hcju/isabelleps huggingface dataset:
C_eq_Sets_mr: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and s :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "set p = set s"   and "singleCombinators p"   and "wellformed_policy1_strong p"   and "wellformed_policy1_strong s"   and "wellformed_policy3Pr p"   and "wellformed_policy3Pr s"   and "allNetsDistinct p" shows "applied_rule_rev Cp x p = applied_rule_rev Cp x s"
left_tree_in_V: fixes s :: "'c"   and t :: "'c" shows "T.left_tree s t \<subseteq> V\<^bsub>T\<^esub>"
derived_blocks_def: shows "\<B>\<^sup>D \<equiv> image_mset ((\<inter>) bl) (remove1_mset bl \<B>)"
row_iarray_def: fixes k :: "nat"   and A :: "'d iarray iarray" shows "row_iarray k A = A !! k"
drop_all_iff: fixes n :: "nat"   and xs :: "'g list" shows "(drop n xs = []) = (length xs \<le> n)"
refine: fixes R_Iec :: "('c \<times> unit) set" shows "(impl_blue_dfs, blue_dfs) \<in> bg_impl_rel_ext R_Iec Id \<rightarrow> \<langle>\<langle>\<langle>Id\<rangle>list_rel \<times>\<^sub>r \<langle>Id\<rangle>list_rel\<rangle>option_rel\<rangle>nres_rel"
inEdges_def: fixes g :: "'a"   and n :: "'node" shows "i.inEdges g n \<equiv> map (\<lambda>(f, d). (f, d, n)) (inEdges2 g n)"
r_inv: fixes x :: "nat" assumes "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
sees': shows "compP2 P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>(max_stack body, max_vars body, compE2 body @ [Return], compxE2 body 0 0)\<rfloor> in D"
quot_of_fract_to_fract: fixes x :: "'b" shows "quot_of_fract (to_fract x) = (x, 1::'b)"
RID\<^sub>L_finite_invariant: fixes r :: "'r"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "revision_step r s s'"   and "finite (LID\<^sub>G s)" shows "finite (LID\<^sub>G s')"
chine_simps_2: shows "C.dom \<nu>\<mu>.chine = \<nu>\<mu>.dom.apex"
Tinv_quasi_measure_preserving: assumes "I.invertible_qmpt" shows "I.Tinv \<in> quasi_measure_preserving M M"
removeAll_commute: fixes x :: "'b"   and y :: "'b"   and l :: "'b list" shows "removeAll x (removeAll y l) = removeAll y (removeAll x l)"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "le b a"   and "le c b" shows "le c a"
infinite_Un: fixes S :: "'a set"   and T :: "'a set" shows "infinite (S \<union> T) = (infinite S \<or> infinite T)"
inC_def: shows "inC \<equiv> S.UP \<circ> Inl"
that_1: fixes b :: "'a" assumes "b \<in> B" shows "representation B v b = 0"
assms_1: shows "bounded {(f ^^ n) x |n. True}"
inj_on_id2: fixes A :: "'g set" shows "inj_on (\<lambda>x. x) A"
is_red_addsE': fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red F p" shows "\<exists>f\<in>F. \<exists>v\<in>keys p. f \<noteq> 0 \<and> lt f adds\<^sub>t v"
orthogonal_clauses_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and a :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner a zero"
emb_sfun_def: shows "EMB('e \<rightarrow>! 'f) = sfun_emb oo sfun_map\<cdot>PRJ('e)\<cdot>EMB('f)"
eq1: shows "adjoint (tensor_P m1 m2) = tensor_P (adjoint m1) (adjoint m2)"
continuous_map_cases_alt: fixes X :: "'b topology"   and P :: "'b \<Rightarrow> bool"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_map (subtopology X (X closure_of {x \<in> topspace X. P x})) Y f"   and "continuous_map (subtopology X (X closure_of {x \<in> topspace X. \<not> P x})) Y g"   and "\<And>x. x \<in> X frontier_of {x \<in> topspace X. P x} \<Longrightarrow> f x = g x" shows "continuous_map X Y (\<lambda>x. if P x then f x else g x)"
step_1_similar: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat (??.Jordan_Normal_Form_Existence.step_1 A) A"
real_arch_inverse: fixes e :: "real" shows "(0 < e) = (\<exists>n. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
