hcju/isabelleps huggingface dataset:
wlsPar_def: fixes P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS" shows "wlsPar P \<equiv> (\<forall>s. \<forall>X\<in>termsOfS P s. wls s X) \<and> (\<forall>xs s. \<forall>A\<in>absOfS P (xs, s). wlsAbs (xs, s) A) \<and> Ball (envsOfS P) wlsEnv"
bin_rsplit_size_sign: fixes n :: "nat"   and xs :: "int list"   and nw :: "nat"   and w :: "int"   and v :: "int" assumes "0 < n"   and "xs = bin_rsplit n (nw, w)"   and "v \<in> set xs" shows "take_bit n v = v"
add_less_same_cancel2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus a b) b = less a zero"
outs_in_Use: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and ins :: "'c list"   and outs :: "'c list"   and V :: "'c" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "(p, ins, outs) \<in> set procs"   and "V \<in> set outs" shows "V \<in> Use (sourcenode a)"
ccSup_subset_mono: fixes B :: "'a set"   and A :: "'a set" assumes "countable B"   and "A \<subseteq> B" shows "Sup A \<le> Sup B"
matrix_inv_left: fixes M :: "(('c, 'd) vec, 'e) vec" assumes "invertible M" shows "matrix_inv M ** M = mat (1::'c)"
q2s: shows "(red F)\<^sup>*\<^sup>* q2 s"
cobounded2: fixes a :: "'b upred"   and b :: "'b upred" shows "b \<sqsubseteq> (a \<and> b)"
zip_map_map_two: fixes list :: "('a \<times> 'b) list" shows "zip (map fst list) (map snd list) = list"
x: shows "\<forall>y. O y x = (\<exists>z. (\<forall>y. F y \<longrightarrow> P z y) \<and> O y z)"
assms_5: shows "z2 = z1 + \<i> * sgn (if 0 < arg BC then - BC else BC)"
cSUP_lessD: fixes f :: "'b \<Rightarrow> 'a multiset"   and A :: "'b set"   and y :: "'a multiset"   and i :: "'b" assumes "subset_mset.bdd_above (f ` A)"   and "Sup (f ` A) \<subset># y"   and "i \<in> A" shows "f i \<subset># y"
simps_7: fixes n :: "nat"   and p :: "pf" shows "dfa_of_pf' n (Forall p) = dfa_of_pf' n (Neg (Exist (Neg p)))"
assms_12: shows "\<forall>x. (x \<in> R_I) = (wf_tuple n I x \<and> (\<forall>(A, X)\<in>Q_I_pos. restrict A x \<in> X) \<and> (\<forall>(A, X)\<in>Q_I_neg. restrict A x \<notin> X))"
nz': fixes y :: "complex" assumes "dist z y < e'" shows "f y \<noteq> 0"
inj_semiring_hom_axioms: shows "inj_semiring_hom hom"
H_fcs_a_neigh: shows "H.face_cycle_set a_neigh = segment G.face_cycle_succ a' a"
trans: fixes P :: "('a, 'b) Proc"   and Q :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set"   and R :: "('a, 'b) Proc" assumes "(P, Q) \<in> encoding.indRelTEQ Enc TRel"   and "(Q, R) \<in> encoding.indRelTEQ Enc TRel" shows "(P, R) \<in> encoding.indRelTEQ Enc TRel"
countable_subset_image: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(countable B \<and> B \<subseteq> f ` A) = (\<exists>A'. countable A' \<and> A' \<subseteq> A \<and> B = f ` A')"
listProd1_def: fixes a :: "'c"   and bs :: "'d list" shows "listProd1 a bs \<equiv> map (Pair a) bs"
delay_bisimulation_conv_bisimulation: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim (\<lambda>s tl s'. False) (\<lambda>s tl s'. False) = bisimulation trsys1 trsys2 bisim tlsim"
image_Int_subset: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'e set" shows "f ` (A \<inter> B) \<subseteq> f ` A \<inter> f ` B"
fold_mset_empty: fixes f :: "'l \<Rightarrow> 'k \<Rightarrow> 'k"   and s :: "'k" shows "fold_mset f s {#} = s"
