hcju/isabelleps huggingface dataset:
concurrent_ops_commute_concurrent_set: fixes prefix :: "'a list"   and suffix :: "'a list"   and x :: "'a" assumes "concurrent_ops_commute (prefix @ suffix @ [x])"   and "concurrent_set x suffix"   and "distinct (prefix @ x # suffix)" shows "apply_operations (prefix @ suffix @ [x]) = apply_operations (prefix @ x # suffix)"
runit_in_vhom: fixes f :: "'d"   and a :: "'d" assumes "D.ide f"   and "src\<^sub>D f = a" shows "\<guillemotleft>\<r>\<^sub>D[f] : f \<star>\<^sub>D a \<Rightarrow>\<^sub>D f\<guillemotright>"
D: shows "offs_num n xs index key mi ma i \<le> offs_next (offs (enum xs index key n mi ma) k) (length xs + k) xs index key mi ma i - offs (enum xs index key n mi ma) k ! i"
fv_trms\<^sub>s\<^sub>s\<^sub>t_subset_1: fixes S :: "('d, 'c) stateful_strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>s\<^sub>t S) \<subseteq> vars\<^sub>s\<^sub>s\<^sub>t S"
set_of_mul_inc: fixes A :: "'b interval"   and A' :: "'b interval"   and B :: "'b interval"   and B' :: "'b interval" assumes "set_of A \<subseteq> set_of A'"   and "set_of B \<subseteq> set_of B'" shows "set_of (A * B) \<subseteq> set_of (A' * B')"
changes_nonneg: fixes xs :: "'b list" shows "0 \<le> changes xs"
Field_Osum: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set" shows "Field (r \<union>o r') = Field r \<union> Field r'"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "domain_semigroup.ds_ord (antidomain_left_monoid.am_d antirange_op) (\<lambda>x y. y \<cdot> x) b a"   and "domain_semigroup.ds_ord (antidomain_left_monoid.am_d antirange_op) (\<lambda>x y. y \<cdot> x) c b" shows "domain_semigroup.ds_ord (antidomain_left_monoid.am_d antirange_op) (\<lambda>x y. y \<cdot> x) c a"
semi_confluence_imp_CR: fixes r :: "('b \<times> 'b) set" assumes "r\<inverse> O r\<^sup>* \<subseteq> r\<^sup>\<down>" shows "CR r"
bv_msb_one_length: fixes w :: "bit list" assumes "bv_msb w = \<one>" shows "0 < length w"
FinalDeny_approximating_in_doubt_allow: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "(\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalDeny" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
positive_iff_decomp: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "positive A = (\<exists>M\<in>carrier_mat n n. M * adjoint M = A)"
M_neg_imp_z_non_zero: fixes v :: "(real, 3) vec" assumes "v \<bullet> (M *v v) < 0" shows "v $ 3 \<noteq> 0"
correct_state_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list" shows "correct_state P \<Phi> \<equiv> \<lambda>(xp, h, frs). case xp of None \<Rightarrow> case frs of [] \<Rightarrow> True | f # fs \<Rightarrow> P \<turnstile> h \<surd> \<and> (let (stk, loc, C, M, pc) = f in \<exists>Ts T mxs mxl\<^sub>0 is xt \<tau>. P \<turnstile> C sees M: Ts\<rightarrow>T = (mxs, mxl\<^sub>0, is, xt) in C \<and> \<Phi> C M ! pc = \<lfloor>\<tau>\<rfloor> \<and> conf_f P h \<tau> is f \<and> conf_fs P h \<Phi> M (length Ts) T fs) | \<lfloor>x\<rfloor> \<Rightarrow> frs = []"
assms_2: shows "u \<in> \<langle>G\<rangle>"
1: shows "f \<bullet> \<one> = \<one> \<ominus> a"
bij_betw_trans: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "bij_betw f A B"   and "bij_betw g B C" shows "bij_betw (g \<circ> f) A C"
intersection_number_def: fixes b1 :: "'c set"   and b2 :: "'c set" shows "b1 |\<inter>| b2 \<equiv> int (card (b1 \<inter> b2))"