is_increasing_def: fixes f :: "nat \<Rightarrow> nat" shows "is_increasing f = (\<forall>n m. m < n \<longrightarrow> f m < f n)"
rept_K: fixes \<mu> :: "('s \<times> ('c \<Rightarrow> real) set) pmf"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "\<mu> \<in> \<K> (abss s)" shows "rept s \<mu> \<in> K s"
sublist_valid_is_valid: fixes as :: "'a list"   and PROB :: "'a set"   and as' :: "'a list" assumes "as \<in> valid_plans PROB"   and "subseq as' as" shows "as' \<in> valid_plans PROB"
discover_def: fixes u :: "'v"   and v :: "'v"   and s :: "('v, 'es) state_scheme" shows "discover u v s \<equiv> let d = discovered s(v \<mapsto> counter s); c = counter s + 1; st = v # stack s; p = pending s \<union> {v} \<times> E `` {v}; t = insert (u, v) (tree_edges s) in s\<lparr>discovered := d, counter := c, stack := st, pending := p, tree_edges := t\<rparr>"
Min_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and x :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq (linorder.Min less_eq A) x"
trms\<^sub>s\<^sub>t_append: fixes A :: "('a, 'b) strand_step list"   and B :: "('a, 'b) strand_step list" shows "trms\<^sub>s\<^sub>t (A @ B) = trms\<^sub>s\<^sub>t A \<union> trms\<^sub>s\<^sub>t B"
Hausdorff_acTr: fixes C :: "'a set" assumes "C \<in> carrier fTo D" shows "S_inductive_set (Iod fTo D {S \<in> carrier fTo D. C \<subseteq> S})"
ide_in_hom: fixes a :: "'a" shows "C.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>C a\<guillemotright>"
simps_2: fixes n1 :: "'c net_state"   and n2 :: "'c net_state" shows "net_ips (SubnetS n1 n2) = net_ips n1 \<union> net_ips n2"
graph_from_list_def: fixes l :: "('b \<times> 'b) list" shows "graph_from_list l = foldr ins_edge l graph_empty"
ccINF_const: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and f :: "'a" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Inf ((\<lambda>i. f) ` A) = f"
strip_while_rev: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "strip_while P (rev xs) = rev (dropWhile P xs)"
plus_bit1_def: fixes x :: "'b bit1"   and y :: "'b bit1" shows "x + y = Abs_bit1' (Rep_bit1 x + Rep_bit1 y)"
fin_string_empty_def: shows "fin_string_empty \<equiv> Abs_fin_string []"
Read: fixes a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and volatile :: "bool"   and \<O>s :: "nat set list"   and i :: "nat"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and \<D> :: "bool"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat" assumes "a \<in> \<O> \<or> a \<in> read_only \<S> \<or> volatile \<and> a \<in> dom \<S>"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> (\<R>s ! j) a \<noteq> Some False"   and "\<not> volatile \<longrightarrow> (\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> dom (\<R>s ! j))"   and "volatile \<longrightarrow> \<not> \<D>" shows "\<O>s,\<R>s,i\<turnstile> (Read volatile a t # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
negneg_I: fixes H :: "fmla set"   and A :: "fmla" assumes "H \<turnstile> A" shows "H \<turnstile> neg (neg A)"
nl31: fixes l' :: "real"   and r' :: "real"   and lr :: "real"   and r :: "real" assumes "1 \<le> l'"   and "1 \<le> r'"   and "1 \<le> lr"   and "1 \<le> r" shows "4 * (l' + r') * (lr + r) \<le> (l' + lr + r' + r)\<^sup>2"
get_ancestors_si_parent_child_rel: fixes h :: "(_) heap"   and child :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "h \<turnstile> get_ancestors_si child \<rightarrow>\<^sub>r ancestors"   and "(ptr, child) \<in> (parent_child_rel h)\<^sup>*" shows "ptr \<in> set ancestors"
omega_comp_fix: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times (omega x) y = inf (times (times x (omega x)) y) y"
