hcju/isabelleps huggingface dataset:
exact_sequence_sum_lemma: fixes G :: "'b monoid"   and h :: "'b \<Rightarrow> 'b"   and A :: "'b monoid"   and C :: "'b monoid"   and k :: "'b \<Rightarrow> 'b"   and B :: "'b monoid"   and D :: "'b monoid"   and g :: "'b \<Rightarrow> 'b"   and i :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and j :: "'b \<Rightarrow> 'b" assumes "comm_group G"   and "h \<in> Group.iso A C"   and "k \<in> Group.iso B D"   and "exact_seq ([D, G, A], [g, i])"   and "exact_seq ([C, G, B], [f, j])"   and "\<And>x. x \<in> carrier A \<Longrightarrow> f (i x) = h x"   and "\<And>x. x \<in> carrier B \<Longrightarrow> g (j x) = k x" shows "(\<lambda>(x, y). i x \<otimes>\<^bsub>G\<^esub> j y) \<in> Group.iso (A \<times>\<times> B) G \<and> (\<lambda>z. (f z, g z)) \<in> Group.iso G (C \<times>\<times> D)"
set_plus_mono2_b: fixes C :: "'a set"   and D :: "'a set"   and E :: "'a set"   and F :: "'a set"   and x :: "'a" assumes "C \<subseteq> D"   and "E \<subseteq> F"   and "x \<in> C + E" shows "x \<in> D + F"
cis_arg: fixes z :: "complex" assumes "z \<noteq> 0" shows "cis (arg z) = sgn z"
add_0_right: fixes a :: "'a \<Rightarrow> bool" shows "(a \<and>* \<box>) = a"
genideal_zero: shows "Idl\<^bsub>poly_ring R\<^esub> {\<zero>\<^bsub>poly_ring R\<^esub>} = {\<zero>\<^bsub>poly_ring R\<^esub>}"
assertLiteral_def: fixes literal :: "Literal"   and decision :: "bool"   and state :: "State" shows "assertLiteral literal decision state \<equiv> Let (state\<lparr>getM := getM state @ [(literal, decision)]\<rparr>) (notifyWatches (opposite literal))"
f_1: shows "f \<in> Pi\<^sub>E ((\<lambda>a. generate G {a}) ` A) id"
hyps_9: assumes "list_all is_Check (unlabel A)" shows "list_all is_Check (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
comp_arr_inv': fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.iso f" shows "f \<odot> Maps.inv f = Maps.cod f"
fminusD1: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "c |\<in>| A |-| B" shows "c |\<in>| A"
map_dual_def: fixes f :: "'c \<Rightarrow> 'd" shows "\<partial>\<^sub>F f = \<partial> \<circ> f \<circ> \<partial>"
exhaustive_int_def: fixes f :: "int \<Rightarrow> (bool \<times> term list) option"   and d :: "natural" shows "exhaustive_class.exhaustive f d = Quickcheck_Exhaustive.exhaustive_int' f (int_of_integer (integer_of_natural d)) (- int_of_integer (integer_of_natural d))"
SUP_enat_add_right: fixes I :: "'c set"   and c :: "enat"   and f :: "'c \<Rightarrow> enat" assumes "I \<noteq> {}" shows "(\<Squnion>i\<in>I. c + f i) = c + \<Squnion> (f ` I)"
partial_magma_axioms: shows "partial_magma H.VxV.comp"
zero_rw: shows "is_zero_iarray (matrix_to_iarray A !! mod_type_class.to_nat i) = is_zero_row i A"
exhaust: fixes y :: "('g, 'f) tllist" obtains x1 :: "'f" where "y = TNil x1"   | x21 :: "'g" and x22 :: "('g, 'f) tllist" where "y = TCons x21 x22"
spmf_Domainp_rel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Domainp (rel_spmf R) = pred_spmf (Domainp R)"
preserves_reflects_arr: fixes f :: "'d \<times> 'c" shows "S.arr (\<Psi> f) = DopxC.arr f"
\<omega>_gt0: shows "[]\<^sub>\<circ> \<subset>\<^sub>\<circ> \<omega>"
assoc_simps_2: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.ide h"   and "Span.src f = Span.trg g"   and "Span.src g = Span.trg h" shows "Span.dom \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<odot>) Maps.PRJ\<^sub>0 Maps.PRJ\<^sub>1 f g h, Dom = Dom ((f \<circ> g) \<circ> h), Cod = Cod (f \<circ> g \<circ> h)\<rparr> = (f \<circ> g) \<circ> h"