tendstcomp: fixes l :: "'a \<times> 'b"   and F :: "'a filter"   and y :: "'c" assumes "((\<lambda>x. (x, f x)) \<longlongrightarrow> l) F"   and "(case l of (x, xa) \<Rightarrow> g x xa) = y" shows "((\<lambda>x. g x (f x)) \<longlongrightarrow> y) F"
post_fst_k_Suc: fixes i :: "nat" assumes "f i" shows "post_fst_k (Suc i) = testN i + post_fst_k i"
infinite_stable_exists: fixes R :: "('a \<times> 'a) set set" assumes "\<forall>r\<in>R. Card_order r" shows "\<exists>A. infinite A \<and> stable |A| \<and> (\<forall>r\<in>R. r <o |A|)"
iso_is_arr: fixes f :: "'s" assumes "local.iso f" shows "arr f"
Liminf_eq: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "Liminf F f = Liminf F g"
THEN_refine_1: fixes wl :: "'V \<Rightarrow> 'W infty option"   and res :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and it :: "('W \<times> 'V) set"   and wl' :: "'V \<Rightarrow> 'W infty option"   and res' :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and w' :: "'W"   and v'a :: "'V" assumes "uinvarm v (\<alpha>w wl) (\<alpha>r res) it (wl', res')"   and "dinvarm (wl, res)"   and "mpath_weight' (res v) + Num w' < mpath_weight' (res' v'a)"   and "path_weight' (\<alpha>r res v) + Num w' < path_weight' (\<alpha>r res' v'a)"   and "v'a \<in> \<alpha>w wl'"   and "\<alpha>r res v \<noteq> None" shows "v'a \<in> dom wl'"
ppow_pdivides: shows "[\<one>, \<ominus> x] [^]\<^bsub>poly_ring R\<^esub> n pdivides [\<one>, \<ominus> x] [^]\<^bsub>poly_ring R\<^esub> alg_mult p x"
FinDimVectorSpace: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V" shows "FinDimVectorSpace (aezfun_scalar_mult.fsmult smult) V"
subtype_def: fixes t :: "bexp set"   and P :: "bexp set"   and t' :: "bexp set" shows "subtype t P t' \<equiv> pred_entailment (P \<union> t') t"
subst_idem_support: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "subst_idem \<theta>" shows "\<theta> supports \<theta> \<circ>\<^sub>s \<delta>"
unit_factor_gauss_int_def: fixes z :: "gauss_int" shows "unit_factor z = (if z = 0 then 0 else if 0 \<le> ImZ z \<and> 0 < ReZ z then 1 else if ReZ z \<le> 0 \<and> 0 < ImZ z then \<i>\<^sub>\<int> else if ImZ z \<le> 0 \<and> ReZ z < 0 then - 1 else - \<i>\<^sub>\<int>)"
atLeastatMost_psubset_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b \<subset> ord.atLeastAtMost less_eq c d) = ((\<not> less_eq a b \<or> less_eq c a \<and> less_eq b d \<and> (less c a \<or> less b d)) \<and> less_eq c d)"
domain: fixes T :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Ta :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "Domainp (pcr_rbt T Ta) = (\<lambda>x. \<exists>y. rel_rbt T Ta x y \<and> is_rbt y)"
nfv_def: fixes \<phi> :: "'a MFOTL.formula" shows "MFOTL.nfv \<phi> = Max (insert 0 (Suc ` fv \<phi>))"
app: fixes n :: "nat"   and t :: "Term_Class.term"   and u :: "Term_Class.term" shows "wellformed' n (app t u) = (wellformed' n t \<and> wellformed' n u)"
post_chainmap: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and \<Pi> :: "('c, 'd) chain" shows "post (chainmap f g \<Pi>) = f (post \<Pi>)"
vars_lift_add: fixes a :: "nat"   and b :: "nat"   and p :: "'a mpoly"   and q :: "'a mpoly" shows "vars (liftPoly a b (p + q)) \<subseteq> vars (liftPoly a b p) \<union> vars (liftPoly a b q)"
not_less: fixes y :: "'a"   and x :: "'a" shows "(\<not> y < x) = (x \<le> y)"
\<psi>\<^sub>1_values_even: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"   and "j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"   and "even i" shows "Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) $$ (i, j) = complex_of_real (1 / sqrt 2 ^ (n + 1))"
