hcju/isabelleps huggingface dataset:
bounded_matrix_for_marginals_ennreal: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and B :: "'b set"   and g :: "'b \<Rightarrow> ennreal"   and R :: "('a \<times> 'b) set" assumes "integral\<^sup>N (count_space A) f = integral\<^sup>N (count_space B) g"   and "integral\<^sup>N (count_space B) g \<noteq> \<top>"   and "\<And>X. X \<subseteq> A \<Longrightarrow> integral\<^sup>N (count_space X) f \<le> integral\<^sup>N (count_space (R `` X)) g"   and "countable A"   and "countable B"   and "R \<subseteq> A \<times> B" obtains h :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal" where "\<And>x y. 0 < h x y \<Longrightarrow> (x, y) \<in> R" and "\<And>x y. h x y \<noteq> \<top>" and "\<And>x. x \<in> A \<Longrightarrow> integral\<^sup>N (count_space B) (h x) = f x" and "\<And>y. y \<in> B \<Longrightarrow> (\<Sum>\<^sup>+ x\<in>A. h x y) = g y"
simps_1: fixes \<A> :: "'b \<Rightarrow> bool"   and k :: "'b" shows "\<A> \<Turnstile> Atom k = \<A> k"
S5Hist: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> $S5 rmhist p \<longrightarrow> (rmhist!p)$ = $(rmhist!p)"
that: fixes bij :: "'a1 \<Rightarrow> 'a2" assumes "bij_betw bij A A'"   and "\<forall>x\<in>A. f x = f' (bij x)" shows "thesis"
lunit_in_vhom: fixes f :: "'b"   and b :: "'b" assumes "D.ide f"   and "trg\<^sub>D f = b" shows "\<guillemotleft>\<l>\<^sub>D[f] : b \<star>\<^sub>D f \<Rightarrow>\<^sub>D f\<guillemotright>"
drop_map: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "map f xs \<up> n = map f (xs \<up> n)"
prems: shows "Fun f X \<cdot> \<theta> = Fun f X"
exists_complex_root_nonzero: fixes z :: "complex"   and n :: "nat" assumes "z \<noteq> 0"   and "n \<noteq> 0" obtains w :: "complex" where "w \<noteq> 0" and "z = w ^ n"
RCOSETS_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" shows "rcosets\<^bsub>G\<^esub> H = (\<Union>a\<in>carrier G. {H #>\<^bsub>G\<^esub> a})"
enc_satisfies_pred_iff_indRelR_satisfies_pred: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" shows "(\<forall>S. Pred (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>))) = (\<forall>(P, Q)\<in>indRelR. Pred (P, Q))"
count_mset: fixes xs :: "'k list"   and x :: "'k" shows "count (mset xs) x = length (filter ((=) x) xs)"
that: fixes x :: "'a" assumes "x \<in> set_spmf p" shows "S (f x) (g x)"
rule0: fixes a :: "'b" shows "a \<equiv> a + (0::'b)"
intro: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec" assumes "SES_valid SES"   and "isViewOn \<V> E\<^bsub>SES\<^esub>" shows "Unwinding SES \<V>"
exp_scaleR_diagonal1: fixes P :: "'a sq_mtx"   and A :: "'a sq_mtx"   and f :: "'a \<Rightarrow> real"   and t :: "real" assumes "mtx_invertible P"   and "A = P\<^sup>-\<^sup>1 * (\<d>\<i>\<a>\<g> i. f i) * P" shows "exp (t *\<^sub>R A) = P\<^sup>-\<^sup>1 * (\<d>\<i>\<a>\<g> i. exp (t * f i)) * P"
abs_harm: fixes n :: "nat" shows "\<bar>harm n\<bar> = harm n"
that: fixes Vs :: "'a list" assumes "set Vs \<subseteq> carrier R"   and "local.independent K Vs"   and "Span K Vs = Span K Us" shows "thesis"
globally_sound_modes_invariant: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem :: "'a \<Rightarrow> 'b"   and k :: "nat"   and cms' :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.globally_sound_mode_use eval (cms, mem)"   and "((cms, mem), k, cms', mem') \<in> sifum_security_init.meval eval" shows "sifum_security_init.globally_sound_mode_use eval (cms', mem')"
del_vert_simps_1: fixes u :: "'a" shows "verts (del_vert u) = verts G - {u}"