subset_in_fset: fixes xs :: "'b fset"   and ys :: "'b fset" shows "(xs |\<subseteq>| ys) = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
rep_eq: fixes x :: "'k \<Rightarrow> bool"   and xa :: "'k multiset" shows "count (filter_mset x xa) = (\<lambda>x. if x x then count xa x else 0)"
dvd_mult_imp_div: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and a :: "'m"   and c :: "'m"   and b :: "'m" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times (times a c) b" shows "dvd.dvd times a (divide b c)"
fact4: shows "Sigma_Algebra.measure (measure_spmf (game \<A> lazy_channel_recv_f')) {x. snd x} \<le> real q / real (2 ^ \<eta>)"
even_bit_succ_iff: 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 n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "bit (plus one a) n = (bit a n \<or> n = 0)"
next_in_A: shows "next_in A u \<in> A \<inter> {u<..}"
poly_eval_power: fixes a :: "'c \<Rightarrow> 'b"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and m :: "nat" shows "poly_eval a (p ^ m) = poly_eval a p ^ m"
hyp: shows "\<forall>l\<le>k. simple_ownership_distinct (fst (c (n + l)))"
cmp_simps'_2: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "src\<^sub>D (\<Phi>\<^sub>G (\<mu>, \<nu>)) = G.map\<^sub>0 (src\<^sub>C \<nu>)"
zdiff_mod_0_imp_mod_eq: fixes b :: "int"   and a :: "int"   and m :: "int" assumes "(b - a) mod m = 0" shows "a mod m = b mod m"
rep_eq: fixes x :: "'a key_order" shows "lt_of_key_order x = ko.lt x"
phiArg_path_ex': fixes g :: "'g"   and p :: "'val"   and q :: "'val" assumes "phiArg g p q"   and "p \<in> allVars g" obtains ns :: "'node list" and m :: "'node" where "g \<turnstile> defNode g q-ns\<rightarrow>m" and "EntryPath g ns" and "q \<in> phiUses g m" and "m \<in> set (predecessors g (defNode g p))"
lift_prefs_def: fixes alts :: "'a set"   and alts' :: "'a set"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "lift_prefs alts alts' R = (\<lambda>x y. x \<in> alts' \<and> y \<in> alts' \<and> (x = y \<or> x \<notin> alts \<or> y \<in> alts \<and> R x y))"
iSince_not_iTrigger_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "finite I" shows "(P t'. t' \<S> t I. Q t) = (\<not> (\<not> P t'. t' \<T> t I. \<not> Q t))"
h1: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A"   and "i < dim_row (take_rows A [0..<length (pivot_positions A)])" shows "take_rows A [0..<length (pivot_positions A)] $$ (i, j) = A $$ (i, j)"
substitutes_total: fixes A :: "'a trm"   and x :: "'a"   and M :: "'a trm" shows "\<exists>X. substitutes A x M X"
fold_update_conv: fixes g :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and m :: "'b \<Rightarrow> 'a option"   and x :: "'b" shows "fold (\<lambda>n m. m(n \<mapsto> g n)) xs m x = (if x \<in> set xs then Some (g x) else m x)"
saux: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "FWNormalisationCore.insort DenyAll p l = DenyAll # p"
ordertype_list_seg: fixes n :: "nat" shows "ordertype {l. length l = n} (lenlex less_than) = \<omega> \<up> ord_of_nat n"
vmc_path_axioms: shows "vmc_path G P' (P $ n) p \<sigma>"
preserves_imp_eq: fixes F :: "'c program"   and v :: "'c \<Rightarrow> 'd"   and act :: "('c \<times> 'c) set"   and s :: "'c"   and s' :: "'c" assumes "F \<in> preserves v"   and "act \<in> Acts F"   and "(s, s') \<in> act" shows "v s = v s'"
nat_nonzero_pos: fixes a :: "nat" assumes "a \<noteq> 0" shows "0 < a"
