hcju/isabelleps huggingface dataset:
fundfold_comp_trivial_f\<C>_list: fixes rs :: "'a permutation list"   and Cs :: "'a set list" assumes "order.greater_eq (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) (set rs)"   and "\<forall>r\<in>set rs. \<forall>C\<in>set Cs. \<forall>(f, g)\<in>fundfoldpairs. r = Abs_induced_automorph f g \<longrightarrow> C \<in> f \<turnstile> \<C>" shows "fold fst (map Spair rs) \<Turnstile> Cs = Cs"
fset_raw_strong_cases: fixes xs :: "'b list" obtains "xs = []"   | ys :: "'b list" and x :: "'b" where "\<not> List.member ys x" and "xs \<approx> x # ys"
step: fixes f :: "('a, 'b, 'c) gpv"   and g :: "('a, 'b, 'c) gpv" assumes "X f g" shows "ord_spmf (rel_generat (=) (=) (rel_fun (=) X)) (the_gpv f) (the_gpv g)"
assms_1: shows "is_zero_row_upt_k i k A"
contr: shows "\<forall>i. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i) \<and> \<eta> \<notin> old (\<sigma> i)"
distinct_25: fixes ccs1 :: "ccs"   and ccs2 :: "ccs"   and name' :: "name"   and ccs' :: "ccs" shows "ccs1 \<parallel> ccs2 \<noteq> \<lparr>\<nu>name'\<rparr>ccs'"
iso_iff_mono_and_retraction: fixes f :: "'s" shows "local.iso f = (local.mono f \<and> retraction f)"
pos_add_strict: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "(0::'a) < a"   and "b < c" shows "b < a + c"
neg_eq_neg1_smult: fixes m :: "'m" assumes "m \<in> M" shows "- (1::'r) \<cdot> m = - m"
measurable_if_dens: fixes \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and M :: "(nat \<Rightarrow> val) measure"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and b :: "bool" assumes "\<delta> \<in> borel_measurable M"   and "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M count_space (range BoolVal))" shows "if_dens \<delta> f b \<in> borel_measurable M"
elims_2: fixes x :: "'a extB"   and xa :: "'a extB" assumes "x \<le> xa" obtains uu_ :: "'a extB" where "xa__ = extB.Bot" and "xa = uu_"   | x :: "'a" and y :: "'a" where "xa__ = extB.Val x" and "xa = extB.Val y" and "x \<le> y"
ode_e_def: fixes ode_ops :: "ode_ops" shows "ode_operations.ode_e ode_ops = ode_expression ode_ops"
valuation_equivTr1: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "\<forall>x\<in>carrier K. 0 \<le> v x \<longrightarrow> 0 \<le> v' x" shows "carrier (Vr K v) \<subseteq> carrier (Vr K v')"
pw_leI: fixes S' :: "'b nres"   and S :: "'b nres" assumes "nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x)" shows "S \<le> S'"
cont_const: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'a" shows "cont luba orda Sup (\<le>) (\<lambda>x. c)"
ext_inject: fixes routing_match :: "'b prefix_match"   and metric :: "nat"   and routing_action :: "'b routing_action"   and more :: "'c"   and routing_match' :: "'b prefix_match"   and metric' :: "nat"   and routing_action' :: "'b routing_action"   and more' :: "'c" shows "(\<lparr>routing_match = routing_match, metric = metric, routing_action = routing_action, \<dots> = more\<rparr> = \<lparr>routing_match = routing_match', metric = metric', routing_action = routing_action', \<dots> = more'\<rparr>) = (routing_match = routing_match' \<and> metric = metric' \<and> routing_action = routing_action' \<and> more = more')"
strict_iff_order: fixes a :: "'c list"   and b :: "'c list" shows "strict_suffix a b = (suffix a b \<and> a \<noteq> b)"
assms_8: shows "n \<equiv> \<Prod>A1 * s1\<^sup>2"
pwt_eq: shows "\<forall>x\<in>vars (apps (Hd \<zeta>) ss). A (PWt x) = B (PWt x)"
fmdrop_idle': fixes x :: "'c"   and B :: "('c, 'd) fmap" assumes "x \<notin> fmdom' B" shows "fmdrop x B = B"
