> M" obtains s :: "'a" where "s \<in> M" and "s \<le> t" and "\<forall>u\<in>M. \<not> u < s"
upd_lemma: fixes n :: "'a"   and p :: "('a, 'b) form"   and e :: "nat \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and x :: "'c list \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c list \<Rightarrow> bool" assumes "new n p" shows "eval e (f(n := x)) g p = eval e f g p"
conjugate_exponent_ennreal_2: fixes p :: "ennreal" assumes "1 \<le> p" shows "1 \<le> conjugate_exponent p"
assms_3: shows "eoutKnowsECorrect A (sE m)"
rev_nth_on: fixes p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and i :: "'c" shows "(p on (ps, i)) = ((\<exists>v. ps !. v = Some p) \<or> p = Nom i)"
key_sets_pwd_seskey: fixes s :: "(agent \<times> msg) set"   and U :: "msg set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "U \<in> key_sets (Pwd n) (crypts (Log -` spied s))" shows "\<exists>SK. U = {SesKey SK} \<and> ((Owner n, Crypt (SesK SK) (Pwd n)) \<in> s \<or> (Asset n, Crypt (SesK SK) (Num 0)) \<in> s)"
simps_6: fixes h :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and \<phi> :: "'c fm"   and xs :: "'d list" shows "interpret h (Neg \<phi>) xs = (\<not> interpret h \<phi> xs)"