simps_5: fixes f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<sigma> \<Rightarrow> 'a"   and x1 :: "\<omega>" shows "(case \<omega>\<upsilon> x1 of \<omega>\<upsilon> x \<Rightarrow> f1 x | \<sigma>\<upsilon> x \<Rightarrow> f2 x) = f1 x1"
RES_refine: fixes S :: "'a set"   and S' :: "'b set"   and R :: "('a \<times> 'b) set" assumes "\<And>s. s \<in> S \<Longrightarrow> \<exists>s'\<in>S'. (s, s') \<in> R" shows "RES S \<le> \<Down> R (RES S')"
stream_enc_cut_same: fixes \<Sigma> :: "'b list"   and I :: "(nat + nat set) list"   and w :: "'b list" assumes "formula \<Sigma>"   and "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P" shows "formula.stream_enc \<Sigma> (w, I) = cut_same (formula.any \<Sigma>, replicate (length I) False) (formula.stream_enc \<Sigma> (w, I)) @- sconst (formula.any \<Sigma>, replicate (length I) False)"
tendsto_diff: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd"   and b :: "'d" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
abs_eq: fixes xa :: "'c"   and x :: "'c" assumes "eq_onp dense xa xa"   and "eq_onp dense x x" shows "Abs_dense xa \<squnion> Abs_dense x = Abs_dense (xa \<squnion> x)"
smooth_on_le: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and l :: "enat" assumes "k-smooth_on S f"   and "l \<le> k" shows "l-smooth_on S f"
fpxs_val_uminus: fixes f :: "'b fpxs" shows "fpxs_val (- f) = fpxs_val f"
lower_character: fixes \<chi> :: "'a \<Rightarrow> complex" assumes "character (G\<lparr>carrier := adjoin G H' a\<rparr>) \<chi>" shows "character (G\<lparr>carrier := H'\<rparr>) (\<lambda>x. if x \<in> H' then \<chi> x else 0)"
AX_def: fixes p :: "'b set" shows "\<^bold>A\<^bold>X p = - \<^bold>E\<^bold>X - p"
comp_left_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "times one x = x"
power4_eq_xxxx: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.monoid_mult one times" shows "power.power one times x 4 = times (times (times x x) x) x"
no_root: fixes a :: "'a poly"   and i :: "nat" assumes "(a, i) \<in> set bs" shows "poly a x \<noteq> (0::'a)"
path_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and a :: "'a" assumes "CFG sourcenode targetnode valid_edge Entry"   and "valid_edge a" shows "CFG.path sourcenode targetnode valid_edge (sourcenode a) [a] (targetnode a)"
bl2wc_Oc_then_Bk: fixes rs :: "nat"   and n :: "nat" shows "bl2wc (Oc \<up> rs @ Bk \<up> n) = bl2wc (Oc \<up> rs)"
item_origin_x_def: shows "item_origin x = charslength (take r p)"
skip_loc_reach: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>Skip \<otimes> annos, mds, mem\<rangle>" shows "c' = Stop \<and> mds' = mds \<oplus> annos \<or> c' = Skip \<otimes> annos \<and> mds' = mds"
perm_sym: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs <~~> ys" shows "ys <~~> xs"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<preceq> b"   and "b \<preceq> c" shows "a \<preceq> c"
resT_mhead: fixes m :: "methd" shows "resT (mhead m) = resT m"
less_funI: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<exists>k. f k < g k \<and> (\<forall>k'<k. f k' = g k')" shows "less_fun f g"
scale_right_distrib: fixes a :: "'a"   and x :: "'b"   and y :: "'b" shows "s1 a (x + y) = s1 a x + s1 a y"
img: shows "f ` {0..<n} = {0..<Suc n} - {j}"
strict_mono_mono: fixes f :: "'a \<Rightarrow> 'b" assumes "OrdinalInduct.strict_mono f" shows "mono f"
P: shows "P = (\<lambda>((as, lm), ss, arity). as = 0)"