leq_zero: fixes x :: "'a" assumes "Field r \<noteq> {}" shows "((x, zero) \<in> r) = (x = zero)"
ZFfunCod: fixes A :: "ZF"   and B :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "|cod|ZFfun A B f = B"
sequentially_even_odd_imp: fixes P :: "nat \<Rightarrow> bool" assumes "\<forall>\<^sub>F N in sequentially. P (2 * N)"   and "\<forall>\<^sub>F N in sequentially. P (2 * N + 1)" shows "eventually P sequentially"
measurable_marg_dens2: fixes x :: "nat"   and y :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "x \<in> V"   and "y \<in> V"   and "x \<noteq> y"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "marg_dens2 \<Y> x y \<rho> \<in> borel_measurable (stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y)))"
Labels_WhileExit: shows "\<langle>Skip,s\<rangle> \<rightarrow> \<langle>c'',s'\<rangle>"
le_supI1: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x a" shows "less_eq x (sup a b)"
cdens_ctxt_invar_empty: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>" shows "cdens_ctxt_invar [] (vs @ vs') \<Gamma> (CReal 1)"
outstanding_non_volatile_writes_unshared_nth_update: fixes i :: "nat"   and sb :: "'a memref list"   and p :: "'a"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "non_volatile_writes_unshared \<S>\<^sub>s\<^sub>b sb" shows "outstanding_non_volatile_writes_unshared \<S>\<^sub>s\<^sub>b (ts\<^sub>s\<^sub>b[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
hermitean_unit_circle: shows "hermitean (1, 0, 0, - 1)"
existence_ivl0_def: fixes x0 :: "'a" shows "Y.existence_ivl0 x0 = ll_on_open.existence_ivl UNIV (\<lambda>_. g) Y 0 x0"
card_insert_le: fixes A :: "'g set"   and x :: "'g" assumes "finite A" shows "card A \<le> card (insert x A)"
take_Suc: fixes xs :: "'g list"   and n :: "nat" assumes "xs \<noteq> []" shows "take (Suc n) xs = hd xs # take n (tl xs)"
term_of_anything: fixes x :: "'a"   and t :: "term" shows "term_of_class.term_of x \<equiv> t"
lcs_red_single_snd_crit_pair: fixes q :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "q \<noteq> 0"   and "component_of_term (lt p) = component_of_term (lt q)" shows "red_single (monomial (- (1::'c)) (term_of_pair (lcs (lp p) (lp q), component_of_term (lt p)))) (snd (crit_pair p q)) q (lcs (lp p) (lp q) - lp q)"
inner_prod_is_sesquilinear: fixes u1 :: "complex vec"   and n :: "nat"   and u2 :: "complex vec"   and v1 :: "complex vec"   and v2 :: "complex vec"   and c1 :: "complex"   and c2 :: "complex"   and c3 :: "complex"   and c4 :: "complex" assumes "dim_vec u1 = n"   and "dim_vec u2 = n"   and "dim_vec v1 = n"   and "dim_vec v2 = n" shows "\<langle>c1 \<cdot>\<^sub>v u1 + c2 \<cdot>\<^sub>v u2|c3 \<cdot>\<^sub>v v1 + c4 \<cdot>\<^sub>v v2\<rangle> = cnj c1 * c3 * \<langle>u1|v1\<rangle> + cnj c2 * c3 * \<langle>u2|v1\<rangle> + cnj c1 * c4 * \<langle>u1|v2\<rangle> + cnj c2 * c4 * \<langle>u2|v2\<rangle>"
assms: shows "find_largest_block block blocks = (m_b, m_e)"
rrr: shows "round_robin_reschedule t0 queue n0 s = \<lfloor>(t, None, \<sigma>')\<rfloor>"
assms_3: fixes i :: "nat" assumes "i < n" shows "v $ i \<le> w $ i"
in_set_remove1D: fixes a :: "'a"   and x :: "'a"   and xs :: "'a list" assumes "a \<in> set (remove1 x xs)" shows "a \<in> set xs"
inverse_2_2: fixes x :: "real_alg_2" assumes "invariant_2 x" shows "invariant_2 (inverse_2 x)"
zero_stalk_def: fixes V :: "'a set" shows "zero_stalk V \<equiv> \<lfloor>V, \<zero>\<^bsub>V\<^esub>\<rfloor>"