STOP_leDT: fixes P :: "'a process" shows "P \<sqsubseteq>\<^sub>D\<^sub>T STOP"
prems_2: shows "\<forall>e\<in>set (a # as). e \<le> unat (- 1)"
da_elim_cases_54: fixes Env :: "env"   and B :: "lname set"   and accC :: "qtname"   and statDeclC :: "qtname"   and stat :: "bool"   and e :: "expr"   and fn :: "vname"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In2 ({accC,statDeclC,stat}e..fn)\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
\<kappa>_SOME_def: shows "\<kappa> = (SOME \<kappa>. \<forall>i l. i < length fs \<longrightarrow> l \<le> i \<longrightarrow> M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<l]) = M.sumlist (map (\<lambda>j. \<kappa> i l j \<cdot>\<^sub>v fs ! j) [0..<l]))"
e0_def: shows "e0 = prj0 (ECC.prod f1 (local.dom f1) \<cdot> ECC.dup (local.dom f1)) (ECC.prod f0 (local.dom f0) \<cdot> ECC.dup (local.dom f0))"
simps_1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "binrelchain P [] = True"
after_summary_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'c zmultiset"   and S :: "'b antichain" assumes "dataflow_topology summary results_in" shows "dataflow_topology.after_summary results_in M S \<equiv> \<Sum>s\<in>set_antichain S. {#results_in t s. t \<in>#\<^sub>z M#}"
\<alpha>_limit_point_eq_rev: fixes x :: "'a"   and p :: "'a" shows "\<alpha>_limit_point x p = rev.\<omega>_limit_point x p"
cbelow_on_plus: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and r :: "'t \<Rightarrow>\<^sub>0 'c"   and z :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "r \<in> dgrad_p_set d m"   and "keys r \<inter> keys z = {}"   and "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) z (\<lambda>a b. red F a b \<or> red F b a) p q" shows "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) (z + r) (\<lambda>a b. red F a b \<or> red F b a) (p + r) (q + r)"
pw_eq_iff: fixes S :: "'a nres"   and S' :: "'a nres" shows "(S = S') = (nofail S = nofail S' \<and> (\<forall>x. inres S x = inres S' x))"
solves_odeD_2: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set"   and t :: "real" assumes "(y solves_ode f) T X"   and "t \<in> T" shows "y t \<in> X"
assms_3: shows "c ! (pc + 1) \<in> A"
sel_1: fixes x1 :: "'ag"   and x2 :: "'ah" shows "fst (x1, x2) = x1"
pointbased_mk_regular1: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "pointbased (mk_regular1 P F)"
ID_restricted_global_subset_unrestricted_1: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r :: "'b" shows "LID\<^sub>G (s(r := None)) \<subseteq> LID\<^sub>G s"
assms_1: shows "common_sudiv_exists one_chain1 one_chain2"
map_try_spmf: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c spmf"   and q :: "'c spmf" shows "map_spmf f (TRY p ELSE q) = TRY map_spmf f p ELSE map_spmf f q"
rel_conversep: fixes R1 :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_prod R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_prod R1 R2)\<inverse>\<inverse>"
gcd_add_mult: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and m :: "'a"   and k :: "'a"   and n :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd m (plus (times k m) n) = gcd m n"
inprod_cvec_g_zero: fixes z :: "complex \<times> complex" assumes "z \<noteq> vec_zero" shows "0 < Re \<langle>z,z\<rangle>"
insertAssertionE: fixes F :: "'b frame"   and \<Psi> :: "'b"   and A\<^sub>F :: "name list"   and \<Psi>' :: "'b" assumes "insertAssertion F \<Psi> = \<langle>A\<^sub>F, \<Psi>'\<rangle>"   and "A\<^sub>F \<sharp>* F"   and "A\<^sub>F \<sharp>* \<Psi>"   and "distinct A\<^sub>F" obtains \<Psi>\<^sub>F :: "'b" where "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>" and "\<Psi>' = \<Psi> \<otimes> \<Psi>\<^sub>F"