locate_def: fixes f :: "'a \<Rightarrow> nat"   and al :: "'a list"   and i :: "nat" shows "locate f al i \<equiv> SOME n_j. locate_pred f al i n_j"
invertible_inverse_closed: fixes u :: "'a" assumes "IA.additive.sub.invertible u"   and "u \<in> I" shows "IA.additive.sub.inverse u \<in> I"
Repr_SelfSubst_axioms: shows "Repr_SelfSubst var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc S"
linear_fm: fixes var :: "nat"   and L :: "real list"   and C :: "real"   and b :: "real mpoly"   and c :: "real mpoly"   and B :: "real"   and F :: "atom fm" assumes "var < length L"   and "C \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "insertion (nth_default 0 (L[var := B / C])) b = B"   and "insertion (nth_default 0 (L[var := B / C])) c = C" shows "eval F (L[var := B / C]) = (\<forall>v. eval (linear_substitution_fm var b c F) (L[var := v]))"
Sretr_sym: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "sym theta" shows "sym (Sretr theta)"
sint_dist: fixes x :: "32 Word.word"   and y :: "32 Word.word" assumes "x \<noteq> y" shows "sint x \<noteq> sint y"
SubstTermP_eq: fixes H :: "fm set"   and v :: "tm"   and tm :: "tm"   and x :: "tm"   and z :: "tm"   and y :: "tm"   and A :: "fm" assumes "H \<turnstile> SubstTermP v tm x z"   and "insert (SubstTermP v tm y z) H \<turnstile> A" shows "insert (x EQ y) H \<turnstile> A"
"apply": fixes x :: "'a perm" shows "(\<langle>$\<rangle>) x \<in> {f. bij f \<and> finite {a. f a \<noteq> a}}"
intros_2: fixes F :: "\<Pi>\<^sub>2"   and y :: "\<kappa>" shows "SimpleExOrEnc (\<lambda>x. \<lparr>F,x,y\<rparr>)"
simps_4: fixes d :: "int" shows "intv'_const (Greater' d) = d"
distinct_1: fixes x1 :: "nat"   and x2 :: "(val \<times> val) list" shows "VNat x1 \<noteq> VFun x2"
toList_ODList: fixes xs :: "'b list" shows "toList (ODList xs) = msort xs"
merge\<^sub>R_def: fixes i :: "nat" shows "merge\<^sub>R i = {uu_. \<exists>r \<nu> s q q' j. uu_ = (r, \<nu>, s) \<and> r q = Some j \<and> j < i \<and> q' = \<delta> q \<nu> \<and> ((\<exists>q''. q' = \<delta> q'' \<nu> \<and> r q'' \<noteq> None \<and> q'' \<noteq> q) \<or> q' = q\<^sub>0) \<and> q' \<notin> F}"
\<phi>_simps_5: shows "cod \<phi> = h \<star> p\<^sub>0"
HEndPhase1_valueChosen2: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and D :: "Disk set"   and b :: "nat"   and p :: "Proc" assumes "HEndPhase1 s s' q"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))" shows "\<forall>d\<in>D. b \<le> bal (disk s' d p) \<and> (\<forall>q. phase s' q = 1 \<and> b \<le> mbal (dblock s' q) \<and> hasRead s' q d p \<longrightarrow> (\<exists>br\<in>blocksRead s' q d. b \<le> bal (block br)))"
dual_Collect: fixes P :: "'b dual \<Rightarrow> bool" shows "{dual x |x. P (dual x)} = {x'. P x'}"
remove1_insort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "class.linorder less_eq less" shows "remove1 x (linorder.insort_key less_eq (\<lambda>x. x) x xs) = xs"
vcard_vinsert_in: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "vcard (vinsert a A) = vcard A"
Mp_smult_2: fixes m :: "int"   and a :: "int"   and f :: "int poly" shows "poly_mod.Mp m (Polynomial.smult a (poly_mod.Mp m f)) = poly_mod.Mp m (Polynomial.smult a f)"
sel_2: fixes x1 :: "'d"   and x2 :: "'c" shows "snd (x1, x2) = x2"
aux: shows "prod id A * (- 1) ^ card E * a ^ card A * (- 1) ^ card E = prod id A * a ^ card A"
ceiling_def: fixes x :: "'c" shows "\<lceil>x\<rceil> = - \<lfloor>- x\<rfloor>"
length_us'_Q: shows "length (map normalize us) \<le> length Q"