Fvars_fm_simps_3: fixes A :: "fm"   and B :: "fm" shows "Fvars (A OR B) = Fvars A \<union> Fvars B"
dg_Set_is_arrI: fixes \<alpha> :: "V"   and S :: "V"   and A :: "V"   and B :: "V" assumes "arr_Set \<alpha> S"   and "S\<lparr>ArrDom\<rparr> = A"   and "S\<lparr>ArrCod\<rparr> = B" shows "S : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B"
Derives1_unique_dest: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list"   and w :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derives1 \<NN> \<TT> \<RR> u i r v"   and "CFG.Derives1 \<NN> \<TT> \<RR> u i r w" shows "v = w"
Nu: shows "\<forall>n\<ge>Nu. \<forall>m\<ge>Nu. dist (u n) (u m) < e / 2"
equiv_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<longleftrightarrow>. q) = (A \<union> {p} \<turnstile> q \<and> A \<union> {q} \<turnstile> p)"
sym_order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "sympartp (\<sqsubseteq>) x y"   and "z \<sqsubseteq> y"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sqsubseteq> x"
xor_compl_right: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> \<sim> y = \<sim> (x \<oplus> y)"
ideal_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "'b set" assumes "class.ring plus zero minus uminus times" shows "ring.ideal plus zero uminus times I = (ring.left_ideal plus zero uminus times I \<and> ring.right_ideal plus zero uminus times I)"
splus_adds_termI_1: fixes t :: "'a"   and v :: "'t"   and u :: "'t" assumes "t adds\<^sub>p v"   and "u adds\<^sub>t v \<ominus> t" shows "t \<oplus> u adds\<^sub>t v"
less_eq_real_alg_def: fixes x :: "real_alg"   and y :: "real_alg" shows "(x \<le> y) = (real_of x \<le> real_of y)"
inv_aval'_correct: fixes s :: "char list \<Rightarrow> int"   and S :: "'av st option"   and e :: "aexp"   and a :: "'av" assumes "s \<in> \<gamma>\<^sub>o S"   and "aval e s \<in> \<gamma> a" shows "s \<in> \<gamma>\<^sub>o (inv_aval' e a S)"
lin_dep_cols_imp_det_0': fixes ws :: "'b vec list"   and n :: "nat" assumes "\<forall>w\<in>set ws. dim_vec w = n"   and "mat_of_cols n ws \<in> carrier_mat n n"   and "module.lin_dep class_ring (module_vec TYPE('b) n) (set (cols (mat_of_cols n ws)))" shows "det (mat_of_cols n ws) = (0::'b)"
adjacent_sym: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v' :: "'a" assumes "valid_unMultigraph G" shows "valid_unMultigraph.adjacent G v v' = valid_unMultigraph.adjacent G v' v"
blues_seen_others_2: fixes w :: "'person \<Rightarrow> 'color"   and p' :: "'person"   and p :: "'person" assumes "w p' = blue"   and "p \<noteq> p'"   and "w p \<noteq> blue" shows "card (local.blues_seen w p) = Suc (card (local.blues_seen w p'))"
FinGroupRepresentation_FG: assumes "of_nat (card G) \<noteq> (0::'a)" shows "FinGroupRepresentation G (*) ActingGroup.group_ring"
subst_ode: fixes I :: "('sf, 'sc, 'sz) interp"   and ODE :: "('sf, 'sz) ODE"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "osafe ODE"   and "ssafe \<sigma>"   and "Oadmit \<sigma> ODE (BVO ODE)" shows "ODE_sem I (Osubst ODE \<sigma>) (fst \<nu>) = ODE_sem (local.adjoint I \<sigma> \<nu>) ODE (fst \<nu>)"
ordertype_WW_3: shows "\<omega> \<up> \<omega> \<le> ordertype WW (lenlex less_than)"
lift_lift: fixes x :: "V"   and y :: "V"   and z :: "V" shows "lift x (lift y z) = lift (x + y) z"
not_tendsto_frequentlyE: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" assumes "\<not> (f \<longlongrightarrow> l) F" obtains S :: "'c set" where "open S" and "l \<in> S" and "\<exists>\<^sub>F x in F. f x \<notin> S"