prems_4: shows "G' \<in> set (H # Hs)"
self_append_conv: fixes xs :: "'g list"   and ys :: "'g list" shows "(xs = xs @ ys) = (ys = [])"
AE_upper_bound_inf_ereal: fixes F :: "'b \<Rightarrow> ereal"   and G :: "'b \<Rightarrow> ereal"   and M :: "'b measure" assumes "\<And>e. 0 < e \<Longrightarrow> AE x in M. F x \<le> G x + ereal e" shows "AE x in M. F x \<le> G x"
ccompare_finite_3_def: shows "ccompare \<equiv> Some compare"
inf_finfun_apply: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "($) (inf f g) = inf (($) f) (($) g)"
un_B: fixes x :: "'c dual" shows "un_B x \<in> UNIV"
PO_m1_inv2r_serv: shows "reach m1 \<subseteq> m1_inv2r_serv"
chain_imp_relpow: fixes S :: "nat \<Rightarrow> 'b"   and r :: "('b \<times> 'b) set"   and i :: "nat"   and j :: "nat" assumes "\<forall>i. (S i, S (Suc i)) \<in> r" shows "(S i, S (i + j)) \<in> r ^^ j"
Nv1_union_Nv2_subsetof_Nv: shows "N\<^bsub>\<V>1\<^esub> \<union> N\<^bsub>\<V>2\<^esub> \<subseteq> N\<^bsub>\<V>\<^esub>"
pos_less_minus_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less a (uminus (divide b c)) = less (times a c) (uminus b)"
SUM: shows "sum t (A x) = t j\<^sub>1 + t j\<^sub>2 + sum t (A x - {j\<^sub>1} - {j\<^sub>2})"
unnecessaryPhi_def: fixes g :: "'g"   and val :: "'val" shows "unnecessaryPhi g val \<equiv> phi g val \<noteq> None \<and> \<not> necessaryPhi' g val"
scf_nonempty: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "lowered.is_pref_profile R" shows "lowered R \<noteq> {}"
coll_scaleR_right_eq: fixes y :: "real \<times> real"   and r :: "real"   and x :: "real \<times> real" shows "coll 0 y (r *\<^sub>R x) = (r = 0 \<or> coll 0 y x)"
seq_chop_1: fixes s :: "'a"   and ss :: "('b \<times> 'a) list"   and ts :: "('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, ss @ ts) \<in> seq ars" shows "(s, ss) \<in> seq ars"
n_absorb2: fixes x :: "'a"   and y :: "'a" shows "n x \<cdot> (n x \<oplus> n y) = n x"
llist_inf_simps_3: fixes x :: "'a"   and xs :: "'a llist"   and y :: "'a"   and ys :: "'a llist" shows "LCons x xs \<sqinter> LCons y ys = (if x = y then LCons x (xs \<sqinter> ys) else LNil)"
exec_def: fixes p :: "prog"   and us :: "nat list" shows "exec p us \<equiv> case supply p us of None \<Rightarrow> None | Some \<E> \<Rightarrow> eval \<E> (body p)"
abs_eq: shows "1 = abs_fract (1::'a, 1::'a)"
succeed_def: fixes i :: "nat" shows "succeed i = {x. \<exists>n. rank x n = Some i \<and> token_run x n \<notin> F - {q\<^sub>0} \<and> token_run x (Suc n) \<in> F}"
s: fixes r :: "'d rexp"   and s :: "'d rexp" assumes "r \<approx> s" shows "s \<approx> r"
prems: shows "\<Gamma> \<turnstile> App q t \<leftrightarrow> u : T\<^sub>2"
size_mset_tree: fixes t :: "('c, 'd) BinomialTree" assumes "tree_invar t" shows "size (tree_to_multiset t) = 2 ^ rank t"
regular_def: fixes S :: "'b list set" shows "regular S \<equiv> \<exists>fsm. S = {xs. accepts fsm xs}"
AbstrLevels_S4opt_A23: fixes i :: "AbstrLevelsID" assumes "sS4opt \<in> AbstrLevel i" shows "sA23 \<notin> AbstrLevel i"
assms_1: shows "x \<succeq>[\<R>] y \<and> (y, x) \<notin> \<R>"
csppa_Formal_in_in_slice: fixes x :: "nat"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and m :: "'node"   and i :: "nat"   and S :: "'node SDG_node set" assumes "x < length fs"   and "Formal_in (m, x + i) \<in> S" shows "csppa m S i fs ! x = fs ! x"
carrier: shows "carrier (substitute (FBinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a)) (thunks \<Delta>) (Texp e\<cdot>a)) \<subseteq> edom (Aheap \<Delta> e\<cdot>a) \<union> edom (\<A>\<^bsub>a\<^esub> (Terms.Let \<Delta> e))"