Inv_tri_ineq: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "dist_perm xs ys"   and "dist_perm ys zs" shows "Inv xs zs \<subseteq> Inv xs ys \<union> Inv ys zs"
simple_def: fixes \<mu> :: "('m \<times> 'w) set" shows "simple \<mu> = (match \<mu> \<and> (\<forall>m w. blocks m w \<mu> \<longrightarrow> w \<notin> Range \<mu>))"
trigonometric_set_mul_integrable: fixes f :: "real \<Rightarrow> real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}" shows "integrable (lebesgue_on {- pi..pi}) (\<lambda>x. trigonometric_set n x * f x)"
mgu_sets_def: fixes AAA :: "('b, 'a) Term.term set set" shows "mgu_sets AAA = map_option subst_of (unify (Pairs AAA) [])"
prems_13: shows "InvariantVarsM (getM state') F0 Vbl"
induct: fixes P :: "('fun, 'var) extstrand_step list \<Rightarrow> bool"   and a0 :: "('fun, 'var) extstrand_step list" assumes "P []"   and "\<And>t S. P S \<Longrightarrow> P (Decomp t # S)"   and "\<And>x S. P S \<Longrightarrow> P (Step x # S)" shows "P a0"
float_def: shows "float = (\<lambda>(a, b). real_of_int a * 2 powr real_of_int b)"
"2_2": shows "Collect (degen cs) \<inter> set ys = {}"
nth_strip_while: fixes xs :: "'b list" shows "Poly_Mapping.nth (strip_while ((=) (0::'b)) xs) = Poly_Mapping.nth xs"
cong_add: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a"   and e :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo d e a" shows "unique_euclidean_semiring.cong modulo (plus b d) (plus c e) a"
idem: fixes a :: "'b \<times> 'c" shows "sup_prod a a = a"
unlr: fixes x :: "'a admS" shows "unlr x \<in> {x. x \<in> admS}"
inject_1: fixes x11 :: "real"   and x12 :: "real list"   and y11 :: "real"   and y12 :: "real list" shows "(Less x11 x12 = Less y11 y12) = (x11 = y11 \<and> x12 = y12)"
lowOut_def: shows "lowOut = low_out \<tau>\<^sub>2"
prod_cases5: fixes y :: "'d \<times> 'e \<times> 'f \<times> 'g \<times> 'h" obtains a :: "'d" and b :: "'e" and c :: "'f" and d :: "'g" and e :: "'h" where "y = (a, b, c, d, e)"
simps_1: fixes \<Gamma> :: "name list"   and name :: "name" shows "nterm_to_term \<Gamma> (Nconst name) = Const name"
lim: shows "(\<lambda>n. of_real (ln (real n)) - of_real (ln (real n + 1))) \<longlonglongrightarrow> (0::'a)"
enumerate_Ex: fixes S :: "nat set"   and s :: "nat" assumes "infinite S"   and "s \<in> S" shows "\<exists>n. enum S n = s"
intro: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. le x x"   and "\<And>x y z. \<lbrakk>le x y; le y z\<rbrakk> \<Longrightarrow> le x z"   and "\<And>x y. \<lbrakk>le x y; le y x\<rbrakk> \<Longrightarrow> x = y" shows "porder le"
set_sorted_list_of_set: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less"   and "finite A" shows "ran\<^sub>l (linorder.sorted_list_of_set less_eq A) = A"
intf_type_def: fixes c :: "'c"   and I :: "'d itself" shows "c ::\<^sub>i I \<equiv> True"
assms_4: assumes "p \<in> sterms \<Gamma> p2" shows "P"
has_as_binary_product_def: fixes a0 :: "'b"   and a1 :: "'b"   and p0 :: "'b"   and p1 :: "'b" shows "J.has_as_binary_product a0 a1 p0 p1 \<equiv> J.ide a0 \<and> J.ide a1 \<and> limit_cone X.COMP J.comp (binary_product_diagram.map J.comp a0 a1) (J.dom p0) (binary_product_diagram.mkCone J.comp p0 p1)"
assms_4: shows "\<forall>(idn, \<tau>)\<in>set l. typ_ok \<Theta> \<tau>"