predicate2D_vimage2p: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and S :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" assumes "R \<le> BNF_Def.vimage2p f g S"   and "R x y" shows "S (f x) (g y)"
compute_SCC_impl_refine: shows "compute_SCC_impl \<le> \<Down> Id compute_SCC"
equivalence_and_implicates: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set" assumes "equivalent S1 S2" shows "implicates S1 = implicates S2"
midpoint_col: fixes M :: "'p"   and A :: "'p"   and B :: "'p" assumes "M Midpoint A B" shows "Col M A B"
f3: shows "\<forall>A a. (\<exists>Aa f. a = (\<Sum>a\<in>Aa. f a *\<^sub>C a) \<and> finite Aa \<and> Aa \<subseteq> A) \<or> a \<notin> cspan A"
order_conv_closed: fixes x :: "'a" assumes "order x" shows "order (x\<^sup>T)"
finite_set_uprod: fixes x :: "'b uprod" shows "finite (set_uprod x)"
that: fixes m :: "nat"   and b :: "'a"   and q :: "'a list" assumes "[hd w] \<^sup>@ m \<cdot> [b] \<cdot> q = w"   and "b \<noteq> hd w"   and "m \<noteq> 0" shows "thesis"
star_rtc_least: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" assumes "Id \<union> x \<union> y O y \<subseteq> y" shows "x\<^sup>* \<subseteq> y"
list_disj_def: fixes fs :: "'a fm list" shows "list_disj fs = foldr or fs FalseF"
right_unique_alt_def: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "right_unique R = (R\<inverse>\<inverse> OO R \<le> (=))"
card_of_ordIso_finite: fixes A :: "'a set"   and B :: "'b set" assumes "ordIso2 (card_of A) (card_of B)" shows "finite A = finite B"
gd_nat_term_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and cmp_term :: "'a nat_term_order" shows "gd_nat_term pair_of_term term_of_pair cmp_term \<equiv> gd_term pair_of_term term_of_pair (\<lambda>s t. le_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (term_of_pair (s, the_min)) (term_of_pair (t, the_min))) (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term) \<and> gd_nat_term_axioms pair_of_term term_of_pair"
out_col: 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 "Tarski_neutral_dimensionless.Out Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet A B C"
red_def: fixes F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" shows "punit.red F p q = (\<exists>f\<in>F. \<exists>t. punit.red_single p q f t)"
eq_SConsD: fixes xs :: "'c stream"   and y :: "'c"   and ys :: "'c stream" assumes "xs = y ## ys" shows "shd xs = y \<and> stl xs = ys"
mempty_left: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and liftemb :: "'e\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'e\<^sub>\<bottom>"   and liftdefl :: "'e itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'e \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'e"   and defl :: "'e itself \<Rightarrow> udom defl"   and mempty :: "'e"   and mappend :: "'e \<rightarrow> 'e \<rightarrow> 'e"   and ys :: "'e" assumes "class.monoid below liftemb liftprj liftdefl emb prj defl mempty mappend" shows "mappend\<cdot>mempty\<cdot>ys = ys"
Set_filter_empty: fixes P :: "'b \<Rightarrow> bool" shows "Set.filter P {} = {}"
inf_Pair_Pair: fixes a :: "'a"   and b :: "'b"   and c :: "'a"   and d :: "'b" shows "inf (a, b) (c, d) = (inf a c, inf b d)"
length_negGorbit_list_sublist: fixes fs :: "'v list"   and gs :: "'g list"   and T :: "'b \<Rightarrow> 'v"   and as :: "'b list" assumes "fs \<in> set (negGorbit_list gs T as)" shows "length fs = length as"