ExecCondEvalError2: fixes \<sigma> :: "char list \<Rightarrow> nat option"   and e\<^sub>2 :: "expr"   and e\<^sub>1 :: "expr"   and s\<^sub>1 :: "stmt"   and s\<^sub>2 :: "stmt" assumes "eval \<sigma> e\<^sub>2 = None" shows "IfEq e\<^sub>1 e\<^sub>2 s\<^sub>1 s\<^sub>2 \<rhd> \<sigma> \<leadsto> None"
hyps_2: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []"   and "InvariantConsistent (getM (applyUnitPropagate state'))"   and "InvariantUniq (getM (applyUnitPropagate state'))"   and "InvariantEquivalentZL (getF (applyUnitPropagate state')) (getM (applyUnitPropagate state')) Phi"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList (applyUnitPropagate state')) (getF (applyUnitPropagate state'))"   and "InvariantWatchListsUniq (getWatchList (applyUnitPropagate state'))"   and "InvariantWatchListsCharacterization (getWatchList (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesEl (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchesDiffer (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state'))"   and "InvariantWatchCharacterization (getF (applyUnitPropagate state')) (getWatch1 (applyUnitPropagate state')) (getWatch2 (applyUnitPropagate state')) (getM (applyUnitPropagate state'))"   and "InvariantConflictFlagCharacterization (getConflictFlag (applyUnitPropagate state')) (getF (applyUnitPropagate state')) (getM (applyUnitPropagate state'))"   and "InvariantQCharacterization (getConflictFlag (applyUnitPropagate state')) (getQ (applyUnitPropagate state')) (getF (applyUnitPropagate state')) (getM (applyUnitPropagate state'))"   and "InvariantUniqQ (getQ (applyUnitPropagate state'))" shows "let state' = exhaustiveUnitPropagate (applyUnitPropagate state') in InvariantEquivalentZL (getF state') (getM state') Phi"
distr_cong: fixes M :: "'b measure"   and K :: "'b measure"   and N :: "'c measure"   and L :: "'c measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "M = K"   and "sets N = sets L"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = g x" shows "distr M N f = distr K L g"
funpow_shift1_nz: fixes p :: "poly"   and n :: "nat" assumes "p \<noteq> 0\<^sub>p" shows "(shift1 ^^ n) p \<noteq> 0\<^sub>p"
elims: fixes x :: "'a list list"   and xa :: "'a list list"   and y :: "'a list" assumes "interact x xa = y" obtains ys :: "'a list list" where "x = []" and "xa = ys__" and "y = concat ys__"   | v :: "'a list" and va :: "'a list list" where "x = v # va" and "xa = []" and "y = concat (v # va)"   | xb :: "'a list" and xs :: "'a list list" and ya :: "'a list" and ys :: "'a list list" where "x = xb # xs" and "xa = ya # ys" and "y = xb @ ya @ interact xs ys"
seq_comp_secure_aux_1: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and P :: "'d option process"   and Q :: "'d option process"   and ws :: "'d option list"   and Y :: "'d option set"   and xs :: "'d option list"   and y :: "'d option"   and ys :: "'d option list" assumes "secure_termination I D"   and "ref_union_closed P"   and "sequential P"   and "secure P I D"   and "secure Q I D"   and "(ws, Y) \<in> seq_comp_failures P Q"   and "ws = xs @ y # ys" shows "(xs @ ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys Y) \<in> seq_comp_failures P Q"
compat_low: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and cms\<^sub>1 :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and cms\<^sub>2 :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list"   and i :: "nat"   and x :: "'b" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.makes_compatible dma eval (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "i \<sqsubset> length cms\<^sub>1"   and "x \<in> sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "dma x = Low"