that: fixes D' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "Z = [D']\<^bsub>v,n\<^esub>" shows "thesis"
intro: fixes a :: "'c"   and G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "a \<in> carrier G"   and "a \<notin> H" shows "finite_comm_group_adjoin_axioms G H a"
infs_any: fixes P :: "'b \<Rightarrow> bool"   and w :: "'b stream" assumes "infs P w" shows "Bex (sset w) P"
run_one_step_basic_block_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x51 :: "tf"   and x52 :: "b_e list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Block x51 x52) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
content_dvd_coeff: fixes p :: "'a poly"   and n :: "nat" shows "content p dvd poly.coeff p n"
pred_set: fixes P :: "'b \<Rightarrow> bool" shows "pred_stream P = (\<lambda>x. Ball (sset x) P)"
tagged_division_Union: fixes I :: "'a set set"   and pfn :: "'a set \<Rightarrow> ('a \<times> 'a set) set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> pfn i tagged_division_of i"   and "\<And>i1 i2. \<lbrakk>i1 \<in> I; i2 \<in> I; i1 \<noteq> i2\<rbrakk> \<Longrightarrow> interior i1 \<inter> interior i2 = {}" shows "\<Union> (pfn ` I) tagged_division_of \<Union> I"
assms_1: shows "type_update (Type x1) (to_ct_list t_in) (Type t_out) = Type x2"
euler_incr_slp_def: fixes ode_ops :: "ode_ops" shows "euler_incr_slp ode_ops = slp_of_fas (euler_incr_fas' ode_ops (D ode_ops))"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "ring_hom hom" shows "comm_ring_hom hom"
lm045: fixes a :: "('c \<times> 'd set) set"   and X :: "'c set" assumes "Domain a \<inter> X \<noteq> {}"   and "a \<in> allocationsUniverse" shows "\<Union> (a `` X) \<noteq> {}"
PiE_empty_range: fixes i :: "'c"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'd set" assumes "i \<in> I"   and "F i = {}" shows "Pi\<^sub>E I F = {}"
blinding_of_blindable_hash: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "bo \<le> vimage2p h h (=)" shows "blinding_of_blindable h bo \<le> vimage2p (hash_blindable h) (hash_blindable h) (=)"
last_in_set: fixes as :: "'b list" assumes "as \<noteq> []" shows "last as \<in> set as"
vfsequence_vcons_vproductD_3: fixes xs :: "V"   and x :: "V"   and n :: "V"   and A :: "V \<Rightarrow> V" assumes "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> vproduct n A"   and "n \<in>\<^sub>\<circ> \<omega>" shows "n = vcard (xs #\<^sub>\<circ> x)"
observations_closed': shows "\<forall>(s, \<mu>)\<in>set \<O>. \<forall>(t, uu_)\<in>set \<mu>. t \<in> set \<O>\<^sub>s"
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.indRelLTPO Enc TRel"   and "(Q, R) \<in> encoding.indRelLTPO Enc TRel" shows "(P, R) \<in> encoding.indRelLTPO Enc TRel"
simple_distributedI: fixes X :: "'a \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> real" assumes "simple_function M X"   and "\<And>x. 0 \<le> P x"   and "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = prob (X -` {x} \<inter> space M)" shows "simple_distributed M X P"
that_2: fixes p :: "nat" assumes "x = of_nat p"   and "prime p"   and "[p = 3] (mod 4)" shows "thesis"
L2_set_cong: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> f x = g x" shows "L2_set f A = L2_set g B"
stateT_cases: fixes y :: "'b\<cdot>('c, 'a) stateT" obtains k :: "'a \<rightarrow> ('b \<times> 'a)\<cdot>'c" where "y = StateT\<cdot>k"
not_Arr_Null: shows "\<not> (B.arr (fst (B.null, B.null)) \<and> B.arr (snd (B.null, B.null)))"
sets_X: fixes n :: "nat" shows "X n \<in> sets (Pi\<^sub>M {0..<n} M)"