da_elim_cases_41: fixes Env :: "env"   and B :: "lname set"   and v :: "Term.var"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>v:=e\<rangle>\<guillemotright> A" obtains E :: "assigned" and vn :: "lname" where "v = LVar vn" and "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "nrm A = insert vn (nrm E)" and "brk A = brk E"   | V :: "assigned" where "\<forall>vn. v \<noteq> LVar vn" and "Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> V" and "Env\<turnstile> nrm V \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
that_3: assumes "y \<notin> S"   and "y \<noteq> x" shows "thesis"
Sup_least: fixes A :: "('c \<times> 'd) set"   and z :: "'c \<times> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> less_eq_prod x z" shows "less_eq_prod (Sup_prod A) z"
assms_1: fixes x :: "'a" shows "norm (f x) \<le> b * norm x"
Splay_Analysis_axioms: shows "Splay_Analysis (root 3 4) (1 / 3)"
A: shows "r \<in> min_elems (D (P \<lbrakk>A\<rbrakk> S))"
lunit_commutes_with_L: fixes f :: "'a" assumes "B.ide f" shows "B.lunit (B.L f) = B.L (B.lunit f)"
AAss_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and i :: "'addr expr1"   and e' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e\<lfloor>i\<rceil> := e') h s s'"
in_mset_pos_in_zmset: fixes x :: "'a"   and M :: "'a zmultiset" assumes "x \<in># mset_pos M" shows "x \<in>#\<^sub>z M"
eta_cases_3: fixes i :: "nat"   and t :: "Core.term"   and P :: "bool" assumes "Bv i \<rightarrow>\<^sub>\<eta> t" shows "P"
sep_add_commute: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "plus x y = plus y x"
matchPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name" assumes "P \<leadsto><Rel> Q"   and "Rel \<subseteq> Rel'"   and "\<And>R S c. (R, S) \<in> Rel \<Longrightarrow> ([c\<frown>c]R, S) \<in> Rel" shows "[a\<frown>b]P \<leadsto><Rel'> [a\<frown>b]Q"
abs_eq: fixes x :: "'b \<times> 'b" shows "upair_inv (abs_uprod x) = (case x of (x, y) \<Rightarrow> if x = y then x else undefined)"
inj_mapD: fixes f :: "'g \<Rightarrow> 'h" assumes "inj (map f)" shows "inj f"
QTrans: shows "Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q'"
truthline_ND_proof: fixes Z :: "'b set"   and F :: "'b formula"   and \<A> :: "'b \<Rightarrow> bool" assumes "line_suitable Z F" shows "line_assm \<A> Z \<turnstile> turn_true \<A> F"
tail_eqI: fixes a :: "'b" shows "tail H a = tail G a"
canon_ang_eqE: fixes \<alpha>\<^sub>1 :: "real"   and \<alpha>\<^sub>2 :: "real" assumes "\<downharpoonright>\<alpha>\<^sub>1\<downharpoonleft> = \<downharpoonright>\<alpha>\<^sub>2\<downharpoonleft>" shows "\<exists>x. \<alpha>\<^sub>1 - \<alpha>\<^sub>2 = 2 * real_of_int x * pi"
ground_action_path_in_paper_2: fixes M :: "object atom formula set"   and \<alpha> :: "ground_action"   and \<alpha>s :: "ground_action list"   and M' :: "object atom formula set" shows "ground_action_path M (\<alpha> # \<alpha>s) M' = (M \<^sup>c\<TTurnstile>\<^sub>= ground_action.precondition \<alpha> \<and> ground_action_path (apply_effect (ground_action.effect \<alpha>) M) \<alpha>s M')"
eq_v: fixes xa :: "'a"   and x :: "'b"   and i :: "nat" assumes "(xa, x) \<in> g i" shows "x \<in> vertices C"
vpairs_vintersection: fixes A :: "V"   and B :: "V" shows "vpairs (A \<inter>\<^sub>\<circ> B) = vpairs A \<inter>\<^sub>\<circ> vpairs B"
simps: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and Q :: "heap \<times> nat set \<Rightarrow> bool"   and h :: "heap"   and as :: "nat set" shows "times_assn_raw P Q (h, as) = (\<exists>as1 as2. as = as1 \<union> as2 \<and> as1 \<inter> as2 = {} \<and> P (h, as1) \<and> Q (h, as2))"