T_Write: fixes p :: "local_expr"   and \<tau> :: "thread_state"   and sh' :: "nat \<Rightarrow> nat"   and e1 :: "local_expr"   and e2 :: "local_expr"   and W' :: "nat set"   and a :: "abs_level" assumes "eval_bool p \<tau>"   and "sh' = (sh \<tau>)(eval_word e1 \<tau> := eval_word e2 \<tau>)"   and "W' = W \<tau> \<union> {eval_word e1 \<tau>}"   and "a \<in> {No_Abst, Eq_Abst}" shows "step_t a (\<tau>, Write e1 e2, p) (\<tau>\<lparr>sh := sh', W := W'\<rparr>)"
ord_power_aux: fixes m :: "nat"   and a :: "nat"   and k :: "nat" shows "ord m (a ^ k) * gcd k (ord m a) = ord m a"
npeSum2_sub_mulx2: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r \<Sigma>\<^sub>e R (\<lambda>i.  (n choose i) \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (n - i)\<^esup> \<cdot>\<^sub>r y^\<^bsup>R i\<^esup>)) n = \<Sigma>\<^sub>e R (\<lambda>i.  (n choose Suc i) \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (n - i)\<^esup> \<cdot>\<^sub>r (y^\<^bsup>R i\<^esup> \<cdot>\<^sub>r y))) n \<plusminus> (\<zero> \<plusminus> x \<cdot>\<^sub>r x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r 1\<^sub>r)"
next_Null: fixes xs :: "[:'r:]" assumes "xs \<noteq> \<bottom>" shows "next\<cdot>xs\<cdot>Null = Null"
filterlim_of_int_at_bot: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter" assumes "filterlim f F at_bot" shows "LIM x at_bot. f (real_of_int x) :> F"
inverse_unique: fixes u :: "'a"   and v' :: "'a"   and v :: "'a" assumes "u + v' = \<zero>"   and "v + u = \<zero>"   and "u \<in> R"   and "v \<in> R"   and "v' \<in> R" shows "v = v'"
pfun_plus_pos: fixes x :: "('a, 'b) pfun"   and y :: "('a, 'b) pfun" assumes "x + y = {}\<^sub>p" shows "x = {}\<^sub>p"
Cons_both_possible_rightE: fixes t' :: "'b ttree"   and x :: "'b"   and xs :: "'b list"   and t :: "'b ttree" assumes "possible t' x"   and "xs \<in> paths (t \<otimes>\<otimes> nxt t' x)" shows "x # xs \<in> paths (t \<otimes>\<otimes> t')"
invertible_left_inverse: fixes u :: "'a set" assumes "quotient.invertible u"   and "u \<in> Partition" shows "quotient.inverse u [\<cdot>] u = Class \<one>"
interchange_rows_i: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and i :: "'d"   and j :: "'d" shows "interchange_rows A i j $h i = A $h j"
trailIsDecreasedByDeciedUnitPropagateAndBacktrack: fixes stateA :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set"   and stateB :: "State" assumes "invariantsHoldInState stateA F0 decisionVars"   and "appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB F0 \<or> appliedBacktrack stateA stateB F0" shows "(getM stateB, getM stateA) \<in> lexLessRestricted (vars F0 \<union> decisionVars)"
mat_semiring: fixes n :: "nat"   and b :: "'d" shows "semiring (mat_ring n b)"
wf_nodes_le_auxi: shows "wf {(y, x). (Sorted_Less.sorted (rev y) \<and> Sorted_Less.sorted (rev x) \<and> set y \<subset> set x) \<and> x \<noteq> y}"
ang_circ_a_def: fixes E :: "complex"   and \<mu>1 :: "complex"   and \<mu>2 :: "complex"   and p1 :: "bool"   and p2 :: "bool" shows "ang_circ_a E \<mu>1 \<mu>2 p1 p2 = \<angle>a (circ_tang_vec \<mu>1 E p1) (circ_tang_vec \<mu>2 E p2)"
mem_infmal_iff: fixes x :: "'a star" shows "(x \<in> Infinitesimal) = (x \<approx> 0)"
valid_implies_valid_prog_hd: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "valid ts" shows "valid_prog (hd_prog p sb)"
hom_induced_id_gen: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'a"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and p :: "int"   and S :: "'a set" assumes "continuous_map X X f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = x"   and "c \<in> carrier (relative_homology_group p X S)" shows "hom_induced p X S X S f c = c"