drefinesI2: fixes B :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and G :: "'c \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and \<phi> :: "'c \<Rightarrow> 'd" assumes "well_def B"   and "\<And>P Q s. \<lbrakk>unitary P; unitary Q; G s; P \<tturnstile> wp A Q\<rbrakk> \<Longrightarrow> (P \<circ> \<phi>) s \<le> wp B (Q \<circ> \<phi>) s" shows "drefines \<phi> G A B"
has_codomain_iff_arr: fixes f :: "('o, 'a) arr" shows "(codomains f \<noteq> {}) = arr f"
continuous_within_tendsto_compose': fixes a :: "'d"   and s :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and x :: "'f \<Rightarrow> 'd"   and F :: "'f filter" assumes "continuous (at a within s) f"   and "\<And>n. x n \<in> s"   and "(x \<longlongrightarrow> a) F" shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
imirror_cut_greater: fixes I :: "nat set"   and t :: "nat" assumes "finite I"   and "t \<le> iMin I + Max I" shows "imirror I \<down>> mirror_elem t I = imirror_bounds (I \<down>< t) (iMin I) (Max I)"
M1_axioms_def: shows "M1_axioms \<equiv> (\<forall>x y. PP x y = (P x y \<and> \<not> P y x)) \<and> (\<forall>x y. P x y \<longrightarrow> P y x \<longrightarrow> x = y)"
alpha_l_beta_l_add_norm: fixes l :: "nat" shows "alpha_l l * alpha_l l + beta_l l * beta_l l = 1"
assms_1: shows "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
parallel_sum_wiring_def: shows "parallel_sum_wiring = lsumr \<circ> map_sum id lassocr_swap_sum \<circ> rsuml"
set_plus_intro: fixes a :: "'b"   and C :: "'b set"   and b :: "'b"   and D :: "'b set" assumes "a \<in> C"   and "b \<in> D" shows "a + b \<in> C + D"
oalist_inv_filter_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and P :: "'a \<times> 'c \<Rightarrow> bool" assumes "oalist_inv xs" shows "oalist_inv (filter_raw P xs)"
map_id: fixes t :: "('d, 'e) F" shows "F id id t = t"
rep_eq: fixes x :: "'b \<Rightarrow> 'b ttree"   and xa :: "'b set"   and xb :: "'b ttree" shows "Rep_ttree (substitute x xa xb) = Collect (substitute' x xa xb)"
Polys_closed_minus: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "p \<in> P[X]"   and "q \<in> P[X]" shows "p - q \<in> P[X]"
assms_1: shows "\<forall>\<^sub>F x in nhds x. f x = g x"
aux_qSwap_ignoreFirst3_def: fixes K :: "'c \<times> 'd \<times> 'd \<times> ('a, 'b, 'c, 'd, 'e) qTerm + 'c \<times> 'd \<times> 'd \<times> ('a, 'b, 'c, 'd, 'e) qAbs" shows "aux_qSwap_ignoreFirst3 K = (case K of Inl (zs, x, y, xa) \<Rightarrow> termIn xa | Inr (zs, x, y, xa) \<Rightarrow> absIn xa)"
closed_segment_is_geodesic_3: fixes x :: "'b"   and y :: "'b" shows "geodesic_segment (closed_segment x y)"
CallParamsThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and es :: "nat exp list"   and es' :: "nat exp list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and vs :: "val list"   and ex :: "nat exp"   and es\<^sub>2 :: "nat exp list"   and M :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>"   and "es' = map Val vs @ throw ex # es\<^sub>2" shows "P \<turnstile>\<^sub>1 \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw ex,s\<^sub>2\<rangle>"
elims: fixes x :: "('c, 'd) atm"   and y :: "Preliminaries.var set" assumes "nvA x = y" obtains T1 :: "'c trm" and T2 :: "'c trm" where "x = Eq T1 T2" and "y = nvT T1 \<union> nvT T2"   | p :: "'d" and Tl :: "'c trm list" where "x = Pr p Tl" and "y = {}"
Inf_eqI: fixes A :: "('c \<times> 'd) set"   and x :: "'c \<times> 'd" assumes "\<And>i. i \<in> A \<Longrightarrow> less_eq_prod x i"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq_prod y i) \<Longrightarrow> less_eq_prod y x" shows "Inf_prod A = x"
assms_4: shows "igSubstAllIPresIGWlsAll MOD"
obs: shows "L - Tmax \<le> exp (- a \<cdot> t) \<cdot> (L - T) \<and> exp (- a \<cdot> t) \<cdot> (L - T) \<le> L - T"