Append_mono: fixes U :: "(symbol \<times> character list) set"   and V :: "(symbol \<times> character list) set"   and P :: "(symbol \<times> character list) list set"   and Q :: "(symbol \<times> character list) list set"   and k :: "nat" assumes "U \<subseteq> V"   and "P \<subseteq> Q" shows "Append U k P \<subseteq> Append V k Q"
Collect_neg_eq: fixes P :: "'b \<Rightarrow> bool" shows "\<lbrace>\<not> \<acute>P\<rbrace> = - \<lbrace>\<acute>P\<rbrace>"
wpo_s_imp_ns: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" assumes "local.wpo.wpo_s s t" shows "local.wpo.wpo_ns s t"
l11_44_2: 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 C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.LtA Bet Cong B A C B C A = Tarski_neutral_dimensionless.Lt Bet Cong B C B A"
geqr: fixes x :: "'a" assumes "x \<in> W \<union> S" shows "g x = r x"
space_stream_space: fixes M :: "'d measure" shows "space (stream_space M) = streams (space M)"
mset_right_cancel_union: fixes a :: "'f"   and A :: "'f multiset"   and B :: "'f multiset" assumes "a \<in># A + B"   and "a \<notin># B" shows "a \<in># A"
finite_Psegments_pos_linear: fixes P :: "real \<Rightarrow> bool"   and b :: "real"   and lb :: "real"   and c :: "real"   and ub :: "real" assumes "finite_Psegments P (b * lb + c) (b * ub + c)"   and "0 < b" shows "finite_Psegments (P \<circ> (\<lambda>t. b * t + c)) lb ub"
sndI: fixes x :: "'a \<times> 'c"   and y :: "'a"   and z :: "'c" assumes "x = (y, z)" shows "snd x = z"
retraction_prop: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "f \<circ> f = f" shows "(f x = x) = (x \<in> range f)"
conv_def: shows "conv \<equiv> foldl (\<lambda>s x. insert x s)"
subset_insert_iff: fixes A :: "'c set"   and x :: "'c"   and B :: "'c set" shows "(A \<subseteq> insert x B) = (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
predicate2D: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and y :: "'f" assumes "P \<le> Q"   and "P x y" shows "Q x y"
FEval_subst: fixes MI :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and A :: "form" shows "\<forall>e f. FEval MI e (subst f A) = FEval MI (e \<circ> f) A"
preserves_reflects_arr: fixes f :: "'a" shows "S.arr (\<rr> f) = S.arr f"
index_mult_mat_vec: fixes i :: "nat"   and A :: "'c mat"   and v :: "'c vec" assumes "i < dim_row A" shows "(A *\<^sub>v v) $ i = row A i \<bullet> v"
tor: shows "(\<sigma>, t) \<in> oreachable (opnet onp p\<^sub>2) (\<lambda>a b. oarrivemsg I a) (other U (net_tree_ips p\<^sub>2))"
grow_memory: fixes \<C> :: "t_context"   and n :: "nat" assumes "memory \<C> = Some n" shows "\<C> \<turnstile> [Grow_memory] : [T_i32] _> [T_i32]"
simps_2: fixes x :: "'a word"   and ys :: "'a word list" shows "signed.strict_sorted (x # ys) = (Ball (set ys) ((<s) x) \<and> signed.strict_sorted ys)"
betaupright_PositiveImplies: fixes va :: "'c zmultiset"   and vb :: "'c zmultiset" assumes "upright (va + vb)"   and "PositiveImplies va (va + vb)" shows "beta_upright va vb"
reachable_awalkI: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" shows "u \<rightarrow>\<^sup>* v"
case_swap: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'c"   and p :: "'e \<times> 'd" shows "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
induct: fixes P :: "pat \<Rightarrow> value \<Rightarrow> bool"   and a0 :: "pat"   and a1 :: "value" assumes "\<And>name v. P (Patvar name) v"   and "\<And>name ps name' vs. (\<And>x y. \<lbrakk>name = name' \<and> length ps = length vs; x \<in> set ps; y \<in> set vs\<rbrakk> \<Longrightarrow> P x y) \<Longrightarrow> P (Patconstr name ps) (Vconstr name' vs)"   and "\<And>v va vb vc. P (Patconstr v va) (Vabs vb vc)"   and "\<And>v va vb vc vd. P (Patconstr v va) (Vrecabs vb vc vd)" shows "P a0 a1"
assms_2: shows "chaos \<sqsubseteq> (d::'a::refinement_lattice)"
plus_eq_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and s :: "'b"   and t :: "'b" assumes "class.ninv_comm_monoid_add plus zero"   and "plus s t = zero" shows "s = zero"
assms: shows "fmdom' (fmrestrict_set vs (snd a)) = {}"
tNode_1: fixes p :: "ref" assumes "x \<in> set_of lt"   and "x \<noteq> Null"   and "low x \<noteq> Null"   and "Dag p low high lt" shows "low x \<in> set_of lt"
arccos_lbound: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "0 \<le> arccos y"
assms_1: shows "H \<turnstile> HaddP x y z"
lr_def: shows "lr \<equiv> Min {r. poly p r = 0 \<and> lb < r}"
hpsafe_Choice_simps: fixes a :: "('a, 'b, 'c) hp"   and b :: "('a, 'b, 'c) hp" shows "hpsafe (a \<union>\<union> b) = (hpsafe a \<and> hpsafe b)"
compute_except_poly_mapping: fixes xs :: "('c, 'd) fmap"   and S :: "'c set" shows "except (Pm_fmap xs) S = Pm_fmap (fmfilter (\<lambda>k. k \<notin> S) xs)"
insert_before_def: fixes ptr :: "(_) object_ptr"   and node :: "(_) node_ptr"   and child :: "(_) node_ptr option" shows "insert_before ptr node child = Heap_Error_Monad.bind (a_ensure_pre_insertion_validity node ptr child) (\<lambda>_. Heap_Error_Monad.bind (if Some node = child then a_next_sibling node else return child) (\<lambda>reference_child. Heap_Error_Monad.bind (get_owner_document ptr) (\<lambda>owner_document. Heap_Error_Monad.bind (adopt_node owner_document node) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes owner_document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes owner_document (remove1 node disc_nodes)) (\<lambda>_. a_insert_node ptr node reference_child))))))"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c llist" shows "(case x21 % x22 of <> \<Rightarrow> f1 | x % xa \<Rightarrow> f2 x xa) = f2 x21 x22"
derf: fixes x :: "real" assumes "a < x"   and "x < b" shows "(f has_real_derivative 0) (at x)"
reduction_upf_def: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv" shows "reduction_upf \<A> = lift_spmf prf_key_gen \<bind> (\<lambda>k. lift_spmf coin_spmf \<bind> (\<lambda>b. inline (intercept_upf k b) \<A> ({}, {}) \<bind> (\<lambda>(uu_, L, D). Generative_Probabilistic_Value.Done ())))"
edr_characterization_class: shows "class.elementary_divisor_ring TYPE('a) = (class.Hermite_ring TYPE('a) \<and> (\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})))"
unit_simps_1: fixes a :: "'a" assumes "obj a" shows "arr \<i>[a]"
HC_case_distinction: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G"   and "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> F \<^bold>\<rightarrow> G" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H G"
MPoly_inject: fixes x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and y :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(MPoly x = MPoly y) = (x = y)"
rv1_Inl_last: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and kl :: "(bd_type_F + bd_type_F) list"   and xb :: "bd_type_F"   and xa :: "'d" shows "\<exists>b1'. rv1 s1 s2 (kl @ [Inl xb]) xa = Inl b1'"
P2_inv_g_rewrite: fixes \<G> :: "'d cyclic_group"   and s :: "nat"   and u1' :: "nat" assumes "ot \<G>"   and "s < order \<G>" shows "inv\<^bsub>\<G>\<^esub> \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> (u1' + (order \<G> - s)) = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> s \<otimes>\<^bsub>\<G>\<^esub> inv\<^bsub>\<G>\<^esub> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> u1')"
idem_normalize: fixes a :: "'b" shows "lcm a a = normalize a"
lm013: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G"   and "finite G" shows "finite (Range a)"
mono_1: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and P' :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "(\<box>P) \<sigma>"   and "\<And>\<sigma>. P \<sigma> \<Longrightarrow> P' \<sigma>" shows "(\<box>P') \<sigma>"
sort_is_sorted: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "all_in_list p l"   and "distinct p"   and "singleCombinators p" shows "FWNormalisationCore.sorted (FWNormalisationCore.sort p l) l"
Rep_coreflexive: fixes x :: "'c coreflexive" shows "Rep_coreflexive x \<in> \<down>(1::'c)"
not_bn0: fixes p :: "fm" assumes "bound0 p" shows "bound0 (not p)"
igFreshClsSTR_def: fixes MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'k, 'l, 'm) model_scheme" shows "igFreshClsSTR MOD \<equiv> igFreshIGVar MOD \<and> igFreshIGAbs1STR MOD \<and> igFreshIGAbs2STR MOD \<and> igFreshIGOpSTR MOD"
inf_b_def: fixes a :: "'a"   and b :: "'a" shows "inf_b a b = -r (b l\<rightarrow> -l (b r\<rightarrow> a))"
PfP_graph_aux_def: shows "PfP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>PfP_aux k s y. x1 = y \<and> x2 = SyntaxN.Ex k (SyntaxN.Ex s (PrfP (Var s) (Var k) y)) \<and> atom k \<sharp> (s, y) \<and> atom s \<sharp> y)"
interpretation_grounds_all: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a" assumes "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "fv (\<theta> v) = {}"
X: shows "X closure_of U \<subseteq> X interior_of T"
oc_projection_def: fixes S :: "'a vec set"   and fi :: "'a vec" shows "oc_projection S fi \<equiv> SOME v. is_oc_projection v S fi"
assms_1: shows "matcher_agree_on_exact_matches \<gamma> common_matcher"
isomorphicI: fixes f :: "'a" assumes "A.iso f" shows "A.isomorphic (A.dom f) (A.cod f)"
wf_bs_I: fixes bs :: "'a list"   and p :: "poly"   and bs' :: "'a list" assumes "wf_bs bs p" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>bs @ bs'\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
abs_in_BasisE: fixes u :: "'a" assumes "\<bar>u\<bar> \<in> Basis" obtains "\<bar>u\<bar> = u"   | "\<bar>u\<bar> = - u"
mcont_enat_le_case: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "enat \<Rightarrow> enat \<Rightarrow> 'c"   and bot :: "'c" assumes "class.ccpo lub ord (mk_less ord)"   and "mcont Sup (\<le>) lub ord (\<lambda>x. f x (eSuc x))"   and "\<And>x. ord bot (f x (eSuc x))" shows "mcont Sup (\<le>) lub ord (\<lambda>x. case x of 0 \<Rightarrow> bot | eSuc x' \<Rightarrow> f x' x)"
add_assocD: fixes y :: "'c"   and z :: "'c"   and x :: "'c" shows "(D y z \<and> D x (y \<oplus> z)) = (D x y \<and> D (x \<oplus> y) z)"
big_unclocked_unchanged_3: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r3 :: "(v, v) result" assumes "evaluate False env s e (s', r3)" shows "clock s = clock s'"
assoc'_naturality: fixes f0 :: "'d"   and f1 :: "'d"   and f2 :: "'d" assumes "arr f0"   and "arr f1"   and "arr f2" shows "((f0 \<otimes> f1) \<otimes> f2) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom f0, local.dom f1, local.dom f2] = \<a>\<^sup>-\<^sup>1[cod f0, cod f1, cod f2] \<cdot> (f0 \<otimes> f1 \<otimes> f2)"
Maclaurin_all_le: fixes diff :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and x :: "real"   and n :: "nat" assumes "diff 0 = f"   and "\<forall>m x. (diff m has_real_derivative diff (Suc m) x) (at x)" shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> f x = (\<Sum>m<n. diff m 0 / fact m * x ^ m) + diff n t / fact n * x ^ n"
product_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "Group.group (product_group I G)"
comm_monoid_set_axioms: shows "comm_monoid_set (\<^bold>*) \<^bold>1"
substClosedSubset: fixes Rel :: "(pi \<times> pi) set" shows "substClosed Rel \<subseteq> Rel"
subset: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G" shows "H \<subseteq> carrier G"
Abs_regular_inverse: fixes y :: "'c" assumes "y \<in> regular_elements" shows "Rep_regular (Abs_regular y) = y"
Domainp_Domain_eq: fixes r :: "('d \<times> 'e) set" shows "Domainp (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> Domain r)"
path_connected_imp_connected_space: fixes X :: "'a topology" assumes "path_connected_space X" shows "connected_space X"
poincare_distance_def: fixes u :: "complex_homo"   and v :: "complex_homo" shows "poincare_distance u v = (THE d. poincare_distance_pred u v d)"
compose_simps_3: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list" shows "compose (\<sigma> # ss) = \<sigma> \<circ>\<^sub>s compose ss"
norm_divide_numeral: fixes a :: "'c"   and w :: "num" shows "norm (a / numeral w) = norm a / numeral w"
abs_triangle_ineq4: fixes abs :: "'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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (minus a b)) (plus (abs a) (abs b))"
preserves_iso: fixes f :: "'b" assumes "D.iso f" shows "D.iso (src\<^sub>D f)"
finsert_absorb2: fixes x :: "'c"   and A :: "'c fset" shows "finsert x (finsert x A) = finsert x A"
restrict_def: fixes r :: "('b \<times> 'b) set"   and S :: "'b set" shows "r \<restriction> S = {(x, y). x \<in> S \<and> y \<in> S \<and> (x, y) \<in> r}"
iffs: fixes I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b"   and I' :: "nat"   and M' :: "nat"   and N' :: "nat"   and S' :: "nat"   and more' :: "'b" shows "(\<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = \<lparr>vars.I = I', M = M', N = N', S = S', \<dots> = more'\<rparr>) = (I = I' \<and> M = M' \<and> N = N' \<and> S = S' \<and> more = more')"
mat_rel_init: shows "HMA_M3 (1\<^sub>m (dim_row A), A, 1\<^sub>m (dim_col A)) (mat (1::'a), A', mat (1::'a))"
topological_basis_iff: fixes "open" :: "'c set \<Rightarrow> bool"   and B :: "'c set set" assumes "class.topological_space open"   and "\<And>B'. B' \<in> B \<Longrightarrow> open B'" shows "topological_space.topological_basis open B = (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
translation_inverse_subset: fixes a :: "'a"   and V :: "'a set"   and S :: "'a set" assumes "(+) (- a) ` V \<subseteq> S" shows "V \<subseteq> (+) a ` S"
assms: shows "\<forall>x y. SMT.trigger (SMT.Symb_Cons (SMT.Symb_Cons (SMT.pat (f x)) (SMT.Symb_Cons (SMT.pat (g y)) SMT.Symb_Nil)) SMT.Symb_Nil) (f x = g y)"
prems: shows "abc_steps_l (0, lm) B (Suc n) = (sa, lm')"
J: shows "{..<brnL cl (length cl)} = (\<Union>n<length cl. {brnL cl n..<brnL cl n + brn (cl ! n)})"
eql_neg_lit_eql_atm: fixes A' :: "'a"   and \<eta> :: "'s"   and A :: "'a" shows "(Neg A' \<cdot>l \<eta> = Neg A) = (A' \<cdot>a \<eta> = A)"
meet_in_inter: fixes l :: "Lines"   and m :: "Lines" assumes "l \<noteq> m" shows "meet_in l m (Pascal_Property.inter l m)"
r_inv: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.group G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
sel: fixes x :: "hmultiset multiset" shows "hmsetmset (HMSet x) = x"