simps_5: fixes bs :: "real list"   and a :: "MIR.num"   and b :: "MIR.num" shows "Inum bs (Add a b) = Inum bs a + Inum bs b"
pumping_lemma: fixes r :: "'a rexp" obtains n :: "nat" where "\<And>z. \<lbrakk>z \<in> lang r; n \<le> length z\<rbrakk> \<Longrightarrow> \<exists>u v w. z = u @ v @ w \<and> length (u @ v) \<le> n \<and> v \<noteq> [] \<and> (\<forall>i. u @ repeat i v @ w \<in> lang r)"
left_inverse_matrix_dim: fixes A :: "'b matrix"   and Y :: "'b matrix" assumes "left_inverse_matrix A Y" shows "ncols A = nrows Y"
presAtm_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and atm :: "'c" assumes "PL_Indis cval indis" shows "PL_Indis.presAtm aval indis atm \<equiv> \<forall>s. (s, aval atm s) \<in> indis"
StringLiteralTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and c :: "char list"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E Literal (StringLiteral c) : \<tau>" obtains "\<tau> = String[1]"
leq_trans: fixes x :: "'a nres"   and y :: "'a nres"   and z :: "'a nres" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
poly_reciprocal: fixes P :: "'b poly"   and p :: "nat"   and x :: "'b" assumes "degree P \<le> p"   and "x \<noteq> (0::'b)" shows "poly (reciprocal_poly p P) x = x ^ p * poly P (inverse x)"
permutes_not_in: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and x :: "'e" assumes "f permutes S"   and "x \<notin> S" shows "f x = x"
min_def: fixes a :: "'e word"   and b :: "'e word" shows "signed.min a b = (if a \<le>s b then a else b)"
step_to_steps: fixes Cal :: "'a processCalculus"   and P :: "'a"   and P' :: "'a" assumes "P \<longmapsto>Cal P'" shows "P \<longmapsto>Cal* P'"
ennreal_le_epsilon: fixes y :: "ennreal"   and x :: "ennreal" assumes "\<And>e. \<lbrakk>y < top; 0 < e\<rbrakk> \<Longrightarrow> x \<le> y + ennreal e" shows "x \<le> y"
lead_coeff_abs_int_poly: fixes p :: "int poly" shows "lead_coeff (abs_int_poly p) = \<bar>lead_coeff p\<bar>"
CAND_OrL_elim: fixes x :: "name"   and y :: "name"   and M :: "trm"   and z :: "name"   and N :: "trm"   and B :: "ty" assumes "x:OrL y.M z.N x \<in> \<parallel>(B)\<parallel>"   and "x:OrL y.M z.N x \<notin> BINDINGn B (\<parallel><B>\<parallel>)" shows "\<exists>B1 B2. B = B1 OR B2 \<and> y:M \<in> \<parallel>(B1)\<parallel> \<and> z:N \<in> \<parallel>(B2)\<parallel>"
simps_5: fixes C :: "char list" shows "default_val (Class C) = Null"
supp_AC: fixes x :: "'b \<times> 'b fs_set \<times> 'a" shows "supp (AC x) = supp x"
rename_literal_def: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a set"   and L :: "'a Literal" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.rename_literal A L = (if atom L \<in> A then complement L else L)"
r_1: shows "(INF i. emeasure (P (J i)) (B i)) = ennreal r"
Abs_oalist_tc_inverse: fixes y :: "('a \<times> 'b) list" assumes "y \<in> Collect tc.oalist_inv_raw" shows "list_of_oalist_tc (Abs_oalist_tc y) = y"
sequence_of_def: fixes A :: "'d set"   and enum :: "nat \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'c" shows "sequence_of A enum f = map (\<lambda>n. f (enum n)) [0..<card A]"
intros: fixes E :: "ctx"   and x :: "name"   and t :: "lam"   and t' :: "lam" shows "E\<lbrakk>App (Lam [x].t) t'\<rbrakk> \<longrightarrow>x E\<lbrakk>t[x::=t']\<rbrakk>"
replace_id: fixes x :: "'c"   and xs :: "'c list" shows "replace x [x] xs = xs"
matrix_vector_mult_add_rdistrib: fixes A :: "(('d, 'f) vec, 'e) vec"   and B :: "(('d, 'f) vec, 'e) vec"   and x :: "('d, 'f) vec" shows "(A + B) *v x = A *v x + B *v x"
