### Nuprl Lemma : comparison-antisym

`∀[T:Type]. ∀cmp:comparison(T). AntiSym(T;x,y.0 ≤ (cmp x y)) supposing ∀x,y:T.  (((cmp x y) = 0 ∈ ℤ) `` (x = y ∈ T))`

Proof

Definitions occuring in Statement :  comparison: `comparison(T)` anti_sym: `AntiSym(T;x,y.R[x; y])` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` uimplies: `b supposing a` anti_sym: `AntiSym(T;x,y.R[x; y])` implies: `P `` Q` comparison: `comparison(T)` and: `P ∧ Q` squash: `↓T` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` decidable: `Dec(P)` or: `P ∨ Q` le: `A ≤ B` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  comparison_wf equal_wf all_wf int_formula_prop_wf int_term_value_minus_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermMinus_wf intformle_wf itermConstant_wf itermVar_wf intformeq_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__equal_int iff_weakening_equal true_wf squash_wf le_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation sqequalHypSubstitution setElimination thin rename productElimination applyEquality lambdaEquality imageElimination lemma_by_obid isectElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry intEquality natural_numberEquality dependent_functionElimination sqequalRule imageMemberEquality baseClosed universeEquality independent_isectElimination independent_functionElimination because_Cache unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll axiomEquality functionEquality

Latex:
\mforall{}[T:Type]
\mforall{}cmp:comparison(T).  AntiSym(T;x,y.0  \mleq{}  (cmp  x  y))  supposing  \mforall{}x,y:T.    (((cmp  x  y)  =  0)  {}\mRightarrow{}  (x  =  y))

Date html generated: 2016_05_14-PM-02_38_36
Last ObjectModification: 2016_01_15-AM-07_39_23

Theory : list_1

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