assms: shows "((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)"
cong: fixes sample :: "'c pmf \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_prob3.pchoose sample \<equiv> monad_prob3.pchoose sample"
simps_6: fixes t1 :: "PrimScheme"   and t2 :: "PrimScheme" shows "sch_to_pr (Pair_op t1 t2) = c_f_pair (sch_to_pr t1) (sch_to_pr t2)"
bind_foldl_spmf_return: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and xs :: "'d list" shows "p \<bind> (\<lambda>x. foldl_spmf f (return_spmf x) xs) = foldl_spmf f p xs"
stereographic_sphere_circle_set'': fixes H :: "circline"   and z :: "riemann_sphere" shows "on_sphere_circle (inv_stereographic_circline H) z = on_circline H (stereographic z)"
ex0_def: shows "Semantics.ex0 \<equiv> \<lambda>F. F dj"
evals_labels: fixes fs :: "(char list \<times> trm) list"   and fs' :: "(char list \<times> trm) list"   and l :: "char list" assumes "fs [\<longmapsto>] fs'" shows "(fs\<langle>l\<rangle>\<^sub>? = \<bottom>) = (fs'\<langle>l\<rangle>\<^sub>? = \<bottom>)"
use_def: fixes l :: "'d \<Longrightarrow> 'c" shows "use l = zoom l State_Monad.get"
prime_card_def: shows "class.prime_card TYPE('a) \<equiv> prime CARD('a)"
B: fixes n :: "nat" shows "((\<lambda>t. if t \<in> {0..real n} then complex_of_real t powr (z - 1) * (1 - complex_of_real t / of_nat n) ^ n else 0) has_integral of_nat n powr z * fact n / pochhammer z (n + 1)) {0..}"
prv_neg_imp_imp_trans: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (neg \<phi>)"   and "bprv (imp \<chi> (imp \<psi> \<phi>))" shows "bprv (imp \<chi> (neg \<psi>))"
hom_mult: fixes x :: "'a"   and y :: "'a" shows "hom (x * y) = hom x * hom y"
Cassoc: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and g :: "'h"   and h :: "'h" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g"   and "g \<approx>>\<^bsub>C\<^esub> h" shows "f ;;\<^bsub>C\<^esub> g ;;\<^bsub>C\<^esub> h = f ;;\<^bsub>C\<^esub> (g ;;\<^bsub>C\<^esub> h)"
lemma_2_4_7_b: fixes a :: "'a"   and n :: "nat"   and b :: "'a" shows "a r-n\<rightarrow> b = a ^ n r\<rightarrow> b"
b: shows "set as = (\<lambda>v. (v, True)) ` set (add_effects_of op) \<union> (\<lambda>v. (v, False)) ` set (delete_effects_of op)"
assms_2: shows "X\<^sub>1 \<approx>\<^sub>L Y\<^sub>1"
block_num_gt_rep: shows "\<Lambda> * (\<v> - 1) div (\<k> - 1) < \<b>"
simps_2: fixes cs :: "'edge list"   and a :: "'edge"   and as :: "'edge list" shows "valid_path_aux cs (a # as) = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> valid_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> valid_path_aux [] as | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> valid_path_aux cs' as | _ \<Rightarrow> valid_path_aux cs as)"
individually_rational_onI: fixes ds :: "'b set"   and X :: "'a set" assumes "CD_on ds X = X"   and "CH X = X" shows "individually_rational_on ds X"
strict_suffix_tl: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "strict_suffix (tl xs) xs"
type_less_x_OclAny: fixes \<tau> :: "'a basic_type" assumes "\<tau> < OclAny" obtains "\<tau> = OclVoid"   | "\<tau> = Boolean"   | "\<tau> = Integer"   | "\<tau> = UnlimitedNatural"   | "\<tau> = Real"   | "\<tau> = String"   | \<E> :: "('a, String.literal) phantom" where "\<tau> = Enum \<E>"   | \<C> :: "'a" where "\<tau> = \<langle>\<C>\<rangle>\<^sub>\<T>"