root_info: fixes p :: "int poly" assumes "irreducible p"   and "0 < degree p" shows "root_info_cond (root_info p) p"
tfilter'_def: fixes b :: "unit \<Rightarrow> 'd" shows "tfilter' b = tfilter (b ())"
d_conc6: fixes x :: "'a"   and y :: "'a" shows "d (x \<parallel> y) = d x \<parallel> d y"
d'_def: shows "d' = Fn\<cdot>(\<Lambda> uu_. Fn\<cdot>(\<Lambda> uu_. \<bottom>))"
harry_step_Falses_True: fixes n :: "nat" assumes "0 < n" shows "harry_step (replicate n False @ [True]) = True # replicate (n - 1) False @ [True]"
totality: fixes ES :: "'b ES_rec"   and E :: "'b set"   and t :: "'b list"   and t' :: "'b list" assumes "EventSystems.total ES E"   and "t \<in> Tr\<^bsub>ES\<^esub>"   and "set t' \<subseteq> E" shows "t @ t' \<in> Tr\<^bsub>ES\<^esub>"
n_t_compl: fixes x :: "'a" shows "n x \<oplus> t x = (1::'a)"
one_smult: fixes m :: "'m" assumes "m \<in> M" shows "(1::'r) \<cdot> m = m"
sigma_algebra_UNIV: fixes a :: "'b set set" assumes "sigma_algebra a" shows "UNIV \<in> a"
assms_1: shows "s \<equiv> init_fin_lift_state Running (start_state J_local_start P C M vs)"
dist_vec_def: fixes x :: "('e, 'f) vec"   and y :: "('e, 'f) vec" shows "dist x y = L2_set (\<lambda>i. dist (x $ i) (y $ i)) UNIV"
complete_wellformed: fixes V :: "nat set" shows "uwellformed (Ugraph_Lemmas.complete V)"
Output: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<longmapsto> K\<langle>N\<rangle> \<prec> P"
assms: shows "(a \<le> b) = (f a \<le> f b)"
domain: fixes DR1 :: "nat \<Rightarrow> bool"   and DR2 :: "nat \<Rightarrow> bool" assumes "Domainp (=) = DR1"   and "Domainp (=) = DR2" shows "Domainp pcr_int = pred_prod DR1 DR2"
numeral_code_1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
Quantity_ext_def: fixes mag :: "'a"   and dim :: "(int, 'b) dimvec"   and more :: "'c" shows "\<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons Quantity_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple mag dim) more"
geodesic_segment_param_1: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "geodesic_segment_param G x 0 = x"
Uvariation_mon: fixes U :: "variable set"   and V :: "variable set"   and \<omega> :: "variable \<Rightarrow> real"   and \<nu> :: "variable \<Rightarrow> real" assumes "U \<subseteq> V"   and "Uvariation \<omega> \<nu> U" shows "Uvariation \<omega> \<nu> V"
Un_commute: fixes A :: "'e set"   and B :: "'e set" shows "A \<union> B = B \<union> A"
hyps_1: shows "rel_prod (=) R c\<^sub>1 c\<^sub>2"
SBReadUnbuffered: fixes sb :: "'c memref list"   and a :: "nat"   and volatile :: "bool"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "'d"   and \<O> :: "'e"   and \<R> :: "'f"   and \<S> :: "'g" assumes "buffered_val sb a = None" shows "(Read volatile a t # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b (is, \<theta>(t \<mapsto> m a), sb, m, \<D>, \<O>, \<R>, \<S>)"
transpose_minus: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "(A - B)\<^sup>T = A\<^sup>T - B\<^sup>T"
cp: shows "\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0"
ces_imp_ends_vertices: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat" assumes "sub_rel_of g subs"   and "SubRel.ces v1 es v2 subs"   and "set es \<subseteq> edges g"   and "v1 \<in> Graph.vertices g" shows "v2 \<in> Graph.vertices g"
poincare_circle_euclidean_def: fixes z :: "complex_homo"   and r :: "real" shows "poincare_circle_euclidean z r = (let R = (cosh r - 1) / 2; z' = to_complex z; cz = 1 - (cmod z')\<^sup>2; k = cz * R + 1 in (z' / cor k, cz * sqrt (R * (R + 1)) / k))"
intros_1: fixes a :: "'b"   and r :: "('b \<times> 'b) set" shows "(a, a) \<in> r\<^sup>*"
has_derivative_inverse': fixes x :: "'a"   and S :: "'a set" assumes "x \<noteq> (0::'a)" shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within S)"
fold_map: fixes g :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "List.fold g (map f xs) = List.fold (g \<circ> f) xs"
max_stutter_sampler_nostuttering: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and k :: "nat"   and n :: "nat" assumes "\<sigma> (max_stutter_sampler \<sigma> (Suc k)) = \<sigma> (max_stutter_sampler \<sigma> k)"   and "max_stutter_sampler \<sigma> k < n" shows "\<sigma> n = \<sigma> (max_stutter_sampler \<sigma> k)"
IH_2: shows "2 ^ min_height r \<le> size1 r"
phiUsesI: fixes n' :: "'node"   and g :: "'g"   and v' :: "'val"   and vs :: "'val list"   and n :: "'node"   and v :: "'val" assumes "n' \<in> set (\<alpha>n g)"   and "phis g (n', v') = Some vs"   and "(n, v) \<in> set (zip (old.predecessors g n') vs)" shows "v \<in> phiUses g n"
coinduct: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b" assumes "mono f"   and "X \<le> f (sup X (gfp f))" shows "X \<le> gfp f"
linear_surjective_isomorphism: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'e set"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B2 :: "'f set"   and f :: "'e \<Rightarrow> 'f" assumes "finite_dimensional_vector_space_pair s1 B1 s2 B2"   and "linear s1 s2 f"   and "surj f"   and "vector_space.dim s2 UNIV = vector_space.dim s1 UNIV" shows "\<exists>f'. linear s2 s1 f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
cong_identity: shows "\<forall>a b c. Cong a b c c \<longrightarrow> a = b"
Assign4: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3\<^sub>' {(\<lambda>(ps, t). x \<in> dom ps \<and> vars a \<subseteq> dom ps \<and> Q (ps(x \<mapsto> paval a ps), t)) \<and>* $ 1} x ::= a { Q}"
eventually_all_finite_mono: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'c filter"   and Q :: "'c \<Rightarrow> bool" assumes "\<forall>i. eventually (P i) F"   and "\<forall>x. (\<forall>i. P i x) \<longrightarrow> Q x" shows "eventually Q F"
Cut_a_redu_elim: fixes a :: "coname"   and M :: "trm"   and x :: "name"   and N :: "trm"   and R :: "trm" assumes "Cut <a>.M x.N \<longrightarrow>\<^sub>a R" shows "(\<exists>M'. R = Cut <a>.M' x.N \<and> M \<longrightarrow>\<^sub>a M') \<or> (\<exists>N'. R = Cut <a>.M x.N' \<and> N \<longrightarrow>\<^sub>a N') \<or> Cut <a>.M x.N \<longrightarrow>\<^sub>c R \<or> Cut <a>.M x.N \<longrightarrow>\<^sub>l R"
mult_cancel_le_iff: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "(\<alpha> * \<beta> \<subseteq>\<^sub>\<circ> \<alpha> * \<gamma>) = (\<beta> \<subseteq>\<^sub>\<circ> \<gamma> \<or> \<alpha> = 0)"
PORE: fixes a :: "Three \<Rightarrow> ValD" assumes "a \<in> unlr POR.delta" obtains "a = (\<lambda>i. ValTT)"   | "a = (\<lambda>i. ValFF)"   | n :: "nat" where "a = (\<lambda>i. ValN\<cdot>n)"   | "a One = \<bottom>"   | "a Two = \<bottom>"   | fs :: "Three \<Rightarrow> ValD \<rightarrow> ValD" where "a = (\<lambda>i. ValF\<cdot>(fs i))" and "\<And>xs. xs \<in> unlr POR.delta \<Longrightarrow> (\<lambda>j. fs j\<cdot>(xs j)) \<in> unlr POR.delta"
psiCasesFresh: fixes x :: "name"   and Cs :: "('f \<times> ('d, 'e, 'f) psi) list" shows "x \<sharp> psiCases Cs = x \<sharp> Cs"
PO_m1_inv0_fin_trans: shows "{m1_inv0_fin} TS.trans m1 {> m1_inv0_fin}"
invh_baldL_invc: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invh l"   and "invh r"   and "bheight l + 1 = bheight r"   and "invc r" shows "invh (baldL l a r) \<and> bheight (baldL l a r) = bheight l + 1"
box_stack_Nil: fixes f :: "'c" shows "Subx.box_stack f [] = []"
E19: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>F \<longrightarrow> \<box>\<diamond>\<box>F"
connected_edgeRtc: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and v :: "nat" shows "Graph.connected c u v = ((u, v) \<in> (Graph.E c)\<^sup>*)"
a_l_coset_carrier: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <+ H"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "y \<in> carrier G"
csq: shows "Maps.commutative_square \<lbrakk>\<lbrakk>t\<^sub>0\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>u\<^sub>1\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>"
inf_le2: fixes x :: "'b"   and y :: "'b" shows "inf x y \<le> y"
mono_pow: fixes f :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "mono f" shows "mono (f ^^ n)"
rep_eq: fixes x :: "complex_homo" shows "to_complex x = to_complex_homo_coords (rep_complex_homo x)"
continuous_on_norm: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. norm (f x))"
cases: fixes x :: "('a, 'b, 'c) SLin_state \<times> ('a, 'e, 'f, 'g) SLin_state_scheme" obtains s1 :: "('a, 'b, 'c) SLin_state" and s2 :: "('a, 'e, 'f, 'g) SLin_state_scheme" where "x = (s1, s2)"
ab: shows "a \<cdot> b - a' \<cdot> b \<in> I"
T_def: shows "T \<equiv> \<lambda>n. ln (real n) + euler_mascheroni + 1 / (2 * real n) - (\<Sum>i = 1..N. bernoulli (2 * i) / real (2 * i * n ^ (2 * i)))"
a2: shows "d i \<cdot> ad r \<le> d q"
case_distrib: fixes h :: "'f \<Rightarrow> 'e"   and f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and f3 :: "'h \<Rightarrow> 'g \<Rightarrow> 'f"   and step :: "('h, 'g) step" shows "h (case step of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = (case step of Done \<Rightarrow> h f1 | Skip x \<Rightarrow> h (f2 x) | Yield x1 x2 \<Rightarrow> h (f3 x1 x2))"
IHAPP_1: shows "(([entry fg p], {#}), w, [u], c') \<in> trcl (trss fg)"
multiplicative_function'_axioms: fixes n :: "nat" shows "multiplicative_function' (ind (is_nth_power n)) (\<lambda>p k. if n dvd k then 1::'b else (0::'b)) (\<lambda>_. if n = 1 then 1::'b else (0::'b))"
affine_hull_insert_span: fixes a :: "'a"   and S :: "'a set" assumes "a \<notin> S" shows "affine hull insert a S = {a + v |v. v \<in> span {x - a |x. x \<in> S}}"
\<tau>exec_movesI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_moves ci P t es h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s'"   and "\<tau>moves2 P h stk es pc xcp" shows "\<tau>exec_moves ci P t es h (stk, loc, pc, xcp) s'"
intros_4: fixes f :: "recf"   and k :: "nat"   and gs :: "recf list"   and m :: "nat"   and n :: "nat" assumes "primerec f k"   and "length gs = k"   and "\<forall>i<length gs. primerec (gs ! i) m"   and "m = n" shows "primerec (Cn n f gs) m"
mono_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<le> f y)"
jonsson1a: shows "(\<exists>f. \<forall>x y. x \<rhd> y = f x \<cdot> y) = (\<forall>x y. x \<rhd> y = (x \<rhd> (1::'a)) \<cdot> y)"
hom_components_of_homogeneous: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "homogeneous p" shows "hom_components p = (if p = 0 then {} else {p})"
right_neutral: fixes a :: "'c \<Rightarrow> 'c" shows "a \<circ> id = a"
restrict_map_insert: fixes f :: "'a \<Rightarrow> 'b option"   and a :: "'a"   and A :: "'a set" shows "f |` insert a A = (f |` A)(a := f a)"
SA_Top: fixes \<Gamma> :: "binding list"   and S :: "POPLmark.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f S" shows "\<Gamma> \<turnstile> S <: Top"
lift_imp_sameDom: fixes f :: "'c \<Rightarrow> 'b"   and inp :: "'a \<Rightarrow> 'c option"   and inp' :: "'a \<Rightarrow> 'c option" assumes "lift f inp = lift f inp'" shows "sameDom inp inp'"