coincidence_ode': fixes ODE :: "('d, 'e) ODE"   and \<nu> :: "(real, 'e) vec"   and \<nu>' :: "(real, 'e) vec"   and I :: "('d, 'f, 'e) interp"   and J :: "('d, 'f, 'e) interp" assumes "osafe ODE"   and "VSagree \<nu> \<nu>' (FVO ODE)"   and "Iagree I J ({Inl x |x. Inl x \<in> SIGO ODE} \<union> {Inr (Inr x) |x. Inr x \<in> SIGO ODE})" shows "ODE_sem I ODE \<nu> = ODE_sem J ODE \<nu>'"
Union_image_empty: fixes A :: "'b set"   and f :: "'c \<Rightarrow> 'b set" shows "A \<union> \<Union> (f ` {}) = A"
frombd_F22_tobd_F22: fixes b :: "'d"   and x :: "('e, 'd, 'f) F2"   and r1 :: "(bd_type_F1 \<times> bd_type_F1) set" assumes "b \<in> F2set2 x" shows "fromCard (F2set2 x) (dir_image (r1 +c bd_F2) Abs_bd_type_F) (toCard (F2set2 x) (dir_image (r1 +c bd_F2) Abs_bd_type_F) b) = b"
bex_UNIV: fixes P :: "'d \<Rightarrow> bool" shows "Bex UNIV P = Ex P"
Lt_lt_conv: fixes x :: "'a"   and y :: "'a" shows "(cmp x y = Lt) = cmp.lt x y"
assms_7: shows "subst_domain \<I> = subst_domain \<theta>"
ide_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "VxVxV.ide f = (ide (fst f) \<and> VxV.ide (snd f))"
new_tv_id_subst: fixes n :: "nat" shows "new_tv n id_subst"
psubstBinp_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set option"   and binp :: "'b \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qAbs set option" shows "binp %%[rho] \<equiv> lift (psubstAbs rho) binp"
pr_iso_test_correct: fixes Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list" assumes "\<forall>F\<in>set Fs\<^sub>1. distinct F"   and "\<forall>F\<in>set Fs\<^sub>2. distinct F"   and "[] \<notin> set Fs\<^sub>2"   and "distinct Fs\<^sub>1"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>1)"   and "distinct Fs\<^sub>2"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)" shows "pr_iso_test Fs\<^sub>1 Fs\<^sub>2 = (\<exists>\<phi>. is_pr_iso \<phi> Fs\<^sub>1 Fs\<^sub>2)"
inv_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "B.iso f"   and "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>B.inv f : b \<rightarrow>\<^sub>B a\<guillemotright>"
circle_arcs_neq: fixes k :: "real"   and n :: "real"   and m :: "real"   and q :: "real" assumes "0 \<le> k"   and "k \<le> 1"   and "0 \<le> n"   and "n \<le> 1"   and "n < k"   and "k + n < 1" shows "subcube k m (1, circle_polar) \<noteq> subcube n q (1, circle_polar)"
shift_compxE2: fixes pc :: "nat"   and e :: "'a expr1"   and pc' :: "nat"   and d :: "nat" shows "shift pc (compxE2 e pc' d) = compxE2 e (pc' + pc) d"
path_component_of_refl: fixes X :: "'a topology"   and x :: "'a" shows "path_component_of X x x = (x \<in> topspace X)"
enc: fixes f :: "'trm \<Rightarrow> 'trm" assumes "f \<in> Ops" shows "tenc f \<in> num"
prems_3: shows "\<forall>i<length cs. call_of_return_node (tl ms ! i) (sourcenode (cs ! i))"
exhaust: fixes y :: "'b DBMEntry" obtains x1 :: "'b" where "y = Le x1"   | x2 :: "'b" where "y = Lt x2"   | "y = \<infinity>"
case_2: fixes f1 :: "nat \<Rightarrow> 'b"   and f2 :: "'b" shows "(case \<infinity> of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f2"
sets_Sup_measure': fixes M :: "'b measure set"   and A :: "'b measure" assumes "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A"   and "M \<noteq> {}" shows "sets (Sup_measure' M) = sets A"
Order_adjunct_ord: fixes a :: "'a" assumes "a \<notin> carrier D" shows "Order (adjunct_ord D a)"
abelian_groupE_1: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"