hyps_1: shows "Array.length h a = Array.length h' a \<and> sorted (take j (Array.get h' a)) \<and> mset (Array.get h a) = mset (Array.get h' a)"
Suc_sqrt_power2_gt: fixes n :: "nat" shows "n < (Suc (local.sqrt n))\<^sup>2"
comp_null_2: fixes f :: "'b" shows "C' f C'.null = C'.null"
list_induct2: fixes xs :: "'d list"   and ys :: "'e list"   and P :: "'d list \<Rightarrow> 'e list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
div_unit_factor: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "divide a (unit_factor a) = normalize a"
additive_strength_inv: fixes f :: "'b \<Rightarrow> 'b"   and xi :: "'b Gromov_completion" assumes "isometry f"   and "Gromov_extension f xi = xi" shows "additive_strength (inv f) xi = - additive_strength f xi"
polysub_normh: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "isnpolyh (p -\<^sub>p q) (min n0 n1)"
add: shows "ell_field_class.add a p\<^sub>1 p\<^sub>2 = p\<^sub>2"
i_shrink_last_def: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat" shows "f \<div>\<^sub>l k \<equiv> i_aggregate f k last"
least: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'a" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "least f = (w (f, 0) = w0 \<and> (\<forall>g. w (g, 0) = w0 \<longrightarrow> pr_weak (g, 0) (f, 0)))"
split_face_def: fixes f :: "face"   and ram\<^sub>1 :: "nat"   and ram\<^sub>2 :: "nat"   and newVs :: "nat list" shows "split_face f ram\<^sub>1 ram\<^sub>2 newVs \<equiv> let vs = vertices f; f\<^sub>1 = [ram\<^sub>1] @ between vs ram\<^sub>1 ram\<^sub>2 @ [ram\<^sub>2]; f\<^sub>2 = [ram\<^sub>2] @ between vs ram\<^sub>2 ram\<^sub>1 @ [ram\<^sub>1] in (Face (rev newVs @ f\<^sub>1) Nonfinal, Face (f\<^sub>2 @ newVs) Nonfinal)"
cases: fixes r :: "obj" obtains tag :: "obj_tag" and "values" :: "vn \<Rightarrow> val option" where "r = \<lparr>tag = tag, values = values\<rparr>"
A6: shows "\<forall>a b. B\<^sub>\<real> a b a \<longrightarrow> a = b"
doesnt_read_mutually_exclusive: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and c :: "'a"   and x :: "'b"   and mds :: "Mode \<Rightarrow> 'b set"   and mem :: "'b \<Rightarrow> 'c"   and c' :: "'a"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.doesnt_read eval c x"   and "(((c, mds), mem), (c', mds'), mem') \<in> eval"   and "\<forall>v. (((c, mds), mem(x := v)), (c', mds'), mem'(x := v)) \<in> eval" shows "\<not> (\<forall>v. (((c, mds), mem(x := v)), (c', mds'), mem') \<in> eval)"
strip_step': fixes S :: "'a Abs_State.st option"   and c :: "'a Abs_State.st option acom" shows "strip (step' S c) = strip c"
hyps: fixes x :: "real_alg_3" shows "real_of_3 x \<in>\<^sub>i get_itvl_3 ((tighten_bounds_3 ^^ n) x)"
key_compare_Gt: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(key_compare ko x y = Gt) = (key_compare ko y x = Lt)"
insert_Diff_if: fixes x :: "'e"   and A :: "'e set"   and B :: "'e set" shows "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
eval: shows "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"
invert_opposite_sides_of_square: fixes f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "B.seq f g"   and "f \<cdot>\<^sub>B g = h \<cdot>\<^sub>B k"   and "B.iso f"   and "B.iso k" shows "B.seq g (B.inv k) \<and> B.seq (B.inv f) h \<and> g \<cdot>\<^sub>B B.inv k = B.inv f \<cdot>\<^sub>B h"
cod_trg: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.cod (Span.trg \<mu>) = Span.trg \<mu>"
list_comb: fixes f :: "sterm"   and xs :: "sterm list" shows "S (list_comb f xs) = (S f \<and> list_all S xs)"
lift_character_lower_character: fixes \<chi> :: "'a \<Rightarrow> complex" assumes "\<chi> \<in> characters (G\<lparr>carrier := adjoin G H' a\<rparr>)" shows "H'.lift_character (\<lambda>x. if x \<in> H' then \<chi> x else 0, \<chi> a) = \<chi>"
hyps_3: shows "Fubx_get F2 g = Some gd"
list_eq_iff_nth_eq: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
assms_3: fixes c :: "'a Abs_State.st option acom" shows "strip (f c) = strip c"
qtable_join: fixes n :: "nat"   and A :: "nat set"   and P :: "'b option list \<Rightarrow> bool"   and Q1 :: "'b option list \<Rightarrow> bool"   and X :: "'b option list set"   and B :: "nat set"   and Q2 :: "'b option list \<Rightarrow> bool"   and Y :: "'b option list set"   and b :: "bool"   and C :: "nat set"   and Q :: "'b option list \<Rightarrow> bool" assumes "qtable n A P Q1 X"   and "qtable n B P Q2 Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A"   and "C = A \<union> B"   and "\<And>x. \<lbrakk>wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> P (restrict A x) \<and> P (restrict B x)"   and "\<And>x. \<lbrakk>b; wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> Q x = (Q1 (restrict A x) \<and> Q2 (restrict B x))"   and "\<And>x. \<lbrakk>\<not> b; wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> Q x = (Q1 (restrict A x) \<and> \<not> Q2 (restrict B x))" shows "qtable n C P Q (join X b Y)"
f2: shows "x \<in> (r |\<^bsub>rel\<^esub> s)"
IH: fixes tvs :: "t list"   and \<C> :: "t_context"   and arb_return :: "t list option"   and arb_labs :: "t list list"   and ts :: "t list"   and ts' :: "t list" assumes "store_typing s \<S>"   and "tvs = map typeof vls"   and "i < length (inst s)"   and "\<C> = (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ tvs, label := arb_labs, return := arb_return\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> es' : ts _> ts' \<and> map typeof vls = map typeof vs'"
H: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) H"
add_mset_remove_trivial_eq: fixes N :: "'b multiset"   and a :: "'b" shows "(N = add_mset a (remove1_mset a N)) = (a \<in># N)"
sdp_intros_5: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "sub_distrib_pconj a"   and "sub_distrib_pconj b"   and "healthy (wp a)"   and "healthy (wp b)"   and "nearly_healthy (wlp b)" shows "sub_distrib_pconj (a \<Sqinter> b)"
cong_right_commutativity: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A B C D" shows "Cong A B D C"
sF: fixes b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and s :: "(char list \<Rightarrow> int option) \<times> nat" shows "wp\<^sub>3\<^sub>' (WHILE b DO c) Q s = ((\<lambda>(ps, n). vars b \<subseteq> dom ps \<and> (if pbval b ps then wp\<^sub>3\<^sub>' c (wp\<^sub>3\<^sub>' (WHILE b DO c) Q) (ps, n) else Q (ps, n))) \<and>* $ 1) s"
kD_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "kD rt \<equiv> {dip. rt dip \<noteq> None}"
R3: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and X :: "'b set"   and Y :: "('b, 'c) ipurge_rec" assumes "\<lparr>Pol = I, Map = D, Doms = U, List = x # xs, ListOp = Some ys, Set = X\<rparr> \<in> ipurge_fail_aux_t_set Y"   and "\<not> (\<exists>u\<in>U. (u, D x) \<in> I)" shows "\<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some (ys @ [x]), Set = X\<rparr> \<in> ipurge_fail_aux_t_set Y"
scale_minus3': fixes a :: "real \<times> real"   and b :: "real \<times> real"   and x :: "real" shows "ccw' 0 a (b - x *\<^sub>R a) = ccw' 0 a b"
simps_1: fixes n :: "nat"   and us :: "'a vec list" shows "gram_schmidt_sub0 n us [] = us"
elims: fixes x :: "'a \<times> 'b \<times> 'c \<times> 'a"   and y :: "'a" assumes "mat_trace x = y" obtains a :: "'a" and b :: "'b" and c :: "'c" and d :: "'a" where "x = (a, b, c, d)" and "y = a + d"
fmrel_cases: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('c, 'a) fmap"   and n :: "('c, 'b) fmap"   and x :: "'c" assumes "fmrel P m n" obtains "fmlookup m x = None" and "fmlookup n x = None"   | a :: "'a" and b :: "'b" where "fmlookup m x = Some a" and "fmlookup n x = Some b" and "P a b"
cmp_imp_equal: fixes x :: "'a"   and y :: "'a" assumes "x \<in> D1"   and "y \<in> D1"   and "cmp1 x y = EQUAL" shows "x = y"
monom_closed: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom P a n \<in> carrier P"
subresultant_prs_locale_def: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly" shows "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2 \<equiv> primitive_remainder_sequence F n \<delta> f k \<beta> \<and> subresultant_prs_locale_axioms F G1 G2"
walk_in_V: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list" assumes "Digraph G"   and "Digraph.walk G xs" shows "set xs \<subseteq> V\<^bsub>G\<^esub>"
asIMOD_igAbsIPresIGWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsDisj SEM" shows "igAbsIPresIGWls (asIMOD SEM)"
all_clauses_Res_inv: fixes S :: "'b literal set set" shows "all_clauses (res S \<union> S) = all_clauses S"
INF_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "\<Sqinter> (C ` A) = \<Sqinter> (D ` B)"
row_transpose_is_col: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and i :: "nat" assumes "mat nr nc m"   and "i < nc" shows "row (Matrix_Legacy.transpose nr m) i = col m i"
Goal: fixes xveca :: "name list"   and Na :: "'a"   and Pa :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> M\<lparr>\<lambda>*xveca Na\<rparr>.Pa \<longmapsto> \<alpha> \<prec> P'"   and "xveca \<sharp>* \<Psi>"   and "xveca \<sharp>* M"   and "xveca \<sharp>* \<alpha>"   and "xveca \<sharp>* P'"   and "distinct xveca"   and "\<And>K Tvec. \<lbrakk>\<Psi> \<turnstile> M \<leftrightarrow> K; set xveca \<subseteq> supp Na; length xveca = length Tvec; distinct xveca\<rbrakk> \<Longrightarrow> Prop (K\<lparr>Na[xveca::=Tvec]\<rparr>) (Pa[xveca::=Tvec])" shows "Prop \<alpha> P'"
assms: shows "qGoodAbs (A $[[x \<and> y]]_zs)"
order_degree: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "order a p \<le> degree p"
eval_nnf: fixes \<phi> :: "atom fm" shows "\<forall>\<Gamma>. eval \<phi> \<Gamma> = eval (nnf \<phi>) \<Gamma>"
\<G>\<^sub>F\<^sub>G_suffix: fixes \<phi> :: "'b ltl"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<^sub>F\<^sub>G \<phi> (suffix i w) = \<G>\<^sub>F\<^sub>G \<phi> w"
forks_state_update_def: fixes fs :: "nat list"   and e :: "dining_event" shows "Up\<^sub>F fs e = (let i = fork e in fs[i := Up\<^sub>f i (fs ! i) e])"
greaterThan_iff: fixes i :: "nat list"   and k :: "nat list" shows "(i \<in> order_vec.greaterThan k) = (k <\<^